The Auslander-Reiten Quiver of Equipped Posets of Finite Growth Representation Type, some Functorial Descriptions and Its Applications
Isa´ıas David Mar´ın Gaviria
Universidad Nacional de Colombia Facultad de Ciencias Departamento de Matematicas´ Bogota,´ D.C. November, 2019
The Auslander-Reiten Quiver of Equipped Posets of Finite Growth Representation Type, some Functorial Descriptions and Its Applications
Isa´ıas David Mar´ın Gaviria
Thesis work to obtain the degree of Doctor in Mathematics
Advisor Professor Octavio Mendoza Hernandez,´ Ph.D. Full professor, National Autonomous University of Mexico-D.F
Coadvisor Agust´ın Moreno Canadas,˜ Ph.D. Associate Professor, National University of Colombia
Research line Representation theory of posets
Research group TERENUFIA-UNAL
Universidad Nacional de Colombia Facultad de Ciencias Departamento de Matematicas´ Bogota,´ D.C. November, 2019
5
Title The Auslander-Reiten Quiver of Equipped Posets of Finite Growth Representation Type, some Functorial Descriptions and Its Applications
Abstract: The theory of representation of partially ordered sets or posets was introduced in the early 1970's as an effort to give an answer to the second Brauer-Thrall conjecture. Recall that one of the main goals of this theory is to give a complete description of the indecomposable objects of the category of representations of a given poset. Perhaps the most useful tool to obtain such classification are the algorithms of differentiation. For instance, Nazarova and Roiter introduced an algorithm known as the algorithm of differentiation with respect to a maximal point which allowed to Kleiner in 1972 to obtain a classification of posets of finite representation type. Soon afterwards between 1974 and 1977, Zavadskij defined the more general algorithm I with respect to a suitable pair of points, this algorithm was used in 1981 by Nazarova and Zavadskij in order to give a criterion for the classification of posets of finite growth representation type. Actually, several years later, Zavadskij himself described the structure of the Auslander-Reiten quiver of this kind of posets, to do that, it was established that such an algorithm is in fact a categorical equivalence.
Since the theory of representation of posets was developed in the 1980's and 1990's for posets with additional structures, for example, for posets with involution or for equipped posets by Bondarenko, Nazarova, Roiter, Zabarilo and Zavadskij among others. It was necessary to define a new class of algorithms to classify posets with these additional structures. In fact, Zavadskij introduced 17 algorithms. Algorithms, I-V (and some additional differentiations) were used by him and Bondarenko to classify posets with involution, whereas algorithms I, VII-XVII were used to classify equipped posets. In particular, algorithms I, VII, VIII and IX were used to classify equipped posets of finite growth representation type without paying attention to the behavior of the morphisms of the corresponding categories. In other words, it was obtained a classification of the objects without proving that the algorithms used to tackle the problems are in fact categorical equivalences, therefore, the main problem of the theory of the algorithms of differentiation consists of giving a detailed description of the behavior of the morphisms under these additive functors, such description allows to give a deep understanding of the Auslander-Reiten quiver of the corresponding categories.
On the other hand, in the last few years has been noted a great interest in the application of the theory of representation of algebras in different fields of computer science, for example, in combinatorics, information security and topological data analysis. Ringel and Fahr, for instance, gave a categorification of Fibonacci numbers by using the 3-Kronecker quiver whereas representation of posets and the theory of posets have been used to analyze tactics of war and cyberwar. Besides, the theory of Auslander has been used to analyze big data via the homological persistent theory.
In this research, it is proved that the algorithms of differentiation VIII-X induce categorical equivalences between some quotient categories, giving a description of the Auslander-Reiten quiver of some equipped posets by using the evolvent associated to these kind of posets. In this work, ideas arising from the theory of representation of equipped posets are used to give a categorification of Delannoy numbers. Actually, such numbers are interpreted as dimensions of some suitable equipped posets. We also interpret the algorithm of differentiation VII as a steganographic algorithm which allows to generate digital watermarks, such an algorithm can be also used to describe the behavior of some kind of informatics viruses, in fact, it is explained how this algorithm describe the infection-detection process when a computer network is affected for this type of malware.
At last but not least, we recall that the theory of representation of equipped posets is a way to deal with the homogeneous biquadratic problem which is an open matrix problem, in this case, with respect to a pair of fields (F,G) with G a quadratic extension of the field F with respect to a polynomial of the form t2 + q, q ∈ F . Actually, explicit solutions to this problem were given by Zavadskij who rediscovered in 2007 the Krawtchouk matrices introducing an interesting Θ-transformation as well. In this research, such Krawtchouk matrices are used in order to give explicit solutions to non-linear systems of differential equations of the form X0(t) + AX2(t) = B, where X(t),X0(t),A and B are n × n square matrices. Tools arising from this solution are called in this work the Zavadskij calculus.
This research was partially supported by COLCIENCIAS convocatoria doctorados nacionales 727 de 2015.
Keywords: Representation theory of partially ordered sets, Representation theory of quivers, Auslander-Reiten theory, Matrix problem, Equipped poset, Equipped posets with Involution, Algorithms of Differentiation, Delannoy numbers, Zavadskij Calculus, Partitions of integer numbers, Integer sequences, Digital Watermark, Virus and Worms. Acceptation Note
Thesis Work
“ mention”
Jury
Jury
Jury
Advisor Octavio Mendoza Hern´andez
Coadvisor Agust´ınMoreno Ca˜nadas
Bogot´a,D.C., November, 2019
Dedicated to
My mom Luz Dary Gaviria and to my wife Andrea Valencia for their constant support and who never gave up on me.
A.M. Ca˜nadas, who put me on the road, who accompanied me throughout the journey and encouraged me in difficult times.
Acknowledgments
I would like to express my special appreciation to my advisor Professor Agustin Moreno, Ph.D. for being such a tremendous mentor for me. I would like to thank you for encourag- ing my research and for allowing me to grow as a research and as a scientist. His advice on both research as well as on my career have been invaluable. I would also like to thank my other advisor, Professor Octavio Mendoza, Ph.D. for his hard work and thorough review of this thesis. I also want to thank him for his brilliant comments and suggestions, so that this work could be read in a clearer and more scientific way, thanks for his support and discussion sessions during my visit research internship in National Autonomous University of Mexico in 2017, many thanks.
A special thanks to my family, words cannot express how grateful I am to my mother Luz Dary, for all of the sacrifices she has made for my well-being on my behalf. Her prayer for me was what sustained me thus far. I would also like to thank my beloved wife Andrea Valencia, MSc. Thanks for always supporting me, for allowing me to sacrifice our family time to finish this research, she took care of me in times where I used to work too much. I specially express my regards to my partners of the TERENUFIA-UNAL research group for the stimulating discussions, for their feedback, for their contributions and mainly for encouraging me throughout this experience. I also want to express my gratitude to my close friends PFF Espinosa and VC Vargas for being such good friends, for always motivating me and for being with me throughout this process, also for providing discussion spaces for the advance of this investigation.
Last but not least, I would like to thank to Professor A.G. Zavadskij, who inspired all this research by leaving us his legacy of research. Finally, I thank God, for letting me to overcome all the difficulties. I have experienced His guidance day by day; He was the one who let me finish my doctorate studies. Thanks My Lord.
Contents
Contents III
List of Tables VII
List of Figures IX
Introduction XIII
1. Partially ordered sets with additional structure 1 1.1 Equipped posets and equipped posets with involution ...... 1 1.2 Categories ...... 5 1.2.1 Auslander-Reiten theory in a Krull-Schmidth category ...... 9 1.3 Complexification and reellification ...... 12 1.4 Representation of equipped posets and equipped posets with involution . . . 15 1.5 The matrix problem ...... 17 1.6 Some indecomposable objects ...... 19 1.7 Some classification theorems ...... 22
2. Algorithms of differentiation 27 2.1 (A, B)-cleaving pairs and the Zavadskij symbol ...... 27 2.2 Algorithm of differentiation I with respect to a suitable pair of points and the completion algorithm ...... 28 2.2.1 Algorithm of differentiation I ...... 29 2.2.2 Completion algorithm ...... 30 2.3 Categorical properties of differentiation VII for equipped posets ...... 32 2.4 Categorical properties of the algorithm of differentiation VIII for equipped posets...... 37 2.5 Categorical properties of the algorithm of differentiation IX for equipped posets...... 44 III CONTENTS IV
2.6 Categorical properties of the algorithm of differentiation X for equipped posets with involution ...... 50
3. The Auslander-Reiten quiver of some equipped posets 58 3.1 The lifting algorithm ...... 58
3.2 The Auslander-Reiten quiver of the equipped posets F13 − F18 ...... 69
3.3 The Auslander-Reiten quiver of a completely weak chain Cn ...... 76 3.4 The preprojective and preinjective components of the critical equipped posets K6 and K8 ...... 78
3.5 The Auslander-Reiten quiver of the equipped poset Yt in the subcategory addN ...... 83
4. Applications of equipped posets to number theory and differential equa- tions 91 4.1 Representation of equipped posets to generate Delannoy numbers ...... 91 4.1.1 Integer partitions ...... 92 4.1.2 An advance to the Andrews problem ...... 92 4.1.3 The category of lattice representations ...... 97 4.1.4 Categorification of Delannoy numbers ...... 102 4.2 The Zavadskij calculus ...... 103 4.2.1 Matrix problems and the solution of some system of differential equa- tions...... 105 4.2.2 The Θ-Transformation ...... 108 4.2.3 Zavadskij calculus to solve some differential equations ...... 110
5. Applications of the algorithm of differentiation VII in cybersecurity 115 5.1 Digital watermarking ...... 115 5.1.1 The wavelets ...... 115 5.1.2 Wavelet transform ...... 117 5.1.3 Digital watermarks based on wavelets ...... 120 5.1.4 DWT-SVD watermark insertion and extraction ...... 122 5.1.5 Image algebra ...... 122 5.1.6 Extracting and embedding digital watermarks with the algorithm of differentiation VII ...... 125 5.1.6.1 Embedding a digital watermark ...... 127 5.2 Viruses and worms ...... 128 5.2.1 Viruses ...... 128 5.2.1.1 Worms ...... 129 CONTENTS V
5.2.1.2 Network worms life cycle ...... 129 5.2.1.3 Trojan horses ...... 130 5.2.1.4 Polymorphic viruses ...... 131 5.2.2 Virus detection mechanisms ...... 131 5.2.2.1 First generation scanners ...... 131 5.2.3 Intentional and accidental interactions ...... 132 5.2.3.1 Cooperations ...... 132 5.2.4 The algorithm of differentiation VII towards secure computer networks132 5.2.4.1 Some preliminary assumptions ...... 133 5.2.4.2 The infection ...... 134 5.2.4.3 Tracking and detection ...... 135 5.2.4.4 Notation ...... 136
A. The Auslander-Reiten quiver of some ordinary posets 139 A.1 Some sincere ordinary and equipped posets ...... 139 A.2 Some ordinary posets of finite representation type ...... 141 A.3 Some ordinary posets of finite growth representation type ...... 152
B. MATLAB routines 155 B.1 Differentiation VII in MATLAB ...... 155 B.2 Wavelet transformation and singular value decomposition ...... 159
Future work 165
List of Tables
3.1 The indecomposable projective representations of the equipped poset F13 and its corresponding dimension vector...... 70
3.2 The indecomposable projective representations of the equipped poset F14 and its corresponding dimension vector...... 70
3.3 The indecomposable projective representations of the equipped poset F15 and its corresponding dimension vector...... 71
3.4 The indecomposable projective representations of the equipped poset F16 and its corresponding dimension vector...... 73
3.5 The indecomposable projective representations of the equipped poset F17 and its corresponding dimension vector...... 74
3.6 The indecomposable projective representations of the equipped poset F18 and its corresponding dimension vector...... 76
3.7 The indecomposable projective representations of the equipped poset Cn and its corresponding dimension vector...... 77 3.8 The indecomposable projective and injective representations of the critical equipped poset K6 and its corresponding dimension vector [67]...... 79 3.9 The indecomposable projective and injective representations of the critical equipped poset K8 and its corresponding dimension vector [67]...... 82
4.1 Compositions of the number n = 4 and D4...... 96
4.2 Compositions of the number n = 5 and D5...... 96
5.1 The summary of pertinent numeric value sets...... 124
A.1 The Hasse diagrams of the sincere trivially equipped posets of finite repre- sentation type...... 139 A.2 The Hasse diagrams of the trivially equipped posets of finite growth repre- sentation type (Kleiner's critical posets)...... 140 A.3 The diagrams of the sincere non-trivially equipped posets of finite represen- tation type...... 140
VII LIST OF TABLES VIII
A.4 The diagrams of the sincere non-trivially equipped posets of finite represen- tation type and its corresponding evolvent...... 140 A.5 The diagrams of the sincere critical non-trivially equipped posets of finite growth representation type (one parameter)...... 141 A.6 The diagrams of the sincere critical non-trivially equipped posets of finite growth representation type (one parameter) and its corresponding evolvent. 141 List of Figures
1.1 Diagram of an equipped poset and some of its subsets...... 4 1.2 Diagram of an equipped poset with involution...... 5 1.3 The Hasse diagram of the critical Kleiner's posets...... 24 1.4 The Hasse diagram of the Nazarova's hypercritical posets...... 25
2.1 The diagram of an (A, B)-cleaving of U0...... 28 2.2 The diagrams of an equipped poset P and its corresponding derivative poset 0 P (a,b) ...... 29 2.3 The diagrams of an equipped poset P and its corresponding completed poset P(a,b)...... 31
2.4 The diagrams of the equipped poset F17 and its completed poset F17(a,b)... 31
2.5 The Auslander-Reiten quiver Γ(F 17) of the equipped poset F 17...... 32 2.6 Diagrams of an equipped poset P and its corresponding derivative poset 0 P (a,b)...... 33 2.7 The matrix representations of an equipped poset and its derived poset. . . . 35 2.8 Diagrams of an equipped poset P and its corresponding derivative poset 0 P (a,b)...... 38 2.9 The lattice associated to the ideals I, I0 and the subspaces R and R0 ac- cording to VIII...... 41 0 2.10 Diagrams of an equipped poset P and its corresponding derived poset P (a,b). 45 2.11 The lattice associated to the ideals I, I0 and the subspaces R and R0 ac- cording to D-IX...... 48 2.12 Diagrams of the equipped posets with involution (P, Θ) and (P0, Θ0)...... 52 2.13 The lattice associated to the ideals I, I0 and the subspaces R and R0 ac- cording to the differentiation X...... 54
3.1 The Auslander-Reiten quiver Γ(F13) of the equipped poset F13 obtained from ./ Γ(F7) via the lifting algorithm ./...... 70
3.2 The Auslander-Reiten quiver Γ(F14) of the equipped poset F14...... 71
IX LIST OF FIGURES X
3.3 The Auslander-Reiten quiver Γ(F15) of the equipped poset F15...... 73
3.4 The Auslander-Reiten quiver Γ(F16) of the equipped poset F16...... 74
3.5 The Auslander-Reiten quiver Γ(F17) of the equipped poset F17...... 75
3.6 The Auslander-Reiten quiver Γ(F18) of the equipped poset F18...... 76
3.7 The diagram associated to a completely weak chain Cn...... 77
3.8 The Auslander-Reiten quiver Γ(Cn) of the equipped poset Cn...... 78
3.9 The preprojective component P(K6) associated to the critical equipped poset K6...... 81
3.10 The preinjective component I(K6) associated to the critical equipped poset K6...... 81
3.11 The preprojective component P(K8) of the critical equipped poset K8..... 82
3.12 The preinjective component I(K8) of the critical equipped poset K8...... 83
3.13 The diagram associate with an equipped poset of type Yt...... 84
3.14 The Auslander-Reiten quiver Γ(Y4) of the equipped poset Y4 in the subcat- egory add N...... 90
4.1 The diagram of compositions of type Dn with 4 ≤ n ≤ 9...... 93
4.2 The diagram of the compositions of D4 and D5 of type D...... 97 4.3 A weak lattice path...... 98 4.4 Product of strong lattice paths...... 98 4.5 Weak and strong products...... 99 4.6 Examples of lattice representations...... 101
4.7 Lattice representation of D5(1)...... 102 4.8 Delannoy numbers and its reticular trajectories...... 103 4.9 Ehrenfest urn model...... 104
5.1 Example of a wavelet decomposition. MATLAB routines describing the process can be found in Appendix B.2 ...... 120 5.2 The diagram of the equipped poset P...... 126 5.3 The matrix presentation of embedding a digital watermark process...... 127 5.4 The diagram associated with the computer network P...... 133
5.5 The diagram associated with the zombie computer network PZ ...... 133 5.6 Infection of a virus in a computer network...... 135 5.7 Antivirus action in the infected computer network...... 137 + 5.8 Differentiation VII allows to recognize infected files, in this case, c1 b and + c2 b. MATLAB routines describing the process can be found in Appendix B.1137 LIST OF FIGURES XI
5.9 Integration is a way of an infection process. In this figure, we show how in- + tegration allows that file c1 b infects files a, c1, c2 and c3. MATLAB routines describing the process can be found in Appendix B.1 ...... 138
A.1 The Auslander-Reiten quiver Γ(F7) of the ordinary poset F7...... 142
A.2 The Auslander-Reiten quiver Γ(F2) of the ordinary poset F2...... 142
A.3 The Auslander-Reiten quiver Γ(F3) of the ordinary poset F3...... 143
A.4 The Auslander-Reiten quiver Γ(F8) of the ordinary poset F8...... 144 A.5 The Auslander-Reiten quiver Γ(2, 2) of the ordinary poset (2, 2)...... 145 A.6 The Auslander-Reiten quiver Γ(F) of the ordinary poset F...... 147 A.7 The Auslander-Reiten quiver Γ(n, n) of the ordinary poset (n, n)...... 151
A.8 The preprojective component P(K1) of the critical ordinary poset K1..... 153
A.9 The preinjective component I(K1) of the critical ordinary poset K1...... 153
Introduction
The theory of representation of equipped posets is a generalization of the theory of rep- resentation of ordinary posets (i.e., posets without additional structures) developed by Nazarova, Roiter and their students in the 1970's in Kiev. The main goal of such theories is to give a complete description of the indecomposable objects and irreducible morphisms of a category of representations rep P of a given poset P [40–47, 56, 67, 68, 71, 72].
According to Simson [56] the poset representation theory allowed to Nazarova and Roiter to obtain a proof of the second Brauer-Thrall conjecture [42,56] which asserts that a finite dimensional algebra over an infinite field k is either representation-finite or there exists an infinite sequence of numbers di ∈ N such that, for each i, there exists an infinite number of nonisomorphic indecomposable modules with k-dimension di [4, 5, 7, 29, 42, 51, 56].
The criterion for posets of finite representation type was obtained by Kleiner in 1972 by using the algorithm of differentiation with respect to a maximal point introduced previously by Nazarova and Roiter. The categorical properties of such an algorithm was given by Gabriel in 1973 [31,36], founding in this way a new line of research in the theory of representation of posets which consists of proving that algorithms of differentiation induce a categorical equivalence between the category of representations of the original poset and the corresponding of the derived poset [14]. Such functors constitute the main tool to classify posets of different types, for instance, the algorithm of differentiation with respect to a maximal point was also used by Nazarova to give a tame representation type criterion for ordinary posets whereas the algorithm of differentiation I with respect to a suitable pair of points introduced by Zavadskij in the period 1974-1977 was used by Nazarova and Zavadskij in 1981 to obtain a finite growth representation type criterion for ordinary posets [43,44,62]. A description of the structure of the Auslander-Reiten quiver for ordinary posets of finite growth representation type was obtained also by Zavadskij in 1991 and 2005 by giving a complete categorical description of algorithms I and completion [65, 69].
In 2000 Zabarilo and Zavadskij introduced equipped posets and gave criteria to classify one-parameter equipped posets [72]. Soon afterwards, in 2003 Zavadskij classified equipped posets of tame representation type by using algorithms of differentiation, I, VII-XVII for this kind of posets and also for equipped posets with involution. In particular, a finite growth representation type criterion was published by Zavadskij in 2005, to do that, he used algorithms of differentiation VII-IX [67, 68]. Zavadskij himself generalized in 2011 the theory of equipped posets to posets with automorphisms and established a bijection between isoclasses of indecomposable representations of the category rep(K,L)(Q, ∆) and XIII INTRODUCTION XIV the Γ-orbits of isoclasses of indecomposables of the category repL Q, where Q is some ordinary poset, ∆ ⊂ Aut Q, and Γ = Gal(L/K) for some Galois field extension K ⊂ L [71], equipped posets were called 2-equipped posets in this generalization.
Since the introduction of the poset representation theory several applications have been given by different authors, for instance, Zavadskij and Kirichenko in 1977 used the Kleiner's finite type representation criterion to give a classification of tiled orders or semimaximal rings [63,64]. Arnold in 2000 also used such criterion to classify finitely generated abelian groups [3]. Furthermore, Rump obtained generalizations (to general orders and semia- belian categories) of the algorithm of differentiation I giving a corresponding categorical description as Gabriel and Zavadskij described before [52,53]. Following these ideas, one of the purposes of this research is to give a complete categorical description of the algorithms of differentiation VII-X for equipped posets. In particular, the categorical description of algorithms of differentiation VII-IX will allow us to describe the Auslander-Reiten quiver of posets of finite growth representation type.
We recall that throughout the years several mathematicians have studied the structure of the Auslander-Reiten quiver for different types of posets, B¨unermann,Ringel and Simson among others [10,20,48,56]. Our point here is to establish that the algorithms of differen- tiation are categorical equivalences by proving explicitly that as additive functors they are full, faithful and dense. Worth noting, that the denseness property arises directly from the work of Zavadskij regarding the classification of equipped posets of tame representation type [13, 15, 16, 19].
We also remind that Zavadskij in his work regarding generalizations of the Kronecker problem introduced an interesting Θ-transformation based on the Krawtchouk matrices (actually, Zavadskij rediscovered such matrices) in order to describe a particular solution of one of the most outstanding matrix problems named the biquadratic homogeneous problem which consists of finding a canonical form for rectangular matrices over a natural (G, G)-bimodule W = G ⊗ G with respect to left elementary transformations of rows and right elementary transformations of columns over G [70].
The Krawtchouk-Zavadskij matrices have many interesting combinatorial properties and have been used in different topics, for example, these matrices have allowed advances in harmonic analysis, statistics, combinatorics, coding theory, representation theory of Lie algebras, quantum theory, probability theory, etc [28].
In this work, we use Krawtchouk-Zavadskij matrices in order to give solutions to some non-linear systems of differential equations, to do that, we introduce a novel matrix rein- terpretation of the classical calculus named the Zavadskij calculus obtained by assuming that the variables in the different exponential functions are Krawtchouk-Zavadskij matrices [18].
We also describe how it is possible to apply the algorithm of differentiation VII to model interactions between informatic worms and viruses. Actually, the process of integration defined also by Zavadskij is a way to interpret an infection stage of a computer network INTRODUCTION XV whereas the differentiation algorithm is a way to detect infected files in these kind of systems.
Main results, contributions, papers and conferences
This research regards the categorical properties of the algorithms of differentiation for some equipped posets and its applications.
Contributions
The following are the main contributions.
1. It is given the categorical properties of algorithms of differentiation VIII-X.
2. It is given a solution for compositions of a combinatorial open problem proposed by Andrews in 1987 regarding integer partitions. To do that, we define a particular equipment for some particular posets.
3. It is presented a combinatorial algorithm to describe the Auslander-Reiten quiver of some posets of finite growth representation type. And it is given a toy-example of a family of posets with a small number of Auslander-Reiten sequences.
4. A categorification in the sense of Ringel and Fahr is given to Delannoy numbers by introducing some new lattice representations.
5. It is given explicit solutions to systems of differential equations of the form X00(t) ± 0 2 αX(t) = 0 and X (t) − X (t)Θk = Θk by using Krawtchouk-Zavadskij matrices and introducing the Zavadskij calculus.
6. Different applications of the algorithm of differentiation VII in cybersecurity are given. In particular, it is used to build schemes of digital watermarks and to simulate behavior of informatic viruses.
Papers
Results of this research allowed us to publish the following papers:
(i) Categorical properties of the algorithm of differentiation D-VIII for equipped posets [15].
(ii) On the algorithm of differentiation D-IX for equipped posets [16].
(iii) Representation of Equipped posets to Generate Delannoy Numbers [17].
(iv) The Zavadskij's Calculus [18].
Conferences
The main results of this research have been presented in the following conferences. INTRODUCTION XVI
1. Maurice Auslander Distinguished Lectures and International Conference, Woods Hole Oceanographic Institute. Woods Hole MA-USA, 04-2019.
2. Primer encuentro de Algebra´ y Topolog´ıaUniversidad Nacional de Colombia. Bo- got´a-Colombia, 01-2018.
3. Coloquio Latinoamericano de Algebra-PUCE.´ Quito-Ecuador, 08-2017.
4. XI International Algebraic Conference in Ukraine Taras Shevchenko National Uni- versity. Kiev-Ukraine, 07-2017.
Research stays
The author is indebted with the following institutions and academics for his warm hospi- tality during his several research stays.
1. Algebra seminar at the Institute of Mathematics, Universidad Nacional Aut´onoma de M´exicoD.C-M´exico,Professor Octavio Mendoza Hern´andez.
2. Algebra seminar at the Department of Mathematics, Valdosta State University Val- dosta, GA USA, Professor Jos´eV´elezMarulanda.
3. Algebra seminar at the Instituto de Matem´aticas, Universidad de Antioquia, Medell´ın-Colombia, Professor Hern´anGiraldo.
This thesis is distributed as follows:
Chapter 1 aims to present a theoretical introduction to the theory of posets with additional structures, equipped posets and equipped posets with involution, as well as, definitions and notation to be used throughout the work.
In Chapter 2, it is given the categorical properties of the algorithms of differentiation VIII-X.
In Chapter 3, an introduction to the Auslander-Reiten theory for category of represen- tations of posets is presented. And it is described a new combinatorial algorithm which allows to build the Auslander-Reiten quiver of some equipped posets of finite growth rep- resentation type. Posets of type Yt with a small number of Auslander-Reiten sequences are also defined in this chapter.
In Chapter 4, we give some applications of the theory of representation of equipped posets. In particular, a categorification of Delannoy numbers is given by introducing some lattice representations. Such procedures allow us to give some advances to a combinatorial open problem posed by Andrews in 1987. Besides, a matrix interpretation of identities of the classical calculus is given by using the Zavadskij's Θ-transformation and its properties in order to give explicit solutions to second order systems of differential equations of the 00 form X (t) ± αX(t) = 0, α ∈ R, and explicit solutions of a particular nonlinear system of 0 2 differential equations of the form X (t) − X (t)Θk = Θk. INTRODUCTION XVII
In Chapter 5, we present some applications of the algorithm of differentiation VII to cybersecurity. In this chapter this algorithm is used to embed and extract fragile digital watermarks, it is also used to model processes of virus infection and its corresponding detection in a computer network.
Finally, the appendix A, contains tables with sincere ordinary posets of finite representa- tion type and some one-parametric sincere equipped posets of finite growth representation type. In appendix B, we present two MATLAB routines, which illustrates how the algo- rithm of differentiation VII is a way to embed a digital watermarking in frequency domain.
CHAPTER 1
Partially ordered sets with additional structure
In this chapter, we present basic definitions regarding posets with additional structures in section 1.1, whereas some elementary notions of the category theory and complexification of vector spaces are defined in sections 1.2 and 1.3, respectively. Categories of representa- tions of ordinary posets and with additional structures are described in section 1.4. Matrix problems and some indecomposable objects of equipped posets and equipped posets with involution are defined in sections 1.5 and 1.6, respectively. Finally, some classical theorems regarding classification of posets are given in section 1.7 [11, 50, 67, 68].
1.1 Equipped posets and equipped posets with involution
In this section, we recall the definition of equipped poset, and equipped poset with invo- lution in the sense of Zavadskij [11, 50, 67, 72]. Worth noting that the current definition of weak and strong point associated to an equipped poset corresponds respectively to the original definition of double and single point given by Zavadskij and Zabarilo. Henceforth, we assume the notation given in [13, 50] for points in an equipped poset.
A partially ordered set (Poset) is an ordered pair (P, ≤) which consists of a not empty set P and a binary relation ≤ on P, called order, such that
(i) ≤ is reflexive, which means that x ≤ x for all x ∈ P;
(ii) ≤ is antisymmetric, that is, x ≤ y and y ≤ x imply that x = y for all x, y ∈ P;
(iii) ≤ is transitive, meaning that x ≤ y and y ≤ z imply that x ≤ z for all x, y, z ∈ P.
The following definition of an equipped poset was given Zavadskij and Zabarilo in [72].
Definition 1. A poset P is said to be equipped if the following three conditions hold:
(1) the points of the set P can be either single or double;
(2) the order relations between points can be either weak or strong;
1 CHAPTER 1. PARTIALLY ORDERED SETS WITH ADDITIONAL STRUCTURE 2
(3) if x < y is a weak relation, then both points x and y are double, and if x < t < y in this case, then the relations x < t and t < y are also weak (and the point t is automatically double).
The following definition is an axiomatic point of view of an equipped poset.
Definition 2. A partially ordered set (P, ≤) is called equipped if it satisfies the following axioms:
(E1) there is a partition of the order ≤ = ∪˙ ¢;
(E2) if x y and x ≤ t ≤ y then x t and t y.
In this case we write (P, ≤, , ¢).
Where all order relations x ≤ y between points x, y ∈ P are separated into strong relations (denoted x ¢ y) and weak relations (denoted x y).
We let P denote an equipped poset (P, ≤, , ¢), x ≤ y denotes an arbitrary relation on P, besides, the order ≤ on P gives raise to the relations ≺ and ¡ of strict inequality: x ≺ y (respectively, x ¡ y) in P if and only if x y (respectively, x ¢ y) and x 6= y.
In general, relations ¢ and are not order relations. These relations are antisymmetric but not reflexive. In particular is not reflexive and ¢ is transitive [50].
Definition 3. If P is an equipped poset then
1. A point x ∈ P is called strong if x ¢ x. We let P◦ denote the subset of strong points of P.
2. A point x ∈ P is called weak if x x. We let P⊗ denote the subset of weak points of P.
The following lemma proves that Definitions 1 and 2 are equivalents.
Lemma 1. For an equipped poset (P, ≤, , ¢) the following facts hold:
(a) If x y then x x and y y;
(b) P = P⊗ ∪˙ P◦;
(c) If x ≤ y ¢ z or x ¢ y ≤ z then x ¢ z;
⊗ (d) If P = ∅ then = ∅ and ≤= ¢.
Proof.
(a) It follows immediately from (E2). CHAPTER 1. PARTIALLY ORDERED SETS WITH ADDITIONAL STRUCTURE 3
(b) Since x ≤ x for all x ∈ P then either x x or x ¢ x thus P ⊆ P⊗ ∪˙ P◦. In order to prove that P⊗ ∪˙ P◦ ⊆ P it is enough to observe that if x ∈ P⊗ ∪˙ P◦ then either x x or x ¢ x then x ≤ x by definition, therefore x ∈ P.
(c) Suppose that x ≤ y ¢ z and x z then by definition of a weak relation it holds that if x ≤ y ≤ z then x y and y z which is a contradiction. The same arguments can be used to prove that x ¢ z if x ¢ y ≤ z.
⊗ (d) If P = ∅ and there exist points x, y ∈ P such that x y then item (a) allows to ⊗ infer that x x and y y thus x, y ∈ P which is a contradiction. Thus = ∅, and therefore ≤= ¢.
⊗ If P = ∅, it is said that the equipment is trivial and the poset P is ordinary. Furthermore, the composition of a strong relation with any other relation is strong, and the relation of a strong point with any other point is a strong relation.
According to Ca˜nadas[13], if P is an equipped poset and a ∈ P, then the subsets of P ∨ O H g denoted by: a , a∧, a , aM, a , aN, a and af, respectively are defined as follows: ∨ a = {x ∈ P | a ≤ x} , a∧ = {x ∈ P | x ≤ a} , O a = {x ∈ P | a ¡ x} , aM = {x ∈ P | x ¡ a} , H ∨ a = a \ a, aN = a∧ \ a, g a = {x ∈ P | a x} , af = {x ∈ P | x a} .
∨ The set a (respectively, a∧) is called the ordinary upper (respectively, lower) cone as- O sociated to the point a ∈ P, and the set a (respectively, aM) is called the strong upper H (respectively, lower) cone associated to the point a ∈ P. Whereas, the sets a and aN are called truncated cones (respectively, upper and lower) associated to the point a ∈ P.
g ◦ g In general, the sets a and af are not cones. Note that, if x ∈ P then x = xf = ∅.
For an equipped poset (P, ≤) and A ⊂ P, we define the subsets, AO, Ag and A∨ as follows AO = S aO, Ag = S ag, A∨ = S a∨. a∈A a∈A a∈A
The sets AM, Af and A∧ are defined similarly.
The diagram of an equipped poset (P, ≤) may be obtained via its Hasse diagram (with strong points denoted ◦ and weak points denoted ⊗). In this case, a new line is added to the line connecting two points x, y ∈ P with x ¡ y if and only if such relation cannot be deduced of any other relations in P.
In Figure 1.1, we show an example of this kind of diagrams. CHAPTER 1. PARTIALLY ORDERED SETS WITH ADDITIONAL STRUCTURE 4
1g = {1, 2, 3, 4, 5} 1O = {6, 7} ⊗ 7 5 2g = {2, 3, 4, 5} 2O = {6, 7} @ e @ 3g = {3, 4, 5} 3O = {6, 7} 4 ⊗ 6
@ e 4g = {4, 5} 4O = ∅ @ 3 ⊗ 5g = {5} 5O = ∅ @ 6g = 6O = {6, 7} @@ ∅ 2 ⊗ 8 7g = 7O = {7} @ ∅ e @@ 8g = ∅ 8O = {3, 4, 5, 6, 7, 8} 1 ⊗ 9 9g = ∅ 9O = {3, 4, 5, 6, 7, 8, 9} e Figure 1.1. Diagram of an equipped poset and some of its subsets.
O g ∨ In this case if A = {4, 6}, then A = {6, 7}, A = {4, 5}, A = {4, 5, 6, 7}, AM = {1, 2, 3, 6, 8, 9}, Af = {1, 2, 3, 4} and A∧ = {1, 2, 3, 4, 6, 8, 9}.
Let P be an equipped poset. A chain (i.e. a linearly order subset) C = {ci ∈ P | 1 ≤ i ≤ n, ci−1 < ci if i ≥ 2} is a weak chain if ci−1 ≺ ci for each i ≥ 2. If c1 ≺ cn, we say that C is a completely weak chain. Moreover, a subset X ⊂ P is completely weak if X = X⊗ and the weak relations are the only relations between points of X. Often, we let {c1 ≺ c2 ≺ · · · ≺ cn} denote a weak chain which consists of points c1, c2, . . . , cn. An ordinary chain C is denoted in the same way (by using the corresponding symbol <)[13, 67]. For example, in Figure 1.1 subsets C1 = {9 < 8 < 3 < 4 < 5} and C2 = {1 ≺ 2 ≺ 3 ≺ 4 ≺ 5} constitute a chain and a completely weak chain, respectively.
For an equipped poset P and A, B ⊂ P, we write A < B if a < b for each a ∈ A and b ∈ B. Notations A ≺ B and A/B are assumed to be defined in the same way.
A convex envelope of a subset A ⊂ P is a subset of the form
[A] = {x ∈ P | a0 ≤ x ≤ a00 for some a0, a00 ∈ A}.
For a ≤ b in P, the closed segment [a, b] = {x ∈ P | a ≤ x ≤ b} is the convex envelope of A = {a, b}. The set [A]\A is the interior of the convex envelope [A], see [67].
The following is the definition of an equipped poset with involution given by Zavadskij [67].
Definition 4. An equipped poset with involution, is an equipped poset (P, ≤), with an involution ∗, satisfying the following two additional conditions:
(iv) on the set of all points P, it is given an involution ∗ : P −! P which preserves strong and weak points (and which is in no connection with the order relation ≤). Hence, strong points are divided into small (x = x∗) and big (x 6= x∗), and weak points are partitioned into weak (x = x∗) and biweak (x 6= x∗);
(v) to each biweak point x it is assigned the number g(x) = g(x∗) ∈ {±1} called its genus (or genus of the pairs x, x∗). CHAPTER 1. PARTIALLY ORDERED SETS WITH ADDITIONAL STRUCTURE 5
In the case x 6= x∗, we called the points x and x∗ equivalents and write x ∼ x∗. The involution ∗ is said to be primitive if it leaves fixed all weak points (i.e. there are no biweak points).
In diagrams of equipped posets with involution, symbols ◦, •, ⊗, depict small, big, weak and biweak points, respectively.
All order relations with a participation of at least one strong point, as well as all weak relations between weak points, are pictured by a single line. But all strong relations between weak points, which are not consequences of some other relations, are pictured by a double line (or by an additional line).
If some group of points is encircled by a contour connected by some (single or double) line with some other points, it means that all points, located inside the contour, have the same order relations with the mentioned other points (determined by the type of the line).
The diagram of an equipped poset with involution is shown in the next Figure 1.2.
b ⊗q ⊗ A # •a∗
P = "! c
B • c∗ a Figure 1.2. Diagram of an equipped poset with involution.
Note that, in this case a ∼ a∗, c ∼ c∗; q = q∗; b = b∗; c∗ ¢ b, a ¢ a∗ ¢ q, a ¢ c b.
1.2 Categories
In this section we recall some basic definitions and notations regarding Auslander-Reiten theory in Krull-Schmidt categories, see [4, 32, 38, 48, 58].
Definition 5. A category R is a class of objects together with the following data [11, 58]:
1. a rule which assigns to any pair (U, V ), of objects in R, a set R(U, V ) whose elements are called the morphisms from U to V . Moreover for any two pairs (U, V ), (W, X) of objects in R, we have that R(U, V ) = R(W, X) 6= ∅ implies that U = W and V = X.
2. for any triplet (U, V, W ) of objects in R, a composition morphism
R(V,W ) × R(U, V ) −! R(U, W ) (g, f) 7−! g ◦ f CHAPTER 1. PARTIALLY ORDERED SETS WITH ADDITIONAL STRUCTURE 6
which is associative, in the sense that f ◦ (g ◦ h) = (f ◦ g) ◦ h, and which admits identity elements in the sense that each set R(V,V ) contains an element 1V such that 1V ◦ f = f, for all f ∈ R(U, V ), and g ◦ 1V = g for all g ∈ R(V,W ).
Henceforth, symbols U ∈ R, and f ∈ R will be used often to denote that U is an object and f is a morphism of a category R, respectively. Furthermore, we will write gf instead of g ◦ f to denote the composition between two morphisms f : U −! V and g : V −! W .
Definition 6. Let R be a category. A category R0 is a subcategory of R if the following four conditions are satisfied:
1. the class of objects of R0 is a subclass of the class of objects of R;
2. if U, V ∈ R0, then R0(U, V ) ⊆ R(U, V );
3. the composition of morphisms in R0 is the same as in R;
0 0 4. for each object U ∈ R, the identity morphism 1U ∈ R (U, U) coincides with the identity morphism 1U ∈ R(U, U).
A subcategory R0 of R is called full if R0(U, V ) = R(U, V ) for all U, V ∈ R0.
In a category R there are special types of morphisms which we define below.
Definition 7. Let f : U −! V be a morphism in a category R.
1. We call f a monomorphism if for all morphisms g, h : W −! U such that fg = fh, it follows that g = h.
2. We call f a epimorphism if for all morphisms g, h : V −! W such that gf = hf it follows that g = h.
3. We call f a isomorphism if there exist a morphism g : V −! U, such that gf = 1U ∼ and fg = 1V . In this case we write U = V . Note that g is uniquely determined by f.
Definition 8. Let R and R0 be categories.
1. A covariant functor F : R −! R0 is a rule which assigns to each object U of R an object F (U) of R0, and to each morphism f : U −! V in R a morphism F (f): 0 F (U) −! F (V ) in R such that F (1U ) = 1F (U) and F (gf) = F (g)F (f). 2. A contravariant functor F : R −! R0 is a rule which assigns to each object U of R an object F (U) of R0, and to each morphism f : U −! V in R a morphism 0 F (f): F (V ) −! F (U) in R such that F (1U ) = 1F (U) and F (gf) = F (f)F (g). 3. A functor F : R −! R0 is an equivalence, if F admits a quasi-inverse, i.e. a functor 0 E : R −! R such that EF ' 1R and FE ' 1R0 . CHAPTER 1. PARTIALLY ORDERED SETS WITH ADDITIONAL STRUCTURE 7
If there exists an equivalence F : R −! R0 of categories, we say that R and R0 are equivalent categories and write R =∼ R0. Remark 1.1. Recall that an equivalence of categories is a covariant functor D : R −! R0 which is dense, full and faithful. It is dense if every object V ∈ R0 has the form D(U), 0 for some U ∈ R. D is full if the morphism D∗ : R(X,Y ) −! R (DX, DY ), ϕ 7−! D(ϕ), is surjective for all objects X and Y of R. If D∗ is an injective morphism for all X and Y , the functor D is called faithful. If R and R0 are equivalent categories, we write R ' R0, see [56].
In this work, we will refer to a covariant functor only as a functor.
The following are the definitions of a k-category and the ideal of a k-category with k a commutative ring.
Definition 9. Let k be a fixed commutative ring. A k-category is a category R whose morphism sets R(U, V ) are endowed with a k-module structure such that the composition morphism are k-bilinear. A k-functor between two k-categories R and S is a functor F : R −! S whose defining morphisms F (U, V ): R(U, V ) −! S(F U, F V ) are k-linear for all U, V ∈ R.
Unless otherwise stated, the functors between k-categories which we consider in the sequel, and especially the equivalences of k-categories, are implicitly supposed to be k-functors.
Definition 10. An object U of a k-category R will be called indecomposable if the en- domorphism algebra R(U, U) = End (U, U) has precisely two idempotents, namely 0 and 1U 6= 0.
Each k-algebra A gives rise to a k-category which has one object Ω, the same for all A, and satisfies Hom (Ω, Ω) = A. In the sequel, we shall identify k-algebras with the associated k-categories. Homomorphisms of algebras then correspond to k-functors. In view of this, k-categories generalize k-algebras. The generalization carries over to (two-sides) ideals, which are defined as follows [32].
Definition 11. An ideal I of a k-category R is a family of subgroups I(U, V ) ⊂ R(U, V ) such that f ∈ I(U, V ) implies gfh ∈ I(Z,W ) for all h ∈ R(Z,U) and g ∈ R(V,W ). Each such ideal I gives rise to a k-quotient category R/I which has the same objects as R and the morphism in the quotient category R/I are defined as (R/I)(U, V ) = R(U, V )/I(U, V ) for all U, V ∈ R.
Let R be a k-category. In order to fix our terminology, we recall the definition of the (direct) sum of two objects U, V ∈ R: First we call summation of U and V in R a i j quintuplet (S, i, j, p, q) consisting of an object S ∈ R and of morphisms U S V such p q that pi = 1U , qj = 1V and ip + jq = 1S. Such summations are known to be “unique up to uniquely determined isomorphisms”. Therefore, whenever a summation of U and V exists, we will suppose that a “canonical” one has been chosen. The object S is then called the sum (or coproduct) of U and V in R and it is denoted by U ⊕ V ; the morphisms p, q are called projections, the morphisms i, j immersions. CHAPTER 1. PARTIALLY ORDERED SETS WITH ADDITIONAL STRUCTURE 8
For a category R, we let hUi | i ∈ IiR denote the ideal consisting of all morphisms passing through finite direct sums of the objects Ui. That is, if ϕ : U ! V ∈ hUi | i ∈ IiR, then f mi g there exist morphisms f, g ∈ R such that ϕ = U −! ⊕Ui −! V with mi = 0 for almost i all i. Definition 12. The k-category R is called additive if U ⊕ V exists for all U, V ∈ R and if R contains a null object, i.e. an element 0 such that 10 = 0. Definition 13. Let R be an additive category, a non-null object U ∈ R is said to be indecomposable provided U ' V ⊕ W implies V = 0 or W = 0 [48].
If [X1],..., [Xk] are equivalence classes of indecomposable objects of a given category R k then we let [X1,X2,...,Xk] denote the union ∪ [Xi]. i=1 Definition 14. Let R be an additive category and f : U −! V be a morphism in R.
(i) A kernel of f is an object Ker f together with a morphism u : Ker f −! U satisfying the following two conditions: 1. f ◦ u = 0, 2. for any object W of R and for any morphism h : W −! U in R such that f ◦h = 0, there exists a unique morphism h0 : W −! Ker f such that h = u ◦ h0. (ii) A cokernel of f is an object Coker f together with a morphism p : V −! Coker f satisfying the following two conditions: 1. p ◦ f = 0, 2. for any object W of R and for any morphism g : V −! W in R such that h◦f = 0, there exists a unique morphism g0 : Coker f −! W such that g = g0 ◦ p.
It is clear that u is a monomorphism and p is an epimorphism [4]. Remark 1.2. Assume that every morphism in R admits a kernel and a cokernel. Then for each morphism f : U −! V in R, there exists a unique morphism f in R making the square in the following diagram u f p Ker f- U- V - Coker f p0 6u0 ? f Coker f - Ker p commutative (that is, f = u0 ◦ f ◦ p0), where u0 : Ker p −! V is the kernel of p and p0 : U −! Coker u is the cokernel of u. Indeed, because p ◦ f = 0, there exists a unique morphism f 0 : U −! Ker p such that f = u0 ◦ f 0. Moreover, because u0 ◦ f 0 ◦ u = f ◦ u = 0 and u0 is a monomorphism, f 0 ◦ u = 0 and hence, by the definition of cokernel, there exists a unique morphism f : Coker u −! Ker p such that f 0 = f ◦ p0. Consequently, the morphism f makes the preceding square commutative. One shows easily that f is unique. The object Ker p is called the image of f and is denoted by Im f [4]. Definition 15. Let R be an additive k-category. An idempotent in R is a morphism e ∈ R(U, U) such that e2 = e. An idempotent e ∈ R(U, U) splits if there are morphisms f : V −! U, g : U −! V in R with gf = 1V and fg = e. CHAPTER 1. PARTIALLY ORDERED SETS WITH ADDITIONAL STRUCTURE 9
1.2.1 Auslander-Reiten theory in a Krull-Schmidth category
In this section we present the basic definitions given by Ringel in [38, 48] to build the Auslander-Reiten quiver of Krull-Schmidt categories (or Krull-Schmidt k-categories).
Definition 16. Let k be a commutative ring. An additive k-category K is called Krull- Schmidt category provided that all idempotents split and the endomorphism ring End(U), of any object U ∈ K, is a semi-perfect ring.
In a Krull-Schmidt category for direct sum decompositions one has the following result of uniqueness.
Theorem 1 (Krull-Schmidt). Let K be a Krull-Schmidt category, let Ui,Vj be indecom- s t posable objects in K with 1 ≤ i ≤ s, 1 ≤ j ≤ t, such that ⊕ Ui ' ⊕ Vj. Then s = t, and i=1 j=1 there is a permutation π of {1, . . . , s} such that Ui ' Vπ(i) for all i.
In a Krull-Schmidt category, we note as Ind K the collections of all indecomposable objects U ∈ K. Let K be a Krull-Schmidt category. According to Ringel a full subcategory L of K which is closed under direct sums and summands will be called an object class in K. Note that, an object class L is itself a Krull-Schmidt category and is uniquely determined by the indecomposable objects belonging to L. Given a set of objects N = {N1,N2,...,Nt} in K, we let hNi denotes the smallest object class of K containing N. This class hNi is given by the direct sums of direct summands of objects in N. Other notation for hNi is also add N. Of course, for a given objects N1,N2,...,Nt in K, hN1,N2,...,Nti = add {N1,N2,...,Nt} denotes the smallest object class of K having N1,N2,...,Nt as objects. Definition 17. Let U, V be indecomposable objects in a Krull-Schmidt category K. The set of all non-invertible morphisms from U to V is called the radical of U and V , and it is denoted by rad (U, V ).
Definition 18. Let U, V, W and Z be objects in a Krull-Schmidt category K.
(i) A morphism f ∈ K(U, V ) is a split monomorphism if there exist a morphism h ∈ K(V,U) such that hf = 1U . (ii) A morphism g ∈ K(W, Z) is a split epimorphism if there exist a morphism h ∈ (Z,W ) such that gh = 1Z .
(iii) A morphism h : W −! Z is irreducible if h is neither split mono, non split epi and, for any factorization h = gf, f is split mono or g is split epi.
We shall refer throughout the document to a morphism that is split monomorphism as split mono or section, and a morphism that is split epimorphism as split epi or retraction, without any distinction.
Of course, for an indecomposable object U, rad (U, U) is just the radical of the local ring s t End (U). Given the direct sum decompositions U = ⊕ Ui and V = ⊕ Vj, any morphism i=1 j=1 CHAPTER 1. PARTIALLY ORDERED SETS WITH ADDITIONAL STRUCTURE 10
s t f : ⊕ Ui −! ⊕ Vj can be written in the matrix form f = (fij), with fij ∈ Hom (Uj,Vi). If i=1 j=1 all Ui, Vj are indecomposable, then f belongs to rad (U, V ) if all fij belong to rad (Uj,Vi). This is equivalent to saying that, for any split mono h : Z −! U and any split epi g : V −! W with Z, W indecomposable, gfh ∈ rad (Z,W ).
Let K be a Krull-Schmidt category and U, V ∈ K. It can be shown that rad2 (U, V ) is given by the set of all the morphisms of the form gf, with f ∈ rad (U, W ) and g ∈ rad (W, V ) for some object W in K. Note that rad2 (U, V ) ⊆ rad (U, V ) ⊆ Hom (U, V ) are k-subspaces, and in fact, End (U) − End (V )-subbimodules of Hom (U, V ). We let denote Irr (U, V ) := rad (U, V )/rad2 (U, V ).
Remark 1.3. Let U and V be indecomposable objects in a Krull-Schmidt category K. It can be proved that a morphism f : U −! V is irreducible if and only if f ∈ rad (U, V )/rad2 (U, V ). Thus, there exists an irreducible morphism from U to V if and only if Irr (U, V ) 6= 0; and hence the bimodule Irr (U, V ) is a measure for the multiplicity of irreducible morphisms.
If K is Krull-Schmidt category and L is a full subcategory, then L itself is a Krull-Schmidt category, provided L is closed under direct sums and direct summands. If it is the case, and U, V are indecomposable objects in L, then radK (U, V ) = radL (U, V ), however the definition of rad2, and therefore of Irr, depends on the whole category K or L. In general, we only have an obvious epimorphism
radL (U, V ) radK (U, V ).
The pair (K, S) is called a Krull-Schmidt category, with short exact sequences, provided K is a Krull-Schmidt category and S is a class of pairs (f, g) of morphisms in K such that f is a kernel of g in K, and g is a cokernel of f in K. The pair (K, S) is an exact category if K is a Krull-Schmidt category with a full embedding K ⊆ A, with A an abelian category, such that S is the set of all short exact sequences belonging to K, and moreover K, is closed under extensions in A [48].
Let (K, S) be a Krull-Schmidt category with short exact sequences. The pair (f, g) ∈ S splits if f is a split monomorphism (i.e., f is section), or equivalently, g is a split epimorphism (i.e., g is a retraction).
Let K be a Krull-Schmidt category and U be an object in K.A source morphism, for U in K, is a morphism f : U ! V in K satisfying the following three properties:
1. f is not split mono,
2. for any not split mono f 0 : U ! V,0 there exists ν : V ! V 0 in K with f 0 = νf, 3. if γ ∈ End V satisfies γf = f, then γ is an isomorphism.
Dually, let W be an indecomposable object in K.A sink morphism, for W in K, is a morphism g : V ! W in K satisfying the following three properties:
1. g is not split epi; CHAPTER 1. PARTIALLY ORDERED SETS WITH ADDITIONAL STRUCTURE 11
2. for any not split epi g0 : V 0 ! W , there exists ν : V 0 ! V with g0 = gν; 3. if γ ∈ End V satisfies g = gγ, then γ is an isomorphism.
The source and sink morphisms were called, by Auslander and Reiten, minimal left almost split and minimal right almost split, respectively.
Let (K, S) be a Krull-Schmidt category with short exact sequences. The pair (f, g) is an almost split sequence, or an Auslander-Reiten sequence, of (K, S) if (f, g) ∈ S and f is a source morphism in K and g is a sink morphism in K.
The Gabriel's quiver ∆(K) of a Krull-Schmidt category K is defined as follows: its vertices are the isomorphism classes [U] of the indecomposable objects U in K, and there is an arrow [U] ! [V ] in case Irr(U, V ) 6= 0. A component of K is the class objects generated by the indecomposable objects belonging to a connected component of ∆(K), see [48].
Given an Auslander-Reiten sequence (f, g) of (K, S), with f : U ! V , g : V ! W , we note that U, W are indecomposable objects such that the isomorphism class [U] of U is uniquely determined by the isomorphism class [W ] of W , and [W ] is uniquely determined − by [U]. Then, [U] is denoted by τ(K,S) [W ] (or U = τ(K,S) W ) and similarly τ(K,S) [U] = [W ]. Moreover, τ(K,S) is called the Auslander-Reiten translation of (K, S).
n di If an object V ∈ K can be decomposed in the form V = ⊕ Vi , with Vi indecomposable i=1 and pairwise non-isomorphic (and di ≥ 1), then dimk Irr(U, Vi) = di = dimk Irr(Vi,W ) for all i. Thus there are di arrows [U] ! [Vi] and di arrows [Vi] ! [W ]. The quiver ∆(K) becomes a translation quiver by adding τ(K,S). The translation quiver (∆(K), τ(K,S)) is called the Auslander-Reiten quiver Γ(K, S) of (K, S) (also denoted as Γ(K)). Note that, in case (K, S) is an exact category, then in order to know that (f, g) ∈ S is an almost split sequence, it is necessary to prove only that f is a source morphism or g is a sink morphism [48].
In this work, we consider that object classes M are Krull-Schmidt categories with short exact sequences belonging to M. In particular, its Auslander-Reiten quiver is defined in −l such a way that, if an indecomposable object U ∈ Γ(M), then its translate τM (U) (or r τM (U)) belongs to Γ(M) as well. The Auslander-Reiten quiver Γ(M) is called a relative Auslander-Reiten quiver.
The τ-orbit of an indecomposable projective (injective) representation P (x)(I(y)) is the −l l set of images τ P (x)(τ I(y)), l ∈ Z, where τ is the Auslander-Reiten translation.
We note that, the relations between objects and morphisms in categories of representations of ordinary posets and equipped posets are similar to the relations between objects and morphisms in categories of representations of quivers and valued quivers. Also note that, the Auslander-Reiten quiver Γ(P) of the category of representation of an equipped poset P is a valued translation quiver. Therefore, following to Simson [56], arrows in Γ(P) are valued arrows 0 (dUV , d ) [U] −−−−−−−−!UV [V ] CHAPTER 1. PARTIALLY ORDERED SETS WITH ADDITIONAL STRUCTURE 12 where 0 dUV = dim Irr (U, V )(U) and dUV = dim(V ) Irr (U, V ), and U = End U/rad End U. This means that there are irreducible morphisms
d d0 U −! V UV and U UV −! V,
0 we write U −! V if dUV = dUV = 1. Henceforth, the following notations will be assumed in valued arrows of an Auslander-Reiten quiver.
•••− − − − − •••− − − − − @ @ @ @ 2 @ 22@ 21(, 2) @2, 1) (2, 1) @ (1, 2)( •• − − − − −@R and@R• means •• − − − −@R − and@R• respectively.
In order to unify the Auslander-Reiten theory for different kind of categories (Krull- Schmidt, abelian, exact, triangulated, etc) Liu [38] introduced the following generalization of the definition of an Auslander-Reiten sequence (he also describes in this work the struc- ture of the Auslander-Reiten quiver of a Krull-Schmidt category). f g Let X −! Y −! Z be a sequence of morphism in a Krull-Schmidt category K. One says that f is a pseudo-kernel of g if
f∗ g∗ Homk (M, X) −! Homk (M, Y) −! Homk (M, Z) is an exact sequence for every object M in K, and that g is a pseudo-cokernel of f if
g∗ f ∗ Homk (Z, N) −! Homk (Y, N) −! Homk (X, N)
f g is exact for every N in K. The sequence X −! Y −! Z is said to be short pseudo-exact if f is a pseudo-kernel of g while g is a pseudo-cokernel of f.
f g Definition 19. A short pseudo-exact sequence X −! Y −! Z in a Krull-Schmidt category K with Y 6= 0 is called an Auslander-Reiten sequence if f is a source morphism and g is a sink morphism.
1.3 Complexification and reellification
In this section, we give definitions of complexification and reellification of a vector space and its respective extension to complexification of linear transformations between vec- tor spaces [58]. Some particular subspaces whose properties are useful in the theory of representation of equipped posets are described as well. [67].
The complexification of a real vector space V is the complex vector space Ve = V × V in which addition + : Ve × Ve −! Ve and scalar multiplication · : C × Ve −! Ve are defined by the following identities [58]. The symbols R and C denote as usual the corresponding real numbers and complex numbers fields: CHAPTER 1. PARTIALLY ORDERED SETS WITH ADDITIONAL STRUCTURE 13
v v0 v + v0 v av − bw + = and (a + ib) = . w w0 w + w0 w bv + aw
We identify V with the real subspace V × {0} of Ve and write simply the vector v instead the column vector (v, 0)t. Then an arbitrary element of Ve can be written in the form
v v w = + i = v + iw, v, w ∈ V . w 0 0
If W is a C-space then the reellification WR of W is a real vector space obtained from W by restricting the scalar multiplication to R × W (sloppily, this is just W considered as a real vector space).
According to Spindler [58], every basis of a real vector space V is also a basis on C of Ve, so that, dimC Ve = dimR V . If B = {wt | t ∈ A} is a basis of W over C then 0 B = B ∪ iB = {wt | t ∈ A} ∪ {iwt | t ∈ A} is a basis of WR over R. Thus, we can conclude that dimR WR = 2 dimC W .
Note that, until now the preceding concepts are presented from a categorical point of view. Let R be the category of all real vector spaces and let C be the category of all complex vector spaces. The complexification functor C : R −! C is covariant, where C is defined by associating to each real vector space V its complexification C(V ) = Ve, and to each R- linear morphism ϕ : V −! W its complexification C(ϕ) = ϕe : Ve −! Wf. Identically, the reellification functor R : C −! R is covariant, where R is defined by associating to each complex vector space W its reellification R(W ) = WR, and to each C-linear morphism ψ : V −! W its reellification R(ψ) = ψR : VR −! WR [58].
Complexification and reellification processes as described above admit a generalization to a pair of fields (F,G) where G is a quadratic extension of F , that is, for a minimal polynomial m(t) = t2 + αt + β with β 6= 0, (α, β ∈ F ), and ξ ∈ G a fixed root of m(t) it holds that G = F (ξ). Definition 20. Let U be a F -vector space. The complexification of U is the G-vector space, denoted by Ue = U ×U, in which addition + : Ue ×Ue −! Ue and scalar multiplication · : G × Ue −! Ue are defined by:
u u0 u + u0 u au − βbv + := and (a + ξb) := , v v0 v + v0 v bu + (a − αb)v with u, u0, v, v0 ∈ U, a + ξb ∈ G.
Let W be a G-subspace of Ue. The reellification of W is the F -vector space, denoted by WF , which is obtained from W by restricting the scalar multiplication to F × W ; so the group structure of WF and W are the same.
As in the case of the pair (R, C), we can write Ue = U ⊕ ξU ' U ⊗F G. These identities u allow us to use the notation w = u + ξv or w = (u, v)t for a vector w = ∈ U. v e CHAPTER 1. PARTIALLY ORDERED SETS WITH ADDITIONAL STRUCTURE 14
Let U0 be an F -space and Ue0 its complexification, which is a G-space. Since the com- plexification of Uf0 is a natural direct sum of F -spaces Ue0 = U0 ⊕ ξU0 = (U0,U0) (by identification U0 = U0 ⊗ 1G and ξU0 = U0 ⊗ ξ), an arbitrary F -subspace (or G-subspace) W of Uf0, possesses the real and imaginary parts, which are defined as follows. The real part of W is Re W := {x ∈ U0 | x + ξy ∈ W, for some y ∈ U0}. The imaginary part is
Im W := {y ∈ U0 | x + ξy ∈ W, for some x ∈ U0}.
The real and imaginary parts coincide if W is a G-space, and both the real and imaginary parts are F -subspaces of U. If B = {wl = xl + ξyl | xl, yl ∈ U}l∈A is a basis of W over G, where A is an index set, xl, yl ∈ U for all l ∈ A. The real and imaginary parts of W can be generated as follows: Re W = hxl | l ∈ AiF and Im W = hyl | l ∈ AiF .
Remark 1.4. Sometimes, we will adopt the notation (X,Y ) for a G-subspace W of the form W = X + ξY , where X = Re W = hxl | l ∈ AiF and Y = Im W = hyl | l ∈ AiF .
Note that, if ϕ : U −! V is an F -linear map between F -spaces U and V then the complexification ϕe = ϕ ⊗F 1G of ϕ is a G-linear map ϕe : Ue −! Ve, defined in such a way that for z = x + ξy ∈ Ue, ϕe(z) = ϕe(x + ξy) = ϕ(x) + ξϕ(y). Remark 1.5. Let mod (F ) and mod (G) be the categories of F -modules and G-modules respectively, then the complexification functor C and the reellification functor R can be defined as G ⊗F and HomG (G, ), respectively. Note that, if G = {a + ξb | a, b ∈ F } is the quadratic extension G = F (ξ) of a field F then B = {1, ξ} is a basis of G over F thus dimF G = 2. In particular, since ξ is root of a minimal polynomial t2 + αt + β then ξ−1 = −(ξβ−1 + αβ−1). As natural if g = a + ξb ∈ G then we assume that a is the real part Re g of g and b is the imaginary part Im g of g.
We use F {} (resp. h iF ), G {} (resp. h iG) to denote the vector space generated by the set that is within the keys (rep. angle brackets) over the fields F and G, respectively.
Let U be an F -space and Ue be its complexification. For any G-subspace W of Ue, it is possible to associate two F -subspaces of U, denoted by W −, W +, and one G-subspace of Ue, denoted by F (W ), and defined as follows:
(i) W + = Re W = Im W ; (ii) W − = F {u ∈ U | u + ξ0 ∈ W } = F {u ∈ U | (u, 0)t ∈ W };
(iii) F (W ) = Wg+.
Note that, W − ⊆ W +, W ⊆ F (W ), F (W ) is called the F -hull of W . Furthermore, if V is an F -subspace of U and Z = Ve, then Z+ = Z− = V . Therefore, V is an F -form of Z. − − Also we can interpret W as W = W ∩ U0. Remark 1.6. Any G-subspace W of Ue can be written as a direct sum of G-subspaces, W = Wg− ⊕H, where H is a complementary subspace of Wg− in W . Therefore, H+ ' W +/W −. ◦ Remark 1.7. Note that, since x ¢ x whenever x ∈ P then Ux ⊂ F (Ux) ⊂ Ux. Therefore, ◦ if x ∈ P then F (Ux) = Ux. CHAPTER 1. PARTIALLY ORDERED SETS WITH ADDITIONAL STRUCTURE 15 1.4 Representation of equipped posets and equipped posets with involution
In this section, we recall the definitions given by Zavadskij et al of the category of repre- sentations of equipped posets with and without an involution defined on its set of points. We also remind that ordinary equipped posets (i.e., equipped posets without involution) are called 2-equipped posets by Bautista and Dorado, and Zavadskij in [6] and [71], re- spectively. It should be noted that Zavadskij gave a generalization of equipped posets over a pair of fields (F,G), where G is a Galois extension of the ground field F [71].
We also recall that according to Zavadskij [70], the theory of representations of equipped posets is a version of the open homogeneous biquadratic problem which consists of finding a canonical form for real matrices of even size 2m × 2n with respect to formally complex transformations of the form: AB X −qY AB Z −qT 7−! , (1.1) CD YX CD TZ where all matrices are over F . The blocks A, B, C and D are all of the same size m × n, matrix blocks X,Y,Z and T have a suitable order such that the corresponding matrices defined by them in (1.1) are non-singular, where (F,G) is a pair of fields, F ⊂ G and G is a quadratic extension of F .
Definition 21. If the field G is a quadratic extension of a field F then a representation of an equipped poset over the pair (F,G) is a system of subspaces of the form
U = (U0 ; Ux | x ∈ P), (1.2) where U0 is a finite dimensional F -space; and for each x ∈ P, Ux is a G-subspace of Uf0, such that,
x y =⇒ Ux ⊂ Uy, x ¢ y =⇒ F (Ux) ⊂ Uy.
We let rep P denote the category whose objects are the representations of an equipped poset P over a pair of fields (F,G). In this case, a morphism ϕ :(U0 ; Ux | x ∈ P) −! (V0 ; Vx | x ∈ P), between two representations U and V , is an F -linear map ϕ : U0 −! V0 such that:
ϕe(Ux) ⊂ Vx, for each x ∈ P, where ϕe : Uf0 −! Vf0 is the complexification of ϕ.
The composition between morphisms of rep P is defined in a natural way and the sum U ⊕ V ∈ rep P is defined as for ordinary posets. Therefore, rep P is a Krull-Schmidt category. A representation U ∈ rep P is indecomposable if U 6= 0 and there is not a direct sum decomposition of U into two non-zero representations. CHAPTER 1. PARTIALLY ORDERED SETS WITH ADDITIONAL STRUCTURE 16
Let P be an equipped poset and U, V ∈ rep P. Then U is a sub-representation of V if and only if the spaces U0,V0,Ux and Vx satisfy the inclusions U0 ⊂ V0 and Ux ⊂ Vx, for each x ∈ P.
Two representations U, V ∈ rep P are said to be isomorphic if and only if there exists an F -isomorphism ϕ : U0 −! V0 such that ϕe(Ux) = Vx, for each x ∈ P. We let Ind P denote a set of representatives of the isomorphism classes of all the indecomposable objects of a category rep P.
The main problem, dealing with the theory of representations of equipped posets, con- sists of classifying indecomposable representations up to isomorphisms and irreducible morphisms of a category rep P of a given equipped poset P.
P P For each x ∈ P, we let Ux denote the radical subspace of Ux, that is, Ux = F (Uz)+ Uz. z¡x z≺x
Let P be an equipped poset. The dimension of a representation U ∈ rep P is the vector d = dim U = (d0 ; dx | x ∈ P), where d0 = dimF U0 and dx = dimG Ux/Ux. A representation U ∈ rep P is sincere if d0 6= 0 and dx 6= 0, for each x ∈ P. In other words, the vector d of a sincere representation U has not null coordinates.
_ + Let X ⊂ P and U ∈ rep P. The subspaces of U0, denoted respectively by UX , UX , U X _ − and (UX ) , are defined as follows:
X + X + UX = Ux,UX = Ux ; x∈X x∈X _ _ \ − \ − U X = Ux, (UX ) = Ux ; (1.3) x∈X x∈X _ + U∅ = 0, U∅ = U0.
+ − Also note that if P is an equipped poset and x, y ∈ P with x ¡ y then Ux ⊂ Uy .
Let P be an equipped poset with involution ∗ which naturally induces an equivalence relation on the points of P and Θ be the set of all equivalence classes on P respect to such an involution. Then classes κ ∈ Θ consist either of one or two points, in the second case it holds that x 6= x∗ and κ = (x, x∗).
Now, we recall the definition of a representation of an equipped poset with involution as given by Zavadskij in [67]. In this case, we let (P, Θ) denote an equipped poset with an involution inducing a set of classes Θ over P, if there is not doubt with the order ≤ and the corresponding equipment, we will write simply P to denote an equipped poset with involution.
Definition 22. Let (P, Θ) be an equipped poset with involution. A representation U of (P, Θ) is a system of vector spaces of the form
U = (U0 ; Uκ | κ ∈ Θ), (1.4) CHAPTER 1. PARTIALLY ORDERED SETS WITH ADDITIONAL STRUCTURE 17 where U0 is a finite dimensional F -vector space and Uf0 is its corresponding complexifica- tion, which is a G-vector space, such that,
if x is a small point =⇒ Ux ⊂ U0; if x is a weak point =⇒ Ux ⊂ Uf0; if x is a big point =⇒ U(x,x∗) ⊂ U0 ⊕ U0; if x is a biweak point =⇒ U(x,x∗) ⊂ Uf0 ⊕ Uf0; + − if x < y =⇒ Ux ⊂ Uy .
A morphism ϕ :(U0 ; Uκ | κ ∈ Θ) −! (V0 ; Vκ | κ ∈ Θ) between two representations U and V , is an F -linear map ϕ : U0 −! V0 such that:
κ ϕ (Uκ) ⊂ Vκ, for each κ ∈ Θ.
κ In the natural sense, if z = (z1, z2) ∈ Uκ, then ϕ (z) = (ϕ(z1), ϕ(z2)).
1.5 The matrix problem
Each equipped poset P naturally defines a matrix problem of mixed type over the pair (F,G). Consider a finite rectangular matrix M separated into vertical stripes Mx, x ∈ P, with Mx being over F (over G) if the point x is strong (weak):
x- y ⊗ ⊗ e e M = GG ··· FF
such partitioned matrices M are called matrix representations of P over (F,G). Their admissible transformations are as follows:
(a) F -elementary row transformations of the whole matrix M;
(b) F -elementary (G-elementary) column transformations of a stripe Mx if the point x is strong (weak);
(c) in the case of a weak relation x ≺ y, additions of columns of the stripe Mx to the columns of the stripe My with coefficients in G; (d) in the case of a strong relation x ¡ y, independent additions both real and imaginary parts of columns of the stripe Mx to real and imaginary parts (in any combinations) of columns of the stripe My with coefficients in F (assuming that, for y strong, there are no additions to the zero imaginary part of My).
Two representations are said to be equivalent or isomorphic if they can be turned into each other with help of the admissible transformations. The corresponding matrix problem of mixed type over the pair (F,G) consists of classifying the indecomposable in the natural sense matrices M, up to equivalence. CHAPTER 1. PARTIALLY ORDERED SETS WITH ADDITIONAL STRUCTURE 18
Remark 1.8. The matrix problem for representations (a)-(d) occurs naturally in the clas- sification of the objects U ∈ rep P up to isomorphisms. In this case, it is associated to the representation U its matrix presentation MU = (Mx ; x ∈ P) defined as follows:
◦ ⊗ If a point x ∈ P (respectively, x ∈ P ), then the columns of the stripe Mx consist of coordinates (with respect to a fixed ordered basis B of U0) of a system of generators G + + of Ux (respectively, G-subspace Ux) modulo its radical subspace Ux (respectively, Ux). Problem (a)-(d) may be obtained by changing basis B and the system of generators G.
The dimension of a partitioned matrix M is the vector d = dim M = (d0; dx | x ∈ P), where d0 is the number of rows in M and dx is the number of columns in the stripe Mx.
Naturally, a matrix representation of an equipped poset with involution is such a repre- ∗ sentation M of an equipped poset that the vertical stripes Mx and Mx (related to the equivalent points x ∼ x∗) have the same numbers of columns. The corresponding matrix problem consists of classifying indecomposable matrices up to equivalence, determined by the listed above transformations of type (a), (c), (d) and also by the following ones (b0 ) − (b000 ) replacing the transformations of type (b):
0 (b ) F -elementary (G-elementary) transformations of columns of a stripe Mx if the point x is small (simple weak); 00 ∗ (b ) the same F -elementary transformations of columns of the stripes Mx and Mx if x ∼ x∗ are big points; 000 (b ) the same (conjugate) G-elementary transformations of columns of the stripes Mx ∗ ∗ and Mx if x ∼ x are biweak points of genus 1 (or genus -1). Remark 1.9. 1. The conjugate G-elementary transformations of columns of the stripes ∗ Mx and Mx are generated by transformations of two types:
(a) multiplications of two corresponding to each other columns of the stripes Mx ∗ and Mx by mutually conjugate complex numbers λ, λ 6= 0; ∗ (b) addition of the i-th column of the stripe Mx to its j-th column of the stripe Mx to its j-th column with the conjugate coefficient λ. 2. Obviously, in the case of a primitive involution, the transformations (b000 ) disappear but the others remain. 3. In a more partial case, under absence of all weak points, we obtain the problem on representations of ordinary posets with involution over F , which was considered (over an arbitrary field) in [67].
Later on, a subset X ⊂ P will be called small (big, weak,...) if all its points are small (big, weak,...). A subset, consisting of two (three, four) mutually incomparable points, is called dyad (triad, tetrad). If the points a, b ∈ P are incomparable then we denote these as a k b.
Also we consider equipped posets with involution being reduced, i.e. satisfying the condi- tion:
(R) each big point is incomparable with some big point or some weak point or with some small dyad. CHAPTER 1. PARTIALLY ORDERED SETS WITH ADDITIONAL STRUCTURE 19 1.6 Some indecomposable objects
In this section, we give some examples of indecomposable objects in the category rep P, where P is an equipped poset or an equipped poset with involution. The matrix presen- tation of this indecomposable objects was defined by Zavadskij in [67].
If there is not confusion, hereinafter P denotes an equipped poset unless otherwise stated.
If P is an equipped poset and A ⊂ P, then P (A) = P (minA) = (P0 ; Px | x ∈ P), where P = F and 0 ( G, if x ∈ A∨, Px = 0, otherwise.
In particular, P (∅) = (F ; 0,..., 0).
If a, b ∈ P with a k b then P (a, b) denotes an indecomposable object such that P (a, b) = (P0 ; Px | x ∈ P) with P0 = F and ( G, if x ∈ a∨ ∪ b∨, Px = 0, otherwise.
If a, b, p ∈ P⊗ then T (a), T (a, b) and T (a, p) denote indecomposable objects with matrix representation of the following form:
a a b a p T (a) = 1 T (a, b) = 1 0 , a ≺ b. T (a, p) = 1 1 , a, p incomparable. ξ ξ 1 ξ ξ
Note that an indecomposable T (a) can be completely described as a representation of P 2 such that, T (a) = (T0 ; Tx | x ∈ P), where T0 = F and
2 Tf0 = G , if x ∈ aO, t Tx = G (1, ξ) , if x ∈ ag, 0, otherwise,
t in this case, (1, ξ) is the column of coordinates with respect to an ordered basis of T0.
On the other hand, representation T (a, b) may be described in such a way that T (a, b) = 2 (T0 ; Tx | x ∈ P), where T0 = F and G{(1, ξ)t}, if a x ≺ b, 2 ∨ Tx = Tf0 = G , if x ∈ aO ∪ b , 0, otherwise. CHAPTER 1. PARTIALLY ORDERED SETS WITH ADDITIONAL STRUCTURE 20
If a ∈ P⊗ and B ⊂ P is a subset completely weak such that a ≺ B, then we let T (a, B) 2 denote the representation of P which satisfies the following conditions with T0 = F : G{(1, ξ)t}, if x ∈ ag \ B, 2 ∨ Tx = Tf0 = G , if x ∈ aO + B , 0, otherwise.
In particular, T (a, ∅) = T (a).
⊗ ◦ If a ∈ P and c ∈ P , where a, c are incomparable points in P, then G1(a, c) and G2(a, c) denote indecomposable objects with matrix representations of the following form:
a c a c 1 0 1 0 1 G1(a, c) = G2(a, c) = ξ 1 ξ 1 0
The indecomposable object G1(a, c) may be defined in such a way that G1(a, c) = 2 (G0 ; Gx | x ∈ P), where G0 = F and G (1, ξ)t , if x ∈ ag, t Gx = G (0, 1) , if x ∈ cO, 0, otherwise.
Whereas, the representation G2(a, c) is defined in such a way that G2(a, c) = (G0 ; Gx | 2 x ∈ P), where G0 = F and G (1, ξ)t , if x ∈ ag, t t 2 Gx = G (0, 1) , (1, 0) = G , if x ∈ cO, 0, otherwise.
If P is an equipped poset, with a primitive involution ∗, and a, b are incomparable points ⊗ in P with a one big point and b ∈ P , then G1(b, a) and G2(b, a) denote indecomposable representations with matrix representations of the following form:
b a a∗ b a a∗ 1 0 0 0 1 1 0 0 0 G1(b, a) = G2(b, a) = ξ 1 0 0 ξ 0 1 0 0
Note that, if P is an equipped poset, U, V, W ∈ rep P and U = V ⊕ W then assuming the natural notations for these representations it holds that U0 = V0 ⊕ W0 and Ux = (Vf0 ∩ Ux) ⊕ (Wf0 ∩ Ux) for each x ∈ P.
The following result is an equipped version of the characterization of indecomposable representations in the ordinary case [3].
Lemma 2. A representation U of an equipped poset P over the pair of fields (F,G) is indecomposable if and only if 0 and 1 are the only idempotents of End U. CHAPTER 1. PARTIALLY ORDERED SETS WITH ADDITIONAL STRUCTURE 21
2 Proof. Let U = (U0; Ux | x ∈ P) be a representation of P, and ϕ = ϕ ∈ End U. Then U = (ϕ(U0); ϕe(Ux) | x ∈ P) ⊕ ((1F − ϕ)(U0); (1^F − ϕ)(Ux) | x ∈ P) is a direct sum of representations. Indeed, to confirm this assertion, we need to prove first that U0 = ϕ(U0) ⊕ (1F − ϕ)(U0). This condition follows from the observations that z = ϕ(z) + (1F − ϕ)(z) for each z ∈ U0, and ϕ(U0) ∩ (1F − ϕ)(U0) = 0, noticing that if 2 w = ϕ(u) = (1F − ϕ)(v) ∈ ϕ(U0) ∩ (1F − ϕ)(U0), then w = ϕ (u) = ϕ(w) = (1F − ϕ)(v) = 2 (ϕ − ϕ )(v) = 0. The second condition is that for each x ∈ P, ϕe(Ux) = ϕ^(U0) ∩ Ux and (1^F − ϕ)(Ux) = (1F ^− ϕ)(U0) ∩ Ux. This condition is satisfied for the idempotent ϕ, since 2 ϕe(Ux) ⊆ ϕ^(U0) ∩ Ux and if w = ϕe(u) ∈ ϕ^(U0) ∩ Ux, then w = ϕe(u) = ϕe (u) = ϕe(w) ∈ ϕe(Ux). Since 1F − ϕ is also idempotent, a similar argument shows that (1^F − ϕ)(Ux) = (1F ^− ϕ)(U0) ∩ Ux, as desired. If ϕ is neither 0 or 1, then U is decomposable. Consequently, if U is indecomposable, then 0 and 1 are the only idempotents of End U.
Conversely, assume that U = V ⊕ W is a direct sum of representations with both V0 and W0 nonzero. Then U0 = V0 ⊕W0 with each Ux = (Vf0 ∩Ux)⊕(Wf0 ∩Ux). Let ϕ = U0 −! V0 be a vector space projection, defined by ϕ(v, w) = v. Then ϕ is a representation morphism of U, because ϕe(Ux) ⊆ Vf0 ∩ Ux for each x ∈ P. Furthermore, ϕ is clearly an idempotent, and ϕ is neither 0 nor 1, since the kernel of ϕ is W0 6= U0 and the image of ϕ is V0 6= U0. It follows that if 0 and 1 are the only idempotents of End U, then U most be indecomposable.
Corollary 1. For any equipped poset P, the following statements hold.
(i) If a, b ∈ P with a k b, then P (a) and P (a, b) are indecomposable representations of P.
(ii) If a, b, p ∈ P with a, b, p ∈ P⊗, a ≺ b and a k p, then T (a), T (a, b) and T (a, p) are indecomposable representations of P.
⊗ ◦ (iii) If a, c ∈ P with a ∈ P , c ∈ P and a k c, then G1(a, c) and G2(a, c) are indecom- posable representations of P.
Proof.
(i) Let ϕ ∈ End P (a) non zero. If λ ∈ F and λ 6= 0, then the image of any λ0 ∈ F under ϕ is given by the identity 0 0 ϕλ(λ ) = λλ . Therefore ϕ is an isomorphism. Thus 0 and 1 are the only idempotents in End P (a) w F . Then, by Lemma 2, P (a) is indecomposable. The same arguments can be used to prove that P (a, b) is indecomposable.
(ii) Let ϕ ∈ End T (a) be non zero. Then ϕ : F 2 ! F 2 is a F -linear transformation. Let 1 0 1 γ 0 B = , be a given fixed base of F 2, with ϕ = and ϕ = 0 1 0 δ 1 ε . Thus, the matrix representation of ϕ, with respect to the given fixed basis ρ CHAPTER 1. PARTIALLY ORDERED SETS WITH ADDITIONAL STRUCTURE 22
γ ε has the shape M = . Since ϕ(U ) ⊆ U , for each x ∈ P it holds that ϕ δ ρ x x 1 1 γ ε 1 1 γ + εξ λ ϕ = λ . Hence = λ and therefore = . So, ξ ξ δ ρ ξ ξ δ + ρξ λξ we get the following system of equations
γ + εξ = λ, δ + ρξ = λξ.
Thus δ = −βε and ρ = γ − αε. Then
γ ε M = ϕ −βε γ − αε
with α, β, γ, ε ∈ F . Note that ϕ = 0 if and only if γ = ε = 0. Moreover, ϕ is invertible if and only if ϕ 6= 0, i.e. if and only if γ 6= 0 or ε 6= 0. Therefore End (T (a)) is a field, and hence by Lemma 2, T (a) is indecomposable.
The same arguments can be used to prove the remaining items. Remark 1.10. Zavadskij proved in [67] that P (∅), P (ci), T (ci) and T (ci, cj), for 1 ≤ i < j ≤ n, are the only indecomposable representations (up to isomorphisms) over the pair
(R, C) of a completely weak chain C = {c1 ≺ · · · ≺ cn}. In fact, if U = (U0 ; Uci | 1 ≤ i ≤ n) is a representation of C over (R, C), then in the corresponding matrix representation to each block Uci , 1 ≤ i ≤ n, can be reduced via admissible transformations to the following standard form:
I I Uc = , i iI
where the columns consist of generators of Uci modulo its radical subspace Uci = Uci−1 with respect to a fixed basis of U0 (in this case, empty cells indicate null coordinates). This result can be generalized in a natural way to the case (F,G) by using a suitable scalar
ξ ∈ G instead of the constant i ∈ C in the matrix presentation of Uci shown above.
1.7 Some classification theorems
In this section, some classical theorems (without proof) of representation of posets are reminded [25,36,56,63,67,68]. By considering instead of the pair (R, C), the pair of poly- nomial rings (R[t], C[t]) or the pair (RhX,Y i, ChX,Y i) of free algebras in two generators, one can define analogously matrix representations of P over such pairs.
Each (R[t], C[t])-representation N, of an equipped poset P, naturally generates (over R) a real series (as a rule, infinite) of (R, C)-representations by substituting any square matrix A over R for the variable t (reduced, if necessary, to some canonical form under the real similarity transformations) and scalar matrices λI of the same size for the numbers CHAPTER 1. PARTIALLY ORDERED SETS WITH ADDITIONAL STRUCTURE 23
λ ∈ C. We call a real series genuine if it contains infinitely many pairwise non-isomorphic indecomposable representations.
If the mentioned (R[t], C[t])-representation N is such that all its strong points stripes Nx are over R (i.e., do not contain the variable t), then one can also generate by N (over C) a complex series of (R, C)-representations by substituting any square matrices A over C for t (usually reduced to the Jordan normal form) and the scalar matrices λI for the scalar entry λ. We call a complex series genuine if it contains infinitely many pairwise non-isomorphic indecomposable representations which are not covered (up to isomorphism of representations) by a finite number of real series.
For a given equipped poset P, denote by µR(d) and µ(d) the least possible numbers, respectively, of a real series and of any series (both real and complex) containing almost all indecomposable (R, C)-representation of P of a given dimension d (considered up to isomorphism).
Obviously, µ(d) ≤ µR(d). Moreover, if µR(d) < ∞, there does not exist any complex series in the dimension d, hence µ(d) = µR(d) (in [67] the complex series (10.10) demonstrates a possibility µ(d) = 1, µR(d) = ∞).
For any (RhX,Y i, ChX,Y i)-representation W , of an equipped poset P, and any pair of square real matrices A, B of equal size denoted by WA,B, the corresponding (R, C)- representations are obtained from W , by substituting the pair A, B for the variables X,Y and the scalar matrices λI of the same size for the scalar λ.
A(RhX,Y i, ChX,Y i)-representation W is called a wild generator if for a complete set of indecomposable and pairwise nonequivalent (under the common real similarity transforma- tions) pairs A, B all the generated (R, C)-representations WA,B are also indecomposable and pairwise nonequivalent [68]. Definition 23. Let P be an equipped poset. We say that:
(i) P is of finite representation type if Ind P < ∞; (ii) P is called tame if µ(d) < ∞ for all dimension vector d;
(iii) we call P of finite growth if supd µR(d) < ∞; (iv) P is called wild if it admits a wild generator, (v) we call P one-parametric (or m-parametric) if µ(d) = 1 (or µ(d) = m) for any chosen dimension d.
Since µ(d) ≤ µR(d), each poset of finite growth is tame. Naturally, the well known defini- tions of ordinary posets of tame type and of finite growth (see [13] and [30] respectively, where the corresponding classes were introduced) coincide with the just given ones in the case of the trivial equipment (which admits the only possibility µ(d) = µR(d) for all d).
Subsequently, the disjoint union of subsets X,Y ∈ P is called sum and denoted by X +Y . The sum X + Y is called cardinal (ordinal) if there is no order relations between points x ∈ X and y ∈ Y (if x < y for all x ∈ X and y ∈ Y , or conversely). CHAPTER 1. PARTIALLY ORDERED SETS WITH ADDITIONAL STRUCTURE 24
By (pe1,..., pek, q1, . . . , ql) we denote an analogous cardinal sum in which l chains are or- dinary with q1, . . . , ql points, and k chains are completely weak with p1, . . . , pk points, respectively. Let (N, m) be a cardinal sum of an ordinary m-point chain and an ordinary four-point subset N = {a < b > c < d}.
The results presented below are classical theorems regarding classification of ordinary posets and posets with additional structure. These theorems will be listed in chronological order of appearance, we use these results to classify equipped posets of type Yt.
The following theorem was proved by Kleiner in [36] by using the algorithm of differen- tiation with respect a maximal point (see also [3, 56]). Soon afterwards, Zavadskij and Kirichenko proved the same result by using the algorithm of differentiation D-I [63].
Unless we mention explicitly, in the following theorems it is assumed that P denotes an ordinary poset.
Theorem 2 (Kleiner 1972). Let P be a finite poset with a partial order ≤ and let k be a field. Then, the following statements are equivalent.
(1) The category repk P is of finite representation type. (2) The poset P does not contain, as a full subposet, any of the following critical Kleiner's posets e
e e
e e e e @ e e e e e e@ e e e e
eK1 : e (1, 1 e, 1, 1) e Ke2 : (2 e, 2, 2) e Ke3 : (1 e, 3, 3) e Ke4 :( eN, 4)e Ke5 : (1 e, 2, 5)e
Figure 1.3. The Hasse diagram of the critical Kleiner's posets.
Theorem 3 (Nazarova 1975 [3]). Let P be a finite poset and k a field. Then repk P has wild representation type if and only if P contains N1,N2,N3,N4 N5 or N6 as a subposet. Where: CHAPTER 1. PARTIALLY ORDERED SETS WITH ADDITIONAL STRUCTURE 25
e e
eN1 e: (1, 1 e, 1, 1 e, 1) e eN2 : (1 e, 1, e1, 2) e e e
e e e
e e e e e @ e e e e e e@ e e e e e Ne3 : (2 e, 2, 3) e Ne4 : (1 e, 3, 4)e eN5 :( eN, 5)e N6 : (1e, 2, 6)e. Figure 1.4. The Hasse diagram of the Nazarova's hypercritical posets.
Theorem 4 (Drozd 1977, Crawley-Boevey 1988). Let Λ be a finite-dimensional algebra over an algebraically closed field. Then the category of Λ-modules has either tame type or wild type, and not both. [25].
Theorem 5 (Nazarova-Zavadskij 1981). A poset P of tame representation type is of finite growth representation type if and only if it does not contain the following subposet
@ f @ f @ f f f f Definition 24. The evolvent Pb of an equipped poset P is the ordinary poset:
[ 0 00 Pb = {x , x }, x∈P where x0 = x00 = x, for a strong point x ∈ P, and x0 6= x00 is a pair of new strong mutually incomparable points (replacing a weak point x), with the following order relations:
(1) if x ≺ y, then x0 < y0 and x00 < y00 ; 0 0 0 00 00 0 00 00 (2) if x C y, then x < y ; x < y ; x < y and x < y .
The following theorems regard the classification of equipped posets [67, 68].
Theorem 6 (Zavadskij 2003). An equipped poset P is tame (wild) if its evolvent Pb is tame (wild).
Theorem 7 (Zavadskij 2005). For an equipped poset P, the following statements are equivalent.
(a) P is of finite growth (over an arbitrary field k).
(b) The evolvent Pb is of finite growth. (c) The set P is not wild and contains none of the following equipped posets: CHAPTER 1. PARTIALLY ORDERED SETS WITH ADDITIONAL STRUCTURE 26
⊗ @ @ ¡ A e e e e @ @ ⊗ ¡ A
e eG1 e e e eG2 e eG3 e e
⊗ ⊗ ⊗ ⊗ ⊗ ¡ A @ ¡ A ⊗ ⊗ ⊗ ⊗ ⊗ @⊗ e e e e G4 G5 G6 G7 (d) P does not contain subsets of the form N1 = (1, 1, 1, 1, 1), N2 = (1, 1, 1, 2), N3 = (2, 2, 3), N4 = (1, 3, 4), N5 = (N, 5), N6 = (1, 2, 6), W1 = (1˜, 1, 1, 1), W2 = (1˜, 1˜, 1), W3 = (1˜, 1˜, 1)˜ , W4 = (1˜, 1, 2), W5 = (2˜, 1, 1), W6 = (1˜, 2)˜ , W7 = (2˜, 3), W8 = (3˜, 2), W9 = (4˜, 1), and posets G1,...,G7. CHAPTER 2
Algorithms of differentiation
In 2003 [67] Zavadskij defined a new group of algorithms of differentiation, VII-XVII and completion to classify equipped posets (with and without involution) of tame repre- sentation type. In particular, he used algorithms of differentiation I and completion for ordinary posets as well as algorithms VII-IX and completion for equipped posets to classify equipped posets of finite growth representation type.
In this chapter, we prove that algorithms of differentiation VIII-X induce categorical equiv- alences between some quotient categories. Worth noting that Ca˜nadasin [13] gave a proof of the algorithm of differentiation VII by using a decomposition introduced by Rodriguez and Zavadskij in [50]. For the sake of clarity and in order to generate applications in computer science, we do not consider such decomposition.
2.1 (A, B)-cleaving pairs and the Zavadskij symbol
In this section, we describe some useful tools arising from the theory of the algorithms of differentiation. In particular, we recall the definition of the Zavadskij symbol which has been used to establish categorical equivalences induced by this kind of functors between category of representations of posets both ordinary and with additional structures [13,69].
The following lattice allows to define a cleaving pair of subspaces in the sense of Zavadskij.
27 CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 28
U0 @ @ A + B W0 @ @ A B @ @ E0 A ∩ B @ @ 0
Figure 2.1. The diagram of an (A, B)-cleaving of U0.
The order relation in this poset is given by the natural inclusion subspaces, where E0 is a complementary subspace of A ∩ B in A, and W0 is a complementary subspace of A + B in U0.
Definition 25. Let U0 be an F -vector space and E0,W0, A, B ⊂ U0. The pair of subspaces (E0,W0) is an (A, B)-cleaving of U0 if the poset of subspaces described in Figure 2.1 is a lattice (with the obvious meets ∩ and sums +).
In other words, (E0,W0) is an (A, B)-cleaving pair of U0 if and only if
U0 = E0 ⊕ W0,A = E0 + (A ∩ B) and B = W0 ∩ (A + B). (2.1)
Set U0, V0 be two arbitrary finite-dimensional F -vector spaces. For any subspaces X ⊂ U0 and Y ⊂ V0, the Zavadskij Symbol [69] [X,Y ] associated to X and Y is a subspace of HomF (U0,V0) such that if ϕ ∈ [X,Y ] then
X ⊂ Ker ϕ and Im ϕ ⊂ Y.
Note that, if X0 ⊂ X and Y ⊂ Y 0; then [X,Y ] ⊂ [X0,Y 0].
Furthermore, for a vector space decomposition U = X ⊕ Y , we let eX denote the idem- potent iπ in End (U), where π : U ! X and i : X ! U are the natural projection and injection, respectively.
2.2 Algorithm of differentiation I with respect to a suitable pair of points and the completion algorithm
In 1991 Zavadskij followed the Gabriel ideas regarding the categorical properties of the algorithm of differentiation with respect to a maximal point describing in [65] the structure of the Auslander-Reiten quiver for ordinary posets of finite growth representation type [65]. To do that, he proved that the algorithm with respect to a suitable pair of points induces a categorical equivalence between some quotient categories. Similar results were given by Rump and Zavadskij in 2000, 2004 and 2005 for generalized versions of the algorithm I in [52, 53, 63]. CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 29
2.2.1 Algorithm of differentiation I
The following is the equipped version of the definition of the algorithm of differentiation I with respect to suitable pair of points [68].
A pair of strong incomparable points (a, b), of an equipped poset P, is called I- suitable O or suitable for differentiation I, if P = a + bM + C where C = {c1 < ··· < cn} is an ordinary chain incomparable with the points a, b. The derived poset of the set P, with 0 0 + − respect to the pair (a, b), is an equipped poset P = P (a,b) = (P\C) + C + C , where − − − + + + C = {c1 < ··· < cn } and C = {c1 < ··· < cn } are new ordinary chains, replacing the − + + − chain C, with the relations ci < ci ; a < ci and ci < b for all 1 ≤ i ≤ n.
I 0 The differentiation functor D(a,b) : rep P −! rep P assigns to each representation U = 0 0 0 0 (U0; Ux | x ∈ P) of P the derivative representation U = (U0; Ux | x ∈ P ) accordingly to the formulas:
0 U0 = U0, 0 U + = Ua + Uci , for 1 ≤ i ≤ n, ci 0 U − = Ub ∩ Uci , for 1 ≤ i ≤ n, (2.2) ci 0 0 Ux = Ux for the remaining points x ∈ P(a,b), 0 ϕ = ϕ for any linear map-morphism, ϕ : U0 ! V0.
0 P(a,b) can be considered as a subposet of the free lattice generated by P. The following Figure 2.2. shows the Hasse diagram for this differentiation.
+ cn b b @ £ d@ £ d cn £ £ + − £ d c @ cn £ d £ £ 2 d @ d £ £ £ £ I £c+ @ £ £ d £ −! £ 1 d @ d £ c £ (a,b) £ − £ 2 £ @ c £ d £ d @ d 2 − c1 c ad d ad d 1
0 Figure 2.2. The diagrams of an equipped poset P and its corresponding derivative poset P (a,b)
Since usually the derivative representation U 0 is decomposable and contains trivial sum- mands P (a), it is convenient to consider (besides U 0) the reduced derivative representation U # such that U 0 ' U # ⊕ P m(a), where m ≥ 0 and U # is free of direct summands P (a). # # There exist an alternative definition of U , namely, U = W = (W0; Wx | x ∈ P), where W0 is any subspace in U0 satisfying the conditions Ua + W0 = U0,(Ua + Ub) ∩ W0 = Ub 0 m(x) and Wx = U x ∩ W0 for all x ∈ P (here m(x) = lx is the multiplicity of a point x). The # representation U does not depend (up to isomorphism), on the choice of W0. CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 30
The inverse (in some sense) operation ", called integration, assigns to each representation W of the set P0 the primitive representation W " of the initial set P such that (W ")# ' W as soon as W contains no direct summands P (a). It holds (see [62, 65]) the following.
I 0 Theorem 8. (a) The functor D(a,b) : rep P −! rep P (a,b) induces an equivalence of the quotient categories
∼ 0 rep P/hP (a),P (a, c1),...,P (a, cn)i ! rep P (a,b)/hP (a)i.
(b) The operations # and " induce mutually inverse bijections
0 Ind P\ [P (a),P (a, c1),...,P (a, cn)] Ind P(a,b)\ [P (a)].
Remark 2.1. It should be noted that Zavadskij proved numerals (a) and (b) of Theorem 8 in [62, 65], for the algorithm of differentiation I and completion, and for the algorithms VII-X which will be defined below, Zavadskij only proved numeral (b) in [67], leaving the numeral (a) without an explicitly proof, we will devote part of this chapter to give the corresponding proof of the numeral (a) for the algorithms of differentiation VII-X, that is, we will prove that all these algorithms of differentiation induce categorical equivalences.
2.2.2 Completion algorithm
In this section, we present the algorithm of completion as Zavadskij defined in [65,67,68].
A pair of weak points a, b weakly comparable a ≺ b, of an equipped poset P, will be called O special if P = a + bM + Σ, where Σ is the interior of the interval [a, b].
The following is the definition of the completion algorithm which is a differentiation with respect to a special pair of points (a, b) of an equipped poset.
The completion of P, with respect to such special pair (a, b), is a transition from P to a slightly different equipped poset P = P(a,b) obtained from P by strengthening the relation between the points a and b, for which, we have the following two kinds of situations:
O (a) P = a + bM, where a, b are incomparable strong points,
O (b) P = a +bM +Σ, where a, b are weak points, a ≺ b and Σ is the interior of the segment [a, b].
In both cases the completed equipped poset P is obtained from P by adding the only one strong relation a C b. In the case (a), this is in fact the classical completion of an ordinary poset (see, [65]). In the case (b), the completion a C b of P conforms to a pair of mutually symmetric completions of the evolvent Pb with respect to ordinary special pairs (a0, b00) and (a00, b0). CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 31
b b A ⊗ A ⊗
Σ completion Σ −−−−−−−! (a,b) ⊗ B ⊗ B a a
Figure 2.3. The diagrams of an equipped poset P and its corresponding completed poset P(a,b).
− Let D(a,b) : rep P −! rep P(a,b) be the functor induced by the algorithm of completion. − This functor is defined as follows: for U = (U0; Ux | x ∈ P) ∈ rep P, D(a,b) (U) := U = (U 0; U x | x ∈ P) ∈ rep P(a,b), where
U 0 = U0,
U b = Ub + F (Ua), (2.3) U x = Ux, for the remaining points x ∈ P(a,b),
ϕ = ϕ, for a linear map-morphism ϕ : U0 −! V0.
It is clear that rep P is a full subcategory of the category rep P. Moreover, the following statement holds, see [65, 67].
Lemma 3. The category rep P coincides with the full subcategory of the category rep P formed by the objects without direct summands of type P (a) in the case (a), and of type T (a) in the case (b). Therefore
Ind P\{P (a)} in the case (a), Ind P = (a,b) Ind P\{T (a)} in the case (b).
Example 1. In this example, we consider that in Figure 2.3 subsets Σ = A = B = ∅, and under these circumstances poset P = F17 (see, Appendix A). We note that the indecomposable representations of F17 are
Ind F17 = {P (∅),P (1),P (2),T (1),T (2),T (1, 2)}.
In the following Figure 2.4 it is shown a completion of the equipped poset F17.
F17 F17 b b ⊗ ⊗ completion −−−−−−−! (a,b) ⊗ ⊗ a a
Figure 2.4. The diagrams of the equipped poset F17 and its completed poset F17(a,b). CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 32
From Lemma 3, we have that all the indecomposable representations of the completed equipped poset F 17 are
Ind F17\[T (a)] Ind F 17 = {P (∅),P (a),P (b),T (a),T (b)}. (2.4)
Indecomposable representations shown in the equivalence (2.4) are obtained from rep F17 as follows:
P (∅) = (F ; 0, 0) ∈ rep F17, then P (∅) = (F ; 0, 0) ' P (∅) ∈ rep F 17,
P (a) = (F ; G, G) ∈ rep F17, then P (a) = (F ; G, G) ' P (a) ∈ rep F 17,
P (b) = (F ; 0,G) ∈ rep F17, then P (b) = (F ; 0,G) ' P (b) ∈ rep F 17, 2 t t 2 t 2 T (a) = (F ; h(1, ξ) iG, h(1, ξ) iG) ∈ rep F17, then T (a) = (F ; h(1, ξ) iG,G ) ' T (a) ∈ rep F 17, 2 t 2 t T (b) = (F ; 0, h(1, ξ) iG) ∈ rep F17, then T (b) = (F ; 0, h(1, ξ) iG) ' T (b) ∈ rep F 17, 2 t 2 2 t 2 T (a, b) = (F ; h(1, ξ) iG,G ) ∈ rep F17, then T (a, b) = (F ; h(1, ξ) iG,G ) ' T (a) ∈ rep F 17.
Note that, T (a, b) = T (a) = T (a).
Regarding the completion functor, Zavadskij proved the followings two results in [69].
− Theorem 9. The completion functor D(a,b) induces the following categorical equivalence of quotient categories.
∼ rep P/hT (a),T (a, b)i ! rep P/hT (a)i.
Corollary 2. Let Γ(R) and Γ(R) be, respectively, the Gabriel's quivers of the categories R = rep P and R = rep P, then
Γ(R) \ [T (a),T (a, b)] ' Γ(R) \ [T (a)].
The following Figure 2.5 shows the Auslander-Reiten quiver of F 17. The Auslander-Reiten quiver of F17 is shown in Figure 3.5. 1 0 2 0 2 1 @ @ 2@ 22@ 2 @R @R Γ(F 17): 0 − − −−1 − − −− 0 1 0 1 0 1 1
Figure 2.5. The Auslander-Reiten quiver Γ(F 17) of the equipped poset F 17.
2.3 Categorical properties of differentiation VII for equipped posets
The differentiation VII is one of the seventeen differentiations developed by Zavadskij to classify (in particular) equipped posets of tame and of finite growth representation type, see [67, 68]. CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 33
The following is the definition of the algorithm of differentiation VII for an equipped poset. This algorithm is defined for a suitable pair of points (a, b), also we present the corresponding properties of VII as described by Zavadskij in [67].
Let P be an equipped poset. A pair of incomparable points (a, b) of the set P, where a is weak and b is strong, is called VII-suitable (suitable for differentiation VII), if P = aO + bM + C, where {c1 ≺ · · · ≺ cn} is a completely weak chain (possibly empty) incomparable with the point b, and a ≺ c1 (note that automatically a ≺ cn).
The derived poset of the set P, with respect to such a pair (a, b), is the equipped poset
0 − + − + P(a,b) = (P \{a + C}) + {a < a } + C + C ,
− + − − − + + + where a is weak, a is strong, C = c1 ≺ · · · ≺ cn and C = c1 ≺ · · · ≺ cn are − + − − + + − completely weak chains, ci ≺ ci for all i; a ≺ c1 ; a < c1 ; cn < b, and the following conditions are satisfied:
− + − + (1) each of the points a , a ,(ci , ci ) inherits all the previous order relations of the point a (ci) with the points of the subset P \{a + C};
0 (2) the order relations in P(a,b) are induced by the relations in its subset P \{a + C}, and − − by the list above relations (note that, in particular, a ≺ cn ).
+ cn b cn b ⊗ ⊗ + @ cn−1 e c e A A n−1 @ − ⊗ ⊗ ⊗ cn c+ @ c2 VII 2 @ − ⊗ −−−! ⊗ ⊗ c (a,b) n−1 c c+ @ 1 1 @ − ⊗ ⊗ ⊗ c B 2 B @ @ − ⊗ a+ ⊗ c @ 1 a e @ ⊗ a− 0 Figure 2.6. Diagrams of an equipped poset P and its corresponding derivative poset P (a,b).
VII The following functor D(a,b) was given by Zavadskij in [67], soon afterwards, it was updated by Rodriguez and Zavadskij in [50].
Let P be an equipped poset with a pair of points (a, b), VII-suitable, the following formulas VII 0 define the differentiation functor D(a,b) : rep P −! rep P(a,b), induced by the algorithm of differentiation VII. Thus for a given representation U = (U0; Ux | x ∈ P) ∈ rep P, we get CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 34
0 0 0 0 the derived representation U = (U0; Ux | x ∈ P(a,b)), if 1 ≤ i ≤ n, where:
0 U0 = U0, 0 Ua− = Ua ∩ Ub, 0 Ua+ = F (Ua), 0 U − = Uci ∩ Ub, ci (2.5) 0 U + = Uci + F (Ua), ci 0 0 Ux = Ux for the remaining points x ∈ P(a,b), 0 ϕ = ϕ, for a linear map-morphism ϕ : U0 −! V0.
0 + 0 0 2 + Note that, P (a) = P (a ) and T (a) = T (a, ci) = P (a ). A representation of P, con- taining no direct summands of the form P (a),T (a) and T (a, ci), will be called reduced. # # # Obviously, P (a) = T (a) = T (a, ci) = 0, for all 1 ≤ i ≤ n. By construction of the reduced derivative representation (see [67]), we have the following Theorem.
0 Theorem 10. For each object W ∈ rep P there exists an object U = W " ∈ rep P such that U 0 ' W ⊕ P m(a+), for some m ≥ 0.
Zavadskij proved that (W ")# ' W ,(U #)" ' U for each reduced representation U of P and 0 each representation W of P where W ' U # [67].
Theorem 11. In the case of the differentiation VII, the operations # and " induce mutually inverse bijections
0 0 + Ind P\[T (a),T (a, ci),P (a) | 1 ≤ i ≤ n] Ind P = Ind P(a,b)\[P (a )].
The following example allows a better understanding of the behavior of the differentiation 0 VII on objects and morphisms of the categories rep F14 and rep F14.
Example 2. Let P = {a, b} be an equipped poset, with P⊗ = {a}, P◦ = {b}, a k b and 0 0 P(a,b) its corresponding derived equipped poset with respect to the algorithm VII. Let M, M 0 be matrix representations of P and P(a,b), respectively. The following are the corresponding associated diagrams. CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 35
P P0 D-VII (a,b) - ⊗ a b (a, b) a+@ b e e@ e @⊗ a− 0 0 0 Ua Ub Ua− Ua+ Ub 1 1 1 1 1 1 1 1 i 1 M = M 0 = 1 1 i 1 1 1 i 1
Figure 2.7. The matrix representations of an equipped poset and its derived poset.
We recall that the matrix presentation M (M 0) is associated to a representation U (U 0) 0 8 0 8 of the poset P (P ). In this setting, U0 = R = U0, therefore Uf0 = C , B = {e1, . . . , e8} is the standard basis of U0 (vectors ej are column vectors).
If ϕ ∈ End U and ψ0 ∈ End U 0 then:
ϕ(e1) = α1e1 + α2e2 + α3e3, ϕ(e2) = β1e1 + β2e2 + β3e3, note that, e1, e2 ∈ Ub.
ϕ(e3) = c1e3, since e3 ∈ Ua ∩ Ub.
ϕ(e4) = γ1e3 + γ2e4 + γ3e5 + γ4e6 + γ5e7 + γ6e8,
ϕ(e5) = δ1e3 + δ2e4 + δ3e5 + δ4e6 + δ5e7 + δ6e8,
ϕ(e6) = 1e3 + 2e4 + 3e5 + 4e6 + 5e7 + 6e8,
ϕ(e7) = λ1e3 + λ2e4 + λ3e5 + λ4e6 + λ5e7 + λ6e8,
ϕ(e8) = ∆1e3 + ∆2e4 + ∆3e5 + ∆4e6 + ∆5e7 + ∆6e8, taking into account that vectors e3, . . . , e8 belong to Ua.
If z = x+iy ∈ Uf0 then in general ϕe(z) = ϕ(x)+iϕ(y) and ϕe(Ux) ⊂ Vx. If z = e7+ie8 ∈ Ua, and λj, ∆j ∈ R for 1 ≤ j ≤ 6, r1, r3, r5 ∈ C then:
ϕe(z) = (λ1 + i∆1)e3 + (λ2 + i∆2)e4 + (λ3 + i∆3)e5 + (λ4 + i∆4)e6 + (λ5 + i∆5)e7+ + (λ6 + i∆6)e8
= r1(e3 + ie4) + r3(e5 + ie6) + r5(e7 + ie8)
= r1e3 + ir1e4 + r3e5 + ir3e6 + r5e7 + ir5e8. CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 36
For j ∈ {1, 3, 5}, the following system of complex equations arise from the identities de- scribed above: λj + i∆j = rj, λj+1 + i∆j+1 = irj.
By solving this system, for j ∈ {1, 3, 5}, we have that ∆j+1 = λj and ∆j = −λj+1.
The same procedures can be used to solve the system of equations for z = e5 + ie6, z = e3 + ie4 ∈ Ua, and δj, j, γj, cj ∈ R for 1 ≤ j ≤ 6. Then:
If z = e5 + ie6 j ∈ {1, 3, 5} δj + ij = rj, δj+1 + ij+1 = irj.
For j ∈ {1, 3, 5}, we have that j+1 = δj and j = −δj+1.
If z = e3 + ie4 j ∈ {1, 3, 5}, then: cj + iγj = rj, cj+1 + iγj+1 = irj. solving the system, we receive that γj+1 = cj and γj = −cj+1. Thus, Mϕ has the form
α1 β1 0 0 0 0 0 0 α2 β2 0 0 0 0 0 0 α3 β3 c 0 δ1 −δ2 λ1 −λ2 0 0 0 c δ2 δ1 λ2 λ1 Mϕ = . 0 0 0 0 δ3 −δ4 λ3 −λ4 0 0 0 0 δ δ λ λ 4 3 4 3 0 0 0 0 δ5 −δ6 λ5 −λ6 0 0 0 0 δ6 δ5 λ6 λ5
0 In order to obtain the matrix Mψ0 associated with the morphism ψ , we proceed in a similar 0 fashion as for Mϕ (without solve any system of equations). For the morphism ψ , we also 0 0 0 need take into account that the intersection of the spaces and the condition ψ (Ux) ⊂ Vx, for each x ∈ P0. Thus,
0 0 0 ψ (e1) = a1e1 + a2e2 + a3e3, ψ (e2) = b1e1 + b2e2 + b3e3, because e1, e2 ∈ U b. 0 0 0 ψ (e3) = ce3, since e3 ∈ U a+ ∩ U b. 0 ψ (e4) = d1e3 + d2e4 + d3e5 + d4e6 + d5e7 + d6e8, 0 ψ (e5) = f1e3 + f2e4 + f3e5 + f4e6 + f5e7 + f6e8, 0 ψ (e6) = g1e3 + g2e4 + g3e5 + g4e6 + g5e7 + g6e8, 0 ψ (e7) = h1e3 + h2e4 + h3e5 + h4e6 + h5e7 + h6e8, 0 0 ψ (e8) = l1e3 + l2e4 + l3e5 + l4e6 + l5e7 + l6e8, note that e4, . . . , e8 belong to Ua+ . Therefore Mψ0 have the following form CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 37
a1 b1 0 0 0 0 0 0 a2 b2 0 0 0 0 0 0 a3 b3 c d1 f1 g1 h1 l1 0 0 0 d2 f2 g2 h2 l2 Mψ0 = . 0 0 0 d3 f3 g3 h3 l3 0 0 0 d f g h l 4 4 4 4 4 0 0 0 d5 f5 g5 h5 l5 0 0 0 d6 f6 g6 h6 l6
We note that the same procedure can be used for more general cases. Actually, we prove the main theorem regarding algorithms VII − X by defining similar correction-morphisms to matrices Mψ0 associated to a morphism of the category of representations of a derived poset. In this particular case, the correction-morphism ω is defined in such a way that 0 0 0 0 ω : U 0 −! U 0 ∈ rep P and ϕ = ψ − ω ∈ rep P. Moreover, ω has the form.
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d1 0 g1 + f2 0 l1 + h2 0 0 0 d2 − c 0 g2 − f1 0 l2 − h1 Mω = . 0 0 0 d3 0 g3 + f4 0 l3 + h4 0 0 0 d 0 g − f 0 l − h 4 4 3 4 3 0 0 0 d5 0 g5 + f6 0 l5 + h6 0 0 0 d6 0 g6 − f5 0 l6 − h5
0 − 0 + It is clear that ω(e1) = ω(e2) = ω(e3) = 0, hence Ub = Ub ⊂ Ker ω and Im ω ⊂ Va+ ⊂ Va , ' − + so by using the Zavadskij s symbol, we have that ω ∈ [Ub ,Va ].
Remark 2.2. Henceforth, if X is an F -subspace of a vector space U0 then, we let λX denote a linear combination of the form λi1 x1 + λi2 x2 + ··· + λik xk for a fixed basis {xi} ⊂ X with λij ∈ F .
2.4 Categorical properties of the algorithm of differentia- tion VIII for equipped posets
In this section, we recall the definition of the algorithm of differentiation VIII and the proof of its categorical properties which Zavadskij described in [67].
A pair of weakly comparable points a ≺ b, of an equipped poset P, is suitable for differen- tiation VIII if P can be written in the form:
O P = a + bM + Σ + {c, a, b}, where Σ is the interior of the completely weak interval [a, b] and c is a strong point incomparable with [a, b].
The derived poset of the set P, with respect to such a pair (a, b), is the equipped poset CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 38
0 0 P = P(a,b), which is obtained from P by replacing the point c for a three-point chain c− < c0 < c+, where c−, c0 are weak points and c+ is a strong point, a ≺ c0 ≺ b and the following conditions are satisfied:
1. each of three points c−, c+ and c0 inherits all the previous order relations of the point c with the points of P\{c};
0 2. the order relations in the whole set P(a,b) are induced by the initial relations in the subset P\{c} and by the mentioned above relations.
The following diagram in the Figure 2.8 shows an equipped poset with a pair of points H (a, b), VIII-suitable and its corresponding derived poset, in this case A = a and B = bN:
b b A ⊗ A ⊗ c+ @ @ e @ Σ c VIII Σ ⊗ c0 e −−−! (a,b) ⊗ B ⊗ B c− a a e 0 Figure 2.8. Diagrams of an equipped poset P and its corresponding derivative poset P (a,b).
Let P be an equipped poset with a pair of points (a, b), VIII-suitable. The following VIII 0 formulas define the differentiation functor D(a,b) : rep P −! rep P(a,b), induced by the algorithm of differentiation VIII. Thus, for a representation given U = (U0; Ux | x ∈ P) ∈ 0 0 0 0 rep P, we get the derived representation U = (U0; Ux | x ∈ P(a,b)), where:
0 U0 = U0, 0 − Uc− = Uc ∩ Ufb , U 0 = U + F (U ), c+ c a (2.6) 0 Uc0 = Ua + Uc ∩ Ub, 0 0 Ux = Ux, for the remaining points x ∈ P , 0 ϕ = ϕ, for any linear map − morphism ϕ : U0 ! V0.
Note that, for the following indecomposable representations of an equipped poset with a pair of points (a, b), VIII-suitable, we have
0 0 0 T (a) = G1(a, c) = G2(a, c) = T (a).
The Theorems 12 and 13 were proved by Zavadskij in [67], which we recall as follows.
# Let U be a reduced (i.e., without direct summands of type T (a), G1(a, c), G2(a, c)) 0 0 # m + representations of a poset P(a,b) for which U = U ⊕ T (a), where 2m = dim(Ua + − − + − # Ub )/Ub . In this case, if (E0,W0) is a (Ua ,Ub )-cleaving pair, then U = W , where CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 39
0 # Wx = Ux ∩Wf0. In this case, U is a representation of the completed (by the relation a¡b) 0 # # # derived poset P(a,b). Obviously, T (a) = G1(a, c) = G2(a, c) = 0.
0 " Theorem 12. For each representation W ∈ rep P(a,b) there exists a representation W ∈ rep P, such that (W ")0 ' W ⊕ T m(a), for some m ≥ 0.
Theorem 13. In the case of the differentiation VIII, the operations # and " induce mu- tually inverse bijections 0 0 Ind P\[T (a),G1(a, c),G2(a, c)] Ind P = Ind P \[T (a)].
The following lemma characterizes the ideal I = hT (a),G1(a, c),G2(a, c)i ⊂ rep P, where P is an equipped poset with a pair of points (a, b), VIII-suitable.
Lemma 4. If U = (U0; Ux | x ∈ P) and V = (V0; Vx | x ∈ P) are representations of an equipped poset P with a pair of points (a, b), VIII-suitable, then the following equivalences hold for a linear map ϕ : U0 ! V0:
− + 1. ϕ ∈ hT (a)i if and only if ϕ ∈ [(Ub + Uc) ,Va ], ϕe(Ub) ⊂ Va. − + 2. ϕ ∈ hG1(a, c)i if and only if ϕ ∈ [Ub ,Va ], ϕe(Ub) ⊂ Va, ϕe(Uc) ⊂ Vc.
− − − − − 3. ϕ ∈ hG2(a, c)i if and only if ϕ ∈ [Ub ,Va ∩ Vc ], Imϕe ⊂ Vga ∩ Vgc .
+ + Proof. It is enough to assume Ub = U0 6= 0. We also assume Va 6= 0 throughout the proof. Furthermore, we adopt the following partitions of spaces Ub and Va:
− − − Ub = Ufb ⊕ Nb; Va = Vga ⊕ Ma ⊕ Na, where Ma = {v = eα + ξeβ ∈ Va | v ∈ Vgb }.
− + If ϕ ∈ [(Ub + Uc) ,Va ] with ϕe(Ub) ⊆ Va, then:
ϕe(Ux) ⊆ F (Va) ⊆ Vx, if x ∈ aO.
ϕe(Ux) ⊆ ϕe(Ub) ⊆ Va ⊆ Vx, for any point x ∈ ag.
Since ϕe(Uc) = 0, the arguments described above allow us to conclude that ϕ ∈ rep P.
− This part of the proof can be finished by considering the cases in which Ub = 0 or Nb = 0.
− + If Ufb = 0 and Nb 6= 0, then U0 = Nb and dimG Nb = m. Therefore, it is possible to + define a representation W ∈ rep P such that W0 = Nb . F (N ) if x ∈ aO, b Wx = Nb if x ∈ ag, 0 otherwise.
+ We also define linear maps f0 : Nb ! W0, f1 : W0 ! V0 such that: CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 40
+ m f0(v) = v for all v ∈ Nb and f1 = ϕ. Since W ' Ta then −1 f0 g0 m m g0 f1 ϕ1 = U −! W −! T (a) ∈ rep P, ϕ2 = T (a) −−! W −! V ∈ rep P and ϕ2ϕ1 = ϕ, m where g0 : W ! T (a) is an isomorphism.
− + In the case Nb = 0, we observe that ϕ = 0. Thus ϕ ∈ [(Ub + Uc) ,Va ] and ϕe(Ub) ⊆ Va imply ϕ ∈ hT (a)i.
m On the other hand, if ϕ ∈ hT (a)i then there exist morphisms ϕ1 : U ! T (a) ∈ rep P m m and ϕ2 : T (a) ! V ∈ rep P, such that ϕ = ϕ2ϕ1. Since, ϕf1(Ub) ⊆ Ta (a) then ϕ1((Ub + − m − + − − Uc) ) ⊆ (Ta (a)) , in particular, ϕ1(Uc ) = ϕ1(Ub ) = 0. Therefore, ϕ((Ub + Uc) ) = 0 − m m m + 2m thus (Ub + Uc) ⊆ Ker ϕ. Furthermore, since Ta (a) = Tb (a) with (Ta (a)) = F it m m + follows ϕf2(Tb ) ⊆ Va. Therefore ϕe(Ub) = ϕf2(ϕf1(Ub)) ⊆ ϕf2(Tb ) ⊆ Va, thus, Im ϕ ⊆ Va . − + With this argument, we conclude ϕ ∈ [(Ub + Uc) ,Va ] and ϕe(Ub) ⊆ Va if and only if ϕ ∈ hT (a)i.
Arguments used above with the additional condition ϕe(Uc) ⊆ Vc allow us to conclude the − + second item, i.e., ϕ ∈ [Ub ,Va ] and ϕe(Ub) ⊆ Va if and only if ϕ ∈ hG1(a, c)i.
The following arguments prove the third item.
m If ϕ ∈ hG2(a, c)i then there exist morphisms ϕ1 : U ! G2 (a, c) ∈ rep P and ϕ2 : m − − G2 (a, c) ! V ∈ rep P, such that ϕ = ϕ2ϕ1. Therefore, ϕ2ϕ1(Ub ) = ϕ2(ϕ1(Ub )) = 0, due − m − − − − that ϕ1(Ub ) ⊆ ((G2 (a, c))b) = 0. Furthermore, ϕe(Nb) = ϕf2ϕf1(Nb) ⊆ Va∩Vgc = Vga ∩Vgc − − thus Imϕ ⊆ Va ∩ Vc .
− − − − − − − On the other hand, if ϕ ∈ [Ub ,Va ∩ Vc ] and Imϕe ⊆ Vga ∩ Vgc , then ϕe(Ux) ⊆ Vga ∩ Vgc ⊆ F (Va) ⊆ Vx if x ∈ aO.
− − ϕe(Ux) ⊆ Vga ∩ Vgc ⊆ Vx if x ∈ ag.
− − Finally, ϕe(Uc) ⊆ Vga ∩ Vgc ⊆ Vc, therefore, ϕ ∈ rep P.
m Now we can use arguments as above to find out morphisms ϕ1 : U ! G2 (a, c), ϕ2 : m m G2 (a, c) ! V ∈ rep P, such that ϕ = ϕ2ϕ1. Note that, the representation W ' G2 (a, c) − + defined for the case Ub = 0, Nb 6= 0 has the form (W0; Wx | x ∈ P), where W0 = Nb and F (N ) if x ∈ aO + cO, b Wx = Nb if x ∈ ag, 0 otherwise.
Therefore,
− − − − − ϕ ∈ [Ub ,Va ∩ Vc ] and Imϕe ⊆ Vga ∩ Vgc if and only if ϕ ∈ hG2(a, c)i.
The following lemma can be proved by using arguments described in the proof of Lemma 4. CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 41
0 0 0 Lemma 5. If U and V are representations of a poset P(a,b) and ϕ : U0 ! V0 is a linear − + morphism, then ϕ ∈ [Ub ,Va ] and ϕe(Ub) ⊆ Va if and only if ϕ ∈ hT (a)iR0 . Remark 2.3. Denote by R = rep P and R0 = rep P0 the categories of representations 0 0 associated with the equipped posets P and P(a,b), respectively. Due to the fact that ϕ = ϕ, we obtain the natural inclusions R(U, V ) ⊂ R0(U 0,V 0) for all objects U, V ∈ R. I denotes the ideal in the category R consisting of morphisms which pass through the objects T (a), 0 0 G1(a, c) and G2(a, c). I denotes the ideal in the category R consisting of morphisms which 0 0 0 pass through the object T (a). Taking into account that T (a) = G1(a, c) = G2(a, c) = T (a), we get also inclusions I(U, V ) ⊂ I0(U 0,V 0) for all objects U, V ∈ R. Thus, for each pair of representations U, V ∈ R, we obtain the following diagram of inclusions.
R0(U 0,V 0) @ @ R(U, V ) I0(U 0,V 0) @ @ I(U, V )
Figure 2.9. The lattice associated to the ideals I, I0 and the subspaces R and R0 according to VIII.
Proposition 1. Let U, V be an arbitrary pair of representations in R. Then, the following identity holds. R(U, V ) ∩ I0(U 0,V 0) = I(U, V ).
Proof. The Remark 2.3 allows us to conclude I(U, V ) ⊆ R(U, V ) ∩ I0(U 0,V 0). So, it is enough to prove R(U, V ) ∩ I0(U 0,V 0) ⊆ I(U, V ) in order to obtain the first identity. To 0 0 0 do that, we suppose that a morphism ψ : U0 ! V0 ∈ R(U, V ) ∩ I (U ,V ) and define the following partition for the space U0:
+ − − + + + + U0 = (Uc ∩ Ub ) ⊕ Tb ⊕ (Uc ∩ Nb ) ⊕ Tb ⊕ Tc ⊕ W0,
− − − + + + + + + + where Tb ⊆ Ub , Tb ∩ Uc = 0, Tb ⊆ Nb , Tb ∩ Uc = 0, Tc ∩ Ub = 0 and W0 is a complementary subspace in U0.
+ + Since by Lemma 4, Imψ ⊆ Va , we can assume Va = V0 and define a partition of the form: + − − − + + + Va = (Va ∩ Vc ) ⊕ Ta ⊕ (Vc ∩ Na ) ⊕ Ta , − − − + + + + + where Ta ⊆ Va , Ta ∩ Vc = 0, Ta ⊆ Na and Ta ∩ Vc = 0.
Lemma 5 allows to build the following linear maps induced by ψ, and by the partition of + the spaces U0 and Va :
ψ1 = e(V −∩V −)ψe(U +∩N +), a c c b (2.7) ψ2 = e − + ψe + + , (Vc ∩Na ) (Uc ∩Nb ) CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 42
ψ3 = e − − ψe + , (Vc ∩Va ) (Tb )
ψ4 = e(T −)ψe(T +), a b (2.8) ψ5 = e − + ψe + , (Vc ∩Na ) (Tb ) ψ6 = e + ψe + , (Ta ) (Tb )
ψ = e − − ψe + , 7 (Vc ∩Va ) (Tc )
ψ8 = e(T −)ψe(T +), a c (2.9) ψ = e − + ψe + , 9 (Vc ∩Na ) (Tc )
ψ = e + ψe + , 10 (Ta ) (Tc )
ψ = e − − ψe , 11 (Vc ∩Va ) (W0)
ψ12 = e(T −)ψe(T +), a c (2.10) ψ = e − + ψe , 13 (Vc ∩Na ) (W0)
ψ = e + ψe , 14 (Ta ) (W0)
Then Lemma 4 allows to conclude that ψ1, ψ3, ψ7, ψ8, ψ11 ∈ hG2(a, c)i, ψ2, ψ5, ψ9, ψ10 ∈ hG1(a, c)i, and ψ4, ψ6, ψ12, ψ13, ψ14 ∈ hT (a)i. As I = hT (a),G1(a, c),G2(a, c)iR, thus 14 P 0 0 0 ψ = ψi ∈ I(U, V ). Therefore, R(U, V ) ∩ I (U ,V ) = I(U, V ). i=1
Proposition 2. Let U, V be an arbitrary pair of representations in R. Then, the following identity holds. R(U, V ) + I0(U 0,V 0) = R0(U 0,V 0).
Proof. The Remark 2.3 allows us to conclude that R(U, V ) + I0(U 0,V 0) ⊆ R0(U 0,V 0). In order to prove the equality, we proceed as follows:
VIII 0 0 0 0 From definition of the functor D(a,b), we can note that for ϕ ∈ R (U ,V ), and for x ∈ 0 {A ∪ B ∪ Σ ∪ {a, b}} ⊂ P, ϕe (Ux) ⊂ Vx, then ϕe(Ux) ⊂ Vx. Therefore, for x ∈ P\{c} and ϕ ∈ R, ϕe(Ux) ⊂ Vx, and ϕe(Uc) ⊂ Vc + F (Va) * Vc, then in general ϕ∈ / R and 0 0 0 R (U ,V ) * R(U, V ).
The following procedure allows to obtain a morphism ϕ ∈ R(U, V ) from a morphism 0 0 0 ψ ∈ R (U ,V ). To get this morphism, we need to do a partition of the vector space U0, as follows. − + + + + − + U0 = Ub ∩ Uc ⊕ Nb ∩ Uc ⊕ Tc ⊕ Tb ⊕ Tb ⊕ W0, − − − + + + + + + + + + where Tb ⊆ Ub , Tb ∩ Uc = 0, Tb ⊆ Nb , Tb ∩ Uc = 0, Tc ⊆ Uc , Tc ∩ Ub = 0, + W0 is a complementary subspace of Tc in U0. Actually, this partition is induced by the (U +,U +)-cleaving pair (T +,W ). Furthermore, T + = T + ⊕ T +. c b c 0 b b1 b2 CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 43
We assume e ∈ T +, if there exists e ∈ N + ∩ U +, such that e + ξe ∈ N for some γ b1 δ b c γ δ x x ∈ U . In this case, T + is a complementary subspace. bf\{b} b2
+ + + The following partition of the space V0 is induced by the (Vc ,Va )-cleaving pair (Xc ,Y0).
− + − + + + + V0 = Va ∩ Vc ⊕ Xa ⊕ Xc ⊕ Nb ∩ Vc ⊕ Xa ⊕ Y0,
− − + + + + where Xa ⊆ Va , Xc ⊆ Vc , Xa ⊆ Na and Y0 is a complementary subspace.
+ + + Note that Xa = (Xa )1 ⊕ (Xa )2, where if Na = G{v = eγj + ξeδj }1≤j≤k, for some positive + + integer k, then (Na )1 = F {eγj },(Na )2 = F {eδj }.
⊗ We use the same notation for any subspace Nx associated to a point x ∈ P . Furthermore, if X is a subspace of a F -vector space with a fixed basis {e1, e2, . . . , et}, then a vector of the form γ1e1+γ2e2+···+γtet will be denoted {γr}X , 1 ≤ r ≤ t. Therefore, if v = eγ +ξeδ ∈ Uf0 0 0 0 and ψ : U0 ! V0 ∈ R (U ,V ), then ψ(eγ) and ψ(eδ) can be written in the following form for suitable sets of indexes:
1 2 ψ(e ) = {γ } − − + {δ } − + {γ } + + {δ } + + + {ε } + + {ε } + + {η } , γ i Va ∩Vc j Xa k Xc l Na ∩Vc m (Xa )1 m (Xa )2 n Y0
0 0 0 0 01 02 0 ψ(e ) = {γ } − − + {δ } − + {γ } + + {δ } + + + {ε } + + {ε } + + {η } , δ i Va ∩Vc j Xa k Xc l Na ∩Vc m (Xa )1 m (Xa )2 n Y0 1 01 2 02 with εm, εm, εm, εm ∈ F .
Let w1, w2 : U0 ! V0 be linear maps induced by ψ, defined in such a way that:
+ + If eγ is a vector of a fixed basis of Nb ∩ Uc , then:
1 2 w (e ) = {δ } − + {ε } + + {ε } + . 1 γ j Xa m (Xa )1 m (Xa )2
If v = e + ξe belongs to a fixed basis of N , with e ∈ T +, then: γ δ b δ b1
0 01 02 w (e ) = {δ } − + {ε } + + {ε } + , 1 δ j Xa m (Xa )1 m (Xa )2 where if ξ2 + αξ + β = 0, then 1 ε01 = − ε2 , for each m, m β m α (2.11) ε02 = ε1 + ε2 , for each m, m m β m w1(t) = 0 for the other basic vectors t ∈ U0.
1 2 + w (e ) = {δ } − + {ε } + + {ε } + if e is a vector of a fixed basis of T , 2 γ j Xa m (Xa )1 m (Xa )2 γ c w2(t) = 0 for the other basic vectors t ∈ U0.
− + Note that, w1, w2 : U0 ! V0 ∈ [Ub ,Va ], with wf1(Ub) ⊆ Va and wf2(Ub) ⊆ Va. Thus, 0 0 0 w = w1 + w2 ∈ hT (a)iR0 , therefore, by Lemma 5, w ∈ I (U ,V ). CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 44
If U = (U0; Ux | x ∈ P) is a representation of an equipped poset P, then: if x ∈ aO then (ψe − we)(Ux) = ψe(Ux) − we(Ux) ⊆ Vx + F (Va) = Vx; if x ∈ ag then (ψe − we)(Ux) = ψe(Ux) − we(Ux) ⊆ Ux + Va = Vx; if x ∈ bM then we(Ux) = 0 and (ψe − we)(Ux) = ψe(Ux) ⊂ Vx.
+ + + + (ψe− we)(Uc) = (ψe− we)(Uc ∩ Ub ⊕ Nb ∩ Uc ⊕ Tc ) ⊆ Vc. Therefore, ϕ = ψ − w ∈ R(U, V ), 0 0 0 0 0 0 0 0 0 and ψ = ϕ + w ∈ R(U, V ) + I (U ,V ), hence R (U ,V ) = R(U, V ) + I (U ,V ). Theorem 14. Let P be an equipped poset with a pair of points (a, b), VIII-suitable. Then, VIII 0 the functor D(a,b) : rep P ! rep P(a,b), defined by formulas (2.6), induces an equivalence between quotient categories: ∼ R/I ! R0/I0, 0 0 0 where R = rep P, R = rep P(a,b), I = hT (a),G1(a, c),G2(a, c)iR and I = hT (a)iR0 .
Proof. Let me make a brief description of the proof as follows:
VIII 1. the density of the functor D(a,b) is guaranteed by Theorems 12 and 13;
VIII 2. the Proposition 2 allows us to conclude that the functor D(a,b) is full;
VIII 3. from Proposition 1, we can infer that the functor D(a,b) is faithful.
Thus, 1.1 allows to conclude the equivalence.
As a consequence of Theorem 14, we get the following corollary regarding Gabriel's quivers of the quotient categories.
Corollary 3. If Γ(R) and Γ(R0) are the Gabriel's quivers of the categories R and R0, then
0 Γ(R) \ [T (a),G1(a, c),G2(a, c)] ' Γ(R ) \ [T (a)].
2.5 Categorical properties of the algorithm of differentia- tion IX for equipped posets
In the present section, we present the definition of the algorithm of differentiation IX giving a proof of its categorical properties [67].
A pair of comparable weak points a ≺ b, of an equipped poset P, is called IX-suitable if P can be written in the form:
O P = a + bM + Σ + {p, a, b}, CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 45 where Σ is the interior of the completely weak interval [a, b] and p is a weak point incom- parable with a, and p ≺ b [67].
The derived poset of the set P, with respect to the pair (a, b), is the equipped poset 0 0 − + P = P(a,b), obtained from P by replacing the point p by a weak two-point chain p ≺ p with the additional relations a ≺ p+ ≺ b and p− ¡ b (plus all the induced relations). The points p−, p+ inherits all the previous order relations of the point p with the points in P\{p}.
The following diagram shows an equipped poset with a pair of points (a, b), IX-suitable, and its corresponding derived poset:
b b A ⊗ A ⊗ @@ J@@ J @ @ + Σ @ IX Σ J @p ⊗ −−−! J⊗ (a,b) p J J ⊗ B ⊗ B ⊗ a a p− 0 Figure 2.10. Diagrams of an equipped poset P and its corresponding derived poset P (a,b).
Let P be an equipped poset with a pair of points (a, b), IX-suitable. The following formulas IX 0 define the differentiation functor D(a,b) : rep P −! rep P(a,b), induced by the algorithm of differentiation IX. Thus for a given representation U = (U0; Ux | x ∈ P) ∈ rep P, we get 0 0 0 0 the derived representation U = (U0; Ux | x ∈ P(a,b)):
0 U0 = U0, 0 − Up− = Up ∩ Ufb , 0 (2.12) Up+ = Up + Ua, 0 Ux = Ux, for the remaining points. 0 ϕ = ϕ, for all F − linear morphism ϕ : U0 ! V0.
IX Note that, for the functor D(a,b) and for indecomposable representations T (a) and T (a, p), we have T 0(a) = T 0(a, p) = T (a). The following arguments were used by Zavadskij in order to describe the integration procedure for the algorithm IX [67].
Representations U in rep P without direct summands T (a) and T (a, p) will be called reduced. A reduced representation U #, for which U 0 = U # ⊕ T m(a), is defined evidently, analogously to the previous cases. Take some complementing pair of subspaces (E0,W0) + − # 0 in U0, with respect to the pair (Ua ,Ub ), and set U = W , where Wx = Ux ∩ W0 (Wx = 0 0 # # Ux ∩ Wf0) for a strong (weak) point x ∈ P . Obviously, T (a) = T (a, p) = 0.
# The representation U does not depend, up to isomorphism, on the choice of E0 and W0 + − 0 and, due to the inclusion Wa ⊂ Wb , is a representation of the set P(a,b) completed by the relation a C b. CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 46
0 " Theorem 15. For each representation W ∈ rep P(a,b), there exists a representation W ∈ rep P such that (W ")0 ' W ⊕ T m(a), for some m ≥ 0.
Theorem 16. In the case of the differentiation IX, the operations # and " induce mutually inverse bijections
0 0 Ind P\[T (a),T (a, p)] Ind P = Ind P \[T (a)].
The following lemma characterizes morphisms which pass through the objects from the ideal I = hT (a),T (a, p)i ⊂ rep P, where P is an equipped poset with a pair of points (a, b), IX-suitable. In Lemmas 6, 7, and Propositions 3, 4, we assume the following partitions for the subspaces Ux, x ∈ ag:
− g − − g Ux = Ufx ⊕ Mx ⊕ Nx, for all x ∈ a \{b}, Mx ⊂ Ufb , Mx ∩ Ux = 0, for all x ∈ a \{b}, P − − g Mb = Mx, Ufb = Hfb ⊕ Mb, Nx ∩ Ub = 0, for all x ∈ a . x∈ag\{b}
Lemma 6. If U = (U0; Ux | x ∈ P) and V = (V0; Vx | x ∈ P) are representations of an equipped poset P with a pair of points (a, b), IX-suitable, then the following equivalences hold for a linear map ϕ : U0 ! V0:
+ 1. ϕ ∈ hT (a)i if and only if ϕ ∈ [Hb,Va ], ϕe(Ub) ⊂ Va; + + 2. ϕ ∈ hT (a, p)i if and only if ϕ ∈ [Hb,Va ∩ Vp ], ϕe(Ub) ⊂ Va ∩ Vp.
Proof. In order to prove the first item, it is enough to adapt arguments used to prove the first item of Lemma 4. In fact, the same arguments can be used if Mb = 0.
+ For the second item, we assume Ub = U0 6= 0.
+ + If ϕ ∈ [Hb,Va ∩ Vp ], with ϕe(Ub) ⊆ Va ∩ Vp then:
ϕe(Ux) ⊆ F (Va) ∩ F (Vp) ⊆ F (Va) ⊆ Vx, if x ∈ aO;
ϕe(Ux) ⊆ ϕe(Ub) ⊆ Va ∩ Vp ⊆ Va ⊆ Vx, for any point x ∈ ag.
Since ϕe(Up) ⊆ ϕe(Ub) ⊆ Va ∩ Vp ⊆ Vp, the arguments described above allow us to conclude that ϕ ∈ rep P.
This part of the proof can be finished by considering the cases in which Nb = 0 or Nb 6= 0 in U0.
− + If Ufb = 0 and Nb 6= 0, then U0 = Nb and dimG Nb = m, for some m > 0. Therefore, it + is possible to define a representation W ∈ rep P such that W0 = Nb and F (N ) if x ∈ aO, b Wx = Nb if x ∈ bf, 0 otherwise. CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 47
+ We also define the linear maps f0 : Nb ! W0 and f1 : W0 ! V0 such that:
+ m f0(v) = v for all v ∈ Nb and f1 = ϕ. Since W ' T (a, p), then −1 f0 g0 m m g0 f1 ϕ1 = U −! W −! T (a, p) ∈ rep P, ϕ2 = T (a, p) −−! W −! V ∈ rep P and m ϕ2ϕ1 = ϕ, where g0 : W ! T (a, p) is an isomorphism.
+ + If Nb = Mb = 0 in U0 or Na ∩ Np = 0 in V0, we note that ϕ = 0.
If Hb = Nb = 0 in U0, we define a representation W = (W0; Wx | x ∈ P) such that P + W0 = Mx and: x∈bf ! P F Mx if x ∈ aO, x∈bf Wx = P Mx if x ∈ bf, x∈b f 0 otherwise.
m If dim Wf0 = m, then W ' T (a, p). Therefore, we can apply the arguments used above to find morphisms ϕ1, ϕ2 ∈ rep P such that ϕ = ϕ2ϕ1.
+ + Thus, ϕ ∈ [Hb,Va ∩ Vp ] and ϕe(Ub) ⊆ Va ∩ Vp implies ϕ ∈ hT (a, p)i.
m On the other hand, if ϕ ∈ hT (a, p)i, there exist morphisms ϕ1 : U ! T (a, p) ∈ rep P and m m ϕ2 : T (a, p) ! V ∈ rep P such that ϕ = ϕ2ϕ1, for some m > 0. Since, ϕf1(Ub) ⊆ Ta (a, p), m − then ϕ1(Hb) ⊆ (Ta (a, p)) , in fact, ϕ1(Hb) = 0. Therefore, ϕ(Hb) = 0, thus Hb ⊆ Ker ϕ. m m m m + 2m Furthermore, since Ta (a, p) = Tb (a, p) = Tp (a, p) with (Ta (a, p)) = F it follows m m + + ϕf2(Tb ) ⊆ Va∩Vp, therefore ϕe(Ub) = ϕf2(ϕf1(Ub)) ⊆ ϕf2(Tb ) ⊆ Va∩Vp and Im ϕ ⊆ Va ∩Vp . + + With this argument, we conclude ϕ ∈ [Hb,Va ∩ Vp ] and ϕe(Ub) ⊆ Va ∩ Vp if and only if ϕ ∈ hT (a, p)i. We are done.
The following lemma can be proved by using arguments described in the proof of Lemma 6.
0 0 0 Lemma 7. If U and V are representations of a poset P(a,b) and ϕ : U0 ! V0 is a linear + morphism, then ϕ ∈ [Hb,Va ] and ϕe(Ub) ⊆ Va if and only if ϕ ∈ hT (a)iR0 . Remark 2.4. Denote by R = rep P and R0 = rep P0, the categories of representations 0 0 associated to the equipped posets P and P(a,b), respectively. Due to the fact that ϕ = ϕ, we obtain the natural inclusions R(U, V ) ⊂ R0(U 0,V 0) for all objects U, V ∈ R. I = 0 0 hT (a),T (a, p)iR and I = hT (a)iR0 denote ideals in the category R and R , respectively. We get also inclusions I(U, V ) ⊂ I0(U 0,V 0) for all objects U, V ∈ R, taking into consideration that T 0(a) = T 0(a, p) = T (a). Thus, for each pair of representations U, V ∈ R, we obtain the following diagram of inclusions CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 48 R0(U 0,V 0) @ @ R(U, V ) I0(U 0,V 0) @ @ I(U, V )
Figure 2.11. The lattice associated to the ideals I, I0 and the subspaces R and R0 according to D-IX.
Proposition 3. Let U, V be an arbitrary pair of representations in R. Then the following identity holds R(U, V ) ∩ I0(U 0,V 0) = I(U, V ).
Proof. Let U, V be arbitrary representations in the category R, and let ϕ be a morphism 0 0 0 − + in R(U, V ) ∩ I (U ,V ). Then ϕ ∈ [Ub ,Va ] with ϕe(Ub) ⊆ Va. Now we define the following + + partitions of the spaces U0 and Va (we assume V0 = Va ):
− − − + − − + + + − + U0 = Ua ∩ Up ⊕ Ua ∩ Np ⊕ Ta ⊕ Tp ⊕ (Na ∩ Np) ⊕ Ta ⊕ Tp ⊕ Up ∩ Na ⊕ X0,
− − − + − − + − where Ta ⊆ Ua , Ta ∩ Up = 0, Tp ⊆ Up , Ua ∩ Tp = 0.
− − + + + + + + Ufa ∩ Ta = 0, Ufp ∩ Tp = 0, Ta ⊆ Na ⊕ Ma , Tp ⊆ Np ⊕ Mp and X0 is a complementary + + + + subspace in U0. Furthermore, Ta ∩ Up = 0 and Tp ∩ Ua = 0.
Now, we consider the next suitable partition to the space V0
− − − + − + + V0 = Va ∩ Vp ⊕ Va ∩ Np ⊕ Xa ⊕ (Na ∩ Np) ⊕ Ta ⊕ Y0.
± The spaces Tx are defined as for the space U0, and Y0 is a complementary subspace in V0.
+ + + We assume the notations X1 = Ta , X2 = Tp , X3 = (Na ∩ Np) , X4 = X0. In Va, − − − + − + + Y1 = Va ∩ Vp , Y2 = Va ∩ Np , Y3 = Xa , Y4 = (Na ∩ Np ), Y5 = Ta Y6 = Y0, and
ϕij = eYj ϕeXi . Then 6 4 6 4 XX XX ϕ = ϕij = eYj ϕeXi . j=1 i=1 j=1 i=1
By Lemma 6, ϕij ∈ hT (a)iR if j = 1, ϕij ∈ hT (a, p)iR otherwise. Therefore ϕ ∈ I(U, V ), thus R(U, V ) ∩ I0(U 0,V 0) ⊆ I(U, V ).
The Remark 2.4 allows us to conclude I(U, V ) ⊆ R(U, V ) ∩ I0(U 0,V 0). This result proves the desired identity. Proposition 4. Let U, V be an arbitrary pair of representations in R. Then, the following identity holds R(U, V ) + I0(U 0,V 0) = R0(U 0,V 0).
IX 0 0 0 Proof. From definition of the functor D(a,b), we can note that for ψ in R (U ,V ), and for x ∈ {A ∪ B ∪ Σ ∪ {a, b}} ⊂ P, ψe(Ux) ⊂ Vx, then ψe(Ux) ⊂ Vx. Therefore, for CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 49 x ∈ P\{p} and ψ ∈ R, ψe(Ux) ⊂ Vx, since ψe(Up) ⊂ Vp + Va * Vp, then in general 0 0 0 ψ∈ / R and R (U ,V ) * R(U, V ). The following procedure allows us to obtain a morphism ϕ ∈ R(U, V ) from a morphism ψ ∈ R0(U 0,V 0). To get this morphism, we assume the same partition, as above for the space U0, and define the following partition for the space V0:
− − − + − + + + − − + V0 = Va ∩ Vp ⊕ Va ∩ Np ⊕ Xa ⊕ (Na ∩ Np) ⊕ Xa ⊕ Xp ⊕ Xp ⊕ Vp ∩ Na ⊕ Y0, where Y0 is a complementary subspace in V0. The spaces Xx are defined as the spaces Tx + in U0, whereas Na,Np ⊆ Vf0 are defined as for space U0. Furthermore, Xp = Xp1 ⊕ Xp2 + + − (Xa = Xa1 ⊕ Xa2 ), where eλ ∈ Xp1 (eλ ∈ Xa1 ) if and only if there exists eζ ∈ Mp ∩ Va + − (eζ ∈ Ma ∩ Vp ) such that v = eζ + ξeλ ∈ Mp (v = eζ + ξeλ ∈ Ma).
+ If Na = G{v = eζj + ξeλj }1≤j≤k, for some positive integer k, then (Na )1 = F {eζj }, + (Na )2 = F {eλj }. We use the same notation for any subspace Nx associated to a point x ∈ ⊗ P . Furthermore, if X is a subspace of a k-vector space with a fixed basis {e1, e2, . . . , et}, then a vector of the form ζ1e1 + ζ2e2 + ··· + ζtet will be denoted {ζr}X , 1 ≤ r ≤ t. 0 0 0 Therefore, if ψ : U0 ! V0 ∈ R (U ,V ), then ψe(Ua + Up) ⊂ Va + Vp; and for any vector − eζ ∈ Up , we have:
ψe(eζ ) = {ζi} − − + {λj} − + {γk} − + {δl} − + {µm} − , Vga ∩Vgp Vga ∩F (Np) Vgp ∩F (Na) Xga Xgp
− − − for suitable index sets. In fact, ψe(Ufp ⊕ Mp) = ψe(Ufb ∩ Up) ⊆ Vp ∩ Vgb ⊆ Vp.
If eζ + ξeλ ∈ Na ∩ Np then:
1 1 2 1 ψ(e ) = {ζ } − − + {λ } − + + {δ } − + {γ } − + + {γ } + {ε } + + ζ i Va ∩Vp j Va ∩Np l Xa k Vp ∩Na k Xa1 n (Na∩Np) 2 1 2 {ε } + + {$ } + {$ } . n (Na∩Np) t Xa2 t Xa2
0 01 0 01 02 01 ψ(e ) = {ζ } − − + {λ } − + + {δ } − + {γ } − + + {γ } + {ε } + + λ i Va ∩Vp j Va ∩Np l Xa k Vp ∩Na k Xa1 n (Na∩Np) 02 01 02 {ε } + + {$ } + {$ } . n (Na∩Np) t Xa2 t Xa2
+ If eζ ∈ Tp then:
1 2 1 2 ψ(e ) = {ζ } − − +{λ } − + +{λ } +{δ } − +{µ } − +{γ } − + +{γ } + ζ i Va ∩Vp j Va ∩Np j Xp1 l Xa m Xp k Vp ∩Na k Xa1 1 2 1 2 1 2 {ε } + + {ε } + + {$ } + {$ } + {ν } + {ν } . n (Na∩Np) n (Na∩Np) t Xa2 t Xa2 s Xp2 s Xp2
We define the F -linear morphisms w1 and w2, as follows:
w1 : U0 ! V0,
+ 1 such that, for all basic vector e ∈ (N ∩ N ) , w (e ) = {λ } − + + {δ } − + θ a p 1 θ j Va ∩Np l Xa 1 2 1 2 {γ } − + + {γ } + {$ } + {$ } , w (t) = 0, for the other basic vectors t k Vp ∩Na k Xa1 t Xa2 t Xa2 1 in U0. w2 : U0 ! V0, + 2 1 2 such that, for all basic vector e ∈ T , w (e ) = {δ } − + {γ } + {$ } + {$ } , θ p 2 θ l Xa k Xa1 t Xa2 t Xa2 + w2(t) = 0, for the other basic vectors t ∈ U0. Thus, if w = w1 + w2 then w ∈ [Hb,Va ] and 0 0 0 we(Ub) ⊂ Va. Therefore, w ∈ I (U ,V ) by Lemma 7. CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 50
Note that, (ψe − we)(Ux) ⊆ F (Va) ⊕ Vx = Vx if x ∈ aO;
(ψe − we)(Ux) ⊆ Vx + Va = Vx, if x ∈ ag;
− − (ψe − we)(Ufp ⊕ Mp) ⊆ Vgb ∩ Vp ⊆ Vp.
If a basic vector v = eζ + ξeλ ∈ Na ∩ Np ⊆ Ub, then:
1 2 0 01 (ψ − w)(v) = {ζi} − − + {ε } + + {ε } + + ξ({ζ } − − + ε } + + e e Va ∩Vp n (Na∩Np) n (Na∩Np) i Va ∩Vp n (Na∩Np) 02 + {εn }(Na∩Np) ) ∈ Vp.
For a basic vector v = eζ + ξeλ ∈ Tp, we have:
1 2 1 1 (ψ−w)(v) = {ζi} − − +{λ } − + +{λ }X +{µm} − +{γ } − + +{ε } + + e e Va ∩Vp j Va ∩Np j p1 Xp k Vp ∩Na n (Na∩Np) 2 1 2 0 01 02 0 {ε } + + {ν } + {ν } + ξ({ζ } − − + {λ } − + + {λ } + {µ } − + n (Na∩Np) s Xp2 s Xp2 i Va ∩Vp j Va ∩Np j Xp1 m Xp 01 01 02 01 02 {γ } − + + {ε } + + {ε } + + {ν } + {ν } ) ∈ V . Therefore, k Vp ∩Na n (Na∩Np) n (Na∩Np) s Xp2 s Xp2 p 0 0 0 (ψe − we)(Up) ⊆ Vp and ϕ = ψ − w ∈ R(U, V ). Thus, ψ = ϕ + w ∈ R(U, V ) + I (U ,V ), hence R0(U 0,V 0) ⊆ R(U, V ) + I0(U 0,V 0).
The Remark 2.4 allows us to conclude that R(U, V ) + I0(U 0,V 0) ⊆ R0(U 0,V 0). With this inclusion, we are done.
0 0 Since Zavadskij proved in [67] that IndP\[T (a),T (a, p)] IndP = IndP \[T (a)]. Then we have automatically the following consequences of Propositions 3, 4, Theorems 15 and 16:
Theorem 17. Let P be an equipped poset with a pair of points (a, b), IX-suitable. Then, IX 0 the functor D(a,b) : rep P ! rep P(a,b), defined by formulas (2.12), induces an equivalence between quotient categories: ∼ R/I ! R0/I0, 0 0 0 where R = rep P, R = rep P(a,b), I = hT (a),T (a, p)iR and I = hT (a)iR0 .
As a consequence of the last theorem, we have the following corollary.
Corollary 4. If Γ(R) and Γ(R0), are the Gabriel's quivers of the categories R and R0, then
Γ(R) \ [T (a),T (a, p)] ' Γ(R0) \ [T (a)].
2.6 Categorical properties of the algorithm of differentia- tion X for equipped posets with involution
Let (P, Θ) = P be an equipped poset with involution ∗ and Θ be the set of all the equivalence classes of its points with respect to this involution. We denoted by rep (P, Θ) CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 51 the category of all the representations of (P, Θ), or simply by rep P if there is no doubt about of the involution and their classes [21, 66, 67].
∗ ∗ Let U = (U0; Uκ | κ ∈ Θ) be a representation in rep P. If x 6= x , then x ∼ x , and we denote (x, x∗) = κ ∈ Θ.
Let (F,G) be the pair of fields we are working on. Let U0 be some finite-dimensional F -vector space, Ue0 its complexification and κ ∈ Θ be some class. We assume the notation, κ κ U0 (Ue0 ) for direct sum of |κ|-copies of U0 (Ue0) numbered by the points x ∈ κ. In this κ κ x x case, the copy of U0 (Ue0) in U0 (Ue0 ) corresponding to a point x, is denoted by U0 (Ue0 ) κ x κ x and usually is identified with U0 (Ue0). So, U0 = U0 = U0 (Ue0 = Ue0 = Ue0) if x is small κ x x∗ 2 κ x x∗ 2 (weak), and U0 = U0 ⊕ U0 = U0 (Ue0 = Ue0 ⊕ Ue0 = Ue0 ) if x is big (biweak).
For each class κ ∈ Θ and each point x ∈ κ, we consider natural injections and projections:
x κ κ x ix :U0 = U0 −! U0 , πx : U0 −! U0 = U0 if x is a small or big point; (2.13) x κ κ x ix :Ue0 = Ue0 −! Ue0 , πx : Ue0 −! Ue0 = Ue0 if x is a weak or biweak point.
κ κ Choosing a subspace Uκ ⊂ U0 (Uκ ⊂ Ue0 ) if κ correspond to a small or big (weak or biweak) point, we attach to it two subspaces in U0 (Ue0) of the form:
− −1 Ux := ix (Uκ), + (2.14) Ux := πx(Uκ).
x x − x − x Identifying U0 (Ue0 ) with U0 (Ue0), we also can assume Ux = Uκ ∩ U0 (Ux = Uκ ∩ Ue0 ). − + Let x be a small or weak point, then κ = {x}. Therefore Ux = Ux , for which points x + we will omit the notations ± and write simply Ux, for a big point a set Ux = {s ∈ U0 | (s, t) ∈ U(x,x∗)}.
P + P Let U κ = iy(Ux ) = exy(U(x,x∗)), where x < y and y ∈ κ. The dimension of a representation U, is the vector dim U = (h0, hκ)κ∈Θ, where h0 = dim U0 over the field F and hκ = dim (Uκ/U κ) over the field G.
Zavadskij defined the algorithm of differentiation X in [67], afterwards, he presented in [70] the following modified version of this differentiation:
A pair of incomparable points (a, b) in P, where a is big (i.e. a 6= a∗) and b is weak, is O called X-suitable (i.e. suitable for differentiation X), if P = a + bM.
The derived equipped poset with involution (P0, Θ0) = P0, with respect to the pair (a, b), is obtained from (P, Θ) in the following way:
∗ ∗ ∗ (a) the point a is replaced by a three-point chain a < q < a0, where a , a0 are big points and q is weak;
(b) the point b is replaced by a two-point chain b0 < b, where b0 is big and b is weak;
(c) an order relation a < b0 is added; CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 52
0 (d)Θ is obtained from Θ by adding two new classes: a non-trivial one {a0, b0} and a trivial one {q}.
Naturally, all the order relations induced by those in P and by the mentioned above are added as well.
The following Figure 2.12 shows an equipped poset with involution (P, Θ) with a pair of points (a, b) X-suitable and its corresponding derived poset (P0, Θ0):
•a0 ⊗ b ∗ a ⊗ b q • ⊗ • b0
A A •a∗ X −−−−! (a,b)
• B • B a a
Figure 2.12. Diagrams of the equipped posets with involution (P, Θ) and (P0, Θ0).
H H 0 0 O 0 Set A = a , B = bN in P and ba = a , B = P \a in P . Let U = (U0; Uκ | κ ∈ Θ) be a representation of the set (P, Θ), where U0 is a finite-dimensional F -space. Consider 2 an ordered sum U0 = U0 ⊕ U0, we can define the coupling of a sequence of n subspaces 2 2 X1,...,Xn ⊂ U0 being a subspace in U0 of the form:
[X1 − X2 − · · · − Xn] = {(t0, tn) | (ti−1, ti) ∈ Xi for some ti}.
The categories R and R0 are described as follows:
− + + − + R ={rep (P, Θ) | UA = Ua ⊂ Ub ,Ub = UB }. 0 0 0 + + − + − + (2.15) R ={rep (P , Θ ) | U ⊂ U 0 ,U = U ,U = U }. a B a0 q b b0 Let P be an equipped poset with involution, and a pair of points (a, b), X-suitable. The X 0 following formulas define the differentiation functor D(a,b) : R −! R(a,b) induced by the algorithm of differentiation X. Thus, for a given representation U = (U0; Uκ | κ ∈ Θ) ∈ R, 0 0 0 0 we get the derived representation U = (U0; Uκ | κ ∈ Θ ):
0 U0 = U0; 0 + Ub = Ub + Ufa ; 0 + U = [U ∗ − U ] + (0,U ); (a0,b0) (a ,a) b a 0 Uq = [U(a∗,a) − Ub − U(a,a∗)]; (2.16) 0 + U(a,a∗) = U(a,a∗) ∩ (UB ,U0); 0 0 Uκ = Uκ for the remaining classes κ ∈ Θ ; 0 ϕ = ϕ for all linear morphism ϕ : U0 ! V0. CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 53
+ + Following [70], if (E0,W0) is a (Ua ,UB )-cleaving pair of U0, then the reduced derived representation U # is defined (uniquely up to isomorphism) by the equality U 0 = U # ⊕ m + + + # P (ba), where m = dimE0 = dim(Ua ,UB )/UB . Its evident form is U = W, with W0 0 κ taken from the cleaving pair and Wκ = Uκ ∩ W0 .
0 0 2 # Obviously, G1(b, a) = P (ba) ⊕ P (b0) and G2(b, a) = P (ba), hence G1(b, a) = P (b0) and # G2(b, a) = 0.
Let W be an object in R0. To construct the primitive object W " ∈ R0, we represent the spaces W(a0,b0),Wq and Wb, respectively, in the form
0 0 W(a0,b0) = W (a0,b0) ⊕ F1,F1 = {(f11, f11),..., (f1p1 , f1p1 )}; + 0 0 g∗ Wq = Wa ⊕ F2,F2 = {(f21, f21),..., (f2p2 , f2p2 )}; W = Wd+ ⊕ H; b b0 where Fi and H are some complements with the choosen bases for Fi. Consider a new F -space E with a base {e , . . . , e }∪{e , e0 , . . . , e , e0 } of dimension m = p +2p . 0 11 1p1 21 21 2p2 2p2 1 2 " Then, set W = (U0; Uκ | κ ∈ Θ) where
U0 = W0 ⊕ E0; ˙ κ∩A ∗ Uκ = Wκ ⊕ E0 for κ 6= {a, a }, {b}; ˙ 0 0 U(a,a∗) = W(a,a∗) + {(e11, f11),..., (e1p1 , f1p1 )} + {(e2j, f2j), (e2j, f2j): j = 1, . . . p2}; + 0 0 ˙ ^0 Ub = WB + {(e11, f11),..., (e1p1 , f1p1 )} + H. (2.17)
The desired isomorphisms (U #)" ' U, for a reduced object U ∈ R (without direct sum- " # 0 mands G2(b, a)) and (W ) ' W , for a reduced object W ∈ R (without direct summands P (ba)) hold. Theorem 18. In the case of the differentiation X, the operations # and " induce mutually inverse bijections 0 Ind R\[G2(b, a)] Ind R \[P (ba)]. Remark 2.5. Let R and R0 be the categories described in the equation (2.15), associated 0 with the equipped posets with involution P and P(a,b), respectively. Due to the fact that ϕ0 = ϕ, we obtain the natural inclusions R(U, V ) ⊂ R0(U 0,V 0) for all objects U, V ∈ R. 0 0 Let I = hG2(b, a)iR and I = hP (ba)iR0 be ideals in the category R and R , respectively. We get also inclusions I(U, V ) ⊂ I0(U 0,V 0) for all objects U, V ∈ R, taking into consideration 0 2 that G2(b, a) = P (ba). Thus, for each pair of representations U, V ∈ R, we obtain the following diagram of inclusions CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 54 R0(U 0,V 0) @ @ R(U, V ) I0(U 0,V 0) @ @ I(U, V )
Figure 2.13. The lattice associated to the ideals I, I0 and the subspaces R and R0 according to the differentiation X.
Proposition 5. Let U and V be arbitrary representations in R. Then, the following identities hold
R(U, V ) + I0(U 0,V 0) = R0(U 0,V 0) and R(U, V ) ∩ I0(U 0,V 0) = I(U, V ).
Proof. The inclusions R(U, V )+I0(U 0,V 0) ⊆ R0(U 0,V 0) and I(U, V ) ⊆ R(U, V )∩I0(U 0,V 0) follows from Remark 2.5. Thus, it suffices to prove R(U, V ) + I0(U 0,V 0) ⊆ R0(U 0,V 0) and R(U, V ) ∩ I0(U 0,V 0) ⊆ I(U, V ) in order to obtain the identities.
0 0 0 0 0 0 0 + 0 + 0 − Firstly, we prove that R(U, V )+I (U ,V ) ⊆ R (U ,V ), with I = [UB +(Ua) , (VA) ]. We note that in general, if (x, y) ∈ U(a,a∗) and (r, s) ∈ Ub, then not necessarily (ψ(x), ψ(y)) ∈ + V(a,a∗) and (ψ(r), ψ(s)) ∈ Vb. However, for any (x, y) ∈ U(a,a∗) ∩ (UB ,U0) it holds that + 0 0 0 (ψ(x), ψ(y)) ∈ V(a,a∗) ∩ (VB ,V0) ⊂ V(a,a∗), provided that ψ : U0 −! V0 ∈ R (U ,V ). Thus, for any pair of vectors of the form (x, y) ∈ U(a,a∗), it is necessary to define a linear map-morphism which can be used to adjust the corresponding images to subspaces V(a,a∗) 2 2 and Vb. To do that, we consider the following partitions of the vector spaces U0 and V0
2 − U0 = U(a,a∗) ∩ Ufb ⊕ U(a,a∗) ∩ Nb ⊕ T(a,a∗) ⊕ Tb ⊕ T0, where
− Ub = Ufb ⊕ Nb, t t t + Nb = h(1, ξ) iG,Nb1 = h(1, 0) iF ,Nb2 = h(0, 1) iF , then Nb = Nb1 + Nb2 , − 0 U(a,a∗) ∩ Ufb ⊆ U(a,a∗), + + + Ua = Ua ∩ UB ⊕ MB, − Tb = Tfb ⊕ Hb, − − Tfb ⊆ Ufb ,Hb ⊂ Nb, + Ua∗ = Ua ⊕ La∗ ,
2 − where T(a,a∗), Tb and T0 are complementary subspaces of U0 = U(a,a∗) ∩ Ufb ⊕ U(a,a∗) ∩ Nb 2 and U(a,a∗) + Ub in U(a,a∗), Ub and U0 , respectively. The same notation is keeping for 2 subspace V0 and the corresponding partition.
Now, we consider the following cases
+ 0 (i) Suppose that (x, y) ∈ U(a,a∗) ∩ (UB ,U0). Then (ψ(x), ψ(y)) ∈ V(a,a∗) = V(a,a∗) ∩ + (VB ,V0) ⊂ V(a,a∗). CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 55
+ (ii) If (x, y) ∈ T(a,a∗), then there exists z ∈ Ub such that (z, x) ∈ Ub. Thus, (y, z) ∈ U 0 , y∈ / U + and (ψ(y), ψ(z)) ∈ V 0 . Assume that vectors {(tj , tj ) : 1 ≤ j ≤ (a0,b0) b (a0,b0) i1 i2 L k} constitute a basis of [U(a,a∗) −Ub] and that {ta } : a ≤ L ≤ m is a basis of subspace + Pm h V . In this case, λ + denotes a linear combination of the form λ t , λ ∈ G. a Va h=1 h a h Therefore,
k X j j (ψ(y), ψ(z)) = λ (t , t ) + (0, λ + ), j i1 i2 Va j=1 k X ψ(y) = λ tj , j i1 j=1 k X j ψ(z) = λ t + λ + . j i2 Va j=1
Then, there exists a unique vector s such that (ψ(y), s) ∈ V ∗ and (s, ψ(z)−λ + ) ∈ (a ,a) Va + Vb, where y∈ / Vb . Thus, if the F -linear map-morphism w1 : U0 −! V0 is defined in such a way that ( ψ(x) − s, if x ∈ MB, w1(x) = 0, otherwise;
+ 0 + 0 − − − then w1 ∈ [UB + (Ua) , (VA) ]. Note that, ψ(UA0 ) ⊆ VA0 besides, if (x, y) ∈ T(a,a∗) then
((ψ − w1)(x), (ψ − w1)(y)) = (ψ(x) − ψ(x) + s, ψ(y)) = (s, ψ(y)) ∈ V(a,a∗).
(iii) If (x, y) ∈ Hb, it holds that
k k X j X j (ψ(x), ψ(y)) = δ t , δ t + λ + . j i1 j i2 Va j=1 j=1
If w2 : U0 −! V0 is a linear map-morphism such that
( + λV + , if y ∈ Hb , w2(y) = a 0, otherwise,
+ 0 + 0 − − − then w2 ∈ [UB + (Ua) , (VA) ]. Note that, ψe(Ufb ) ⊆ Vgb , and for (x, y) ∈ Hb, it holds that
k k k X j X j X j ((ψ−w )(x), (ψ−w )(y)) = ψ(x), δ t + λ + − λ + = δ t , δ t ∈ V . 2 2 j i2 Va Va j i1 j i2 b j=1 j=1 j=1 CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 56
(iv) Suppose now, that (x, y) ∈ U(a,a∗) ∩ Nb, with y ∈ La∗ . Then (y, x) ∈ U(a∗,a) and (x, y) ∈ U . Thus, (y, y) ∈ U 0 and (ψ(y), ψ(y)) ∈ V 0 ,(ψ(x), ψ(y)) ∈ V 0. b (a0,b0) (a0,b0) b
k k X j X j (ψ(x), ψ(y)) = γ t , γ t + λ + j i1 j i2 Va j=1 j=1
Pk j Pk j with γ t , γ t ∈ V and (ψ(y), ψ(y) − λ + ) ∈ [V ∗ − V ]. Hence, j=1 j i1 j=1 j i2 b Va (a ,a) b there exist t such that (t unique) (ψ(y), t ) ∈ V ∗ and (t , ψ(y) − λ + ) ∈ V , we 1 1 1 (a ,a) 1 Va b write (in V )
− V(a,a∗) ∩ Ufb = T1,
V(a,a∗) ∩ Nfb = T2,
T(a,a∗) = T3,
then (t1, ψ(y)) = λT1 + λT2 + λT3 , where
1 2 λT1 = (r1, r1), 1 2 λT2 = (r2, r2), 1 2 λT3 = (r3, r3),
1 2 ri ∈ Re Ti; ri ∈ Im Ti are linear combinations of all elements of the basis of the corresponding subspace (Re Ti= real part of Ti, Im Ti= imaginary part of Ti).
Define the linear map-morphism w3 : U0 −! V0 such that ψ(x) − r1 − r1 if x ∈ M , 1 2 b 2 2 + w3(x) = ψ(x) − r1 − r2 if x ∈ La∗ ∩ Nb , 0 otherwise.
Then
(ψκ − wκ)(x, y) = ((ψ − w)(x), (ψ − w)(y)) 1 1 2 2 = (ψ(x) − ψ(x) + r1 + r2, ψ(y) − ψ(y) + r1 + r2) − ∈ V(a,a∗) ∩ Vgb + V(a,a∗) ∩ Nb, if x ∈ Mb, and y ∈ La∗ .
κ κ + 0 + 0 − Thus (ψ − w )(x, y) ∈ V(a,a∗) ∩ Vb, with w3 ∈ [UB + (Ua) , (VA) ].
+ 0 + 0 − (v) Define w = w1 + w2 + w3 ∈ [UB + (Ua) , (VA) ]. It is easy to see that + 0 + 0 − H [UB + (Ua) , (VA) ] ' hP (a )iR0 . Then, by construction, the linear morphism κ κ κ κ (ψ − w )(Uκ) ⊆ Vκ, for any class k ∈ Θ. In particular, (ψ − w )(U(a,a∗)) = κ (ψ − w) (U(a,a∗)) ⊆ V(a,a∗) and (ψe − we)(Ub) = (ψ^− w)(Ub) ⊆ Vb. There- fore, ϕ = ψ − w ∈ R(U, V ), which proves that ψ ∈ R(U, V ) + I0(U 0,V 0), thus R(U, V ) + I0(U 0,V 0) = R0(U 0,V 0).
0 0 0 + − In order to prove that R(U, V ) ∩ I (U ,V ) ⊆ I(U, V ), with I = [UB ,VA ], (it is easy to see + − 0 0 0 that [UB ,VA ] ' hG2(b, a)iR), we take a morphism ϕ ∈ R(U, V ) ∩ I (U ,V ). Then as ϕ ∈ CHAPTER 2. ALGORITHMS OF DIFFERENTIATION 57
0 0 0 0 m m 0 I (U ,V ), ϕ can be factored through morphisms ϕ1 : U −! P (ba) and ϕ2 : P (ba) −! V that pass through powers of the representation P (ba). Thus ϕ = ϕ2ϕ1 with ϕ = ϕ1, and + + + 0 + 0 − ϕ2 = id, but as Pa = PB = 0, hence ϕ2ϕ1(UB ) = 0, since ϕ ∈ [UB + (Ua) , (VA) ], we 0 − − + − have that Im ϕ ⊂ (VA) . Then Im ϕ ⊂ VA , therefore ϕ ∈ [UB ,VA ] = I(U, V ), with this argument, we are done.
Theorem 19. Let P be an equipped poset with involution, with a pair of points (a, b), X 0 X-suitable. Then, the functor D(a,b) : R −! R , defined by formulas (2.16), induces an equivalence between quotient categories
∼ R/I ! R0/I0 ,
0 where I = hG2(b, a)iR and I = hP (ba)iR0 .
Proof. It suffices to bear in mind the following facts.
X 1. the density of the functor D(a,b) is guaranteed by Theorem 18.
X 2. the Proposition 5 allows us to conclude that the functor D(a,b) is full and faithful.
As a consequence of Theorem 19, we get the following corollary regarding Gabriel quivers of the corresponding quotient categories.
Corollary 5. If Γ(R) and Γ(R0) are the Gabriel quivers of the categories R and R0, then
0 Γ(R) \ [G2(b, a)] ' Γ(R ) \ [P (ba)]. CHAPTER 3
The Auslander-Reiten quiver of some equipped posets
In this chapter, we recall some facts regarding the Auslander-Reiten theory in a category of representations of a poset according to the ideas of Arnold [3]. We also introduce an combinatorial algorithm1 to construct the Auslander-Reiten quiver of some sincere non- trivial equipped posets, giving some examples of its use. The theory of representation of some special posets which we call of type Yt is described as well. In particular, it is com- puted the Auslander-Reiten quiver associated to a suitable subcategory add N of rep Yt. Worth noting that the Auslander-Reiten quivers of equipped posets of finite representation type have been given in [56] by using some valued quivers and that the Auslander-Reiten theory for Krull-Schmidt categories is developed in [38].
The algorithm introduced in this work can be used as a computational tool (in the sense of [23] and [33]) which allows us to describe explicitly the Auslander-Reiten quiver of some equipped posets of finite representation type and the preprojective and preinjective components of some equipped posets of finite growth representation type.
3.1 The lifting algorithm
In this section, we give an algorithm to build explicitly the Auslander-Reiten quiver of some sincere equipped posets, to do that, we start by reminding some useful facts regarding the Auslander-Reiten theory in categories of representations of equipped posets according to the ideas of Arnold [3].
Let P be an equipped poset, U, V, W ∈ rep P, ϕ ∈ Hom (U, V ), and ψ ∈ Hom (V,W ) with U = (U0; Ux | x ∈ P), V = (V0; Vx | x ∈ P) and W = (W0; Wx | x ∈ P). Then the following sequence of morphisms
ϕ ψ 0 / U / V / W / 0
1The Author is indebted with Professor Daniel Simson for his helpful comments regarding valued quivers which allowed to give the description in a more accuracy fashion.
58 CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 59 is an exact sequence if the induced sequence
ϕe ψe 0 / Ux / Vx / Wx / 0 is exact for each x in P ∪ {0}, i.e., for each x, ϕe : Ux ! Vx is one-to-one, ψe : Vx ! Wx is onto, and Ker ψe = Im ϕe.
A representation morphism ϕ : U ! V is a monomorphism if ϕ : U0 ! V0 is one-to- one, and an epimorphism if the ϕ-linear transformation ϕe : Vx ! Wx is onto for each x ∈ P ∪ {0}. The kernel of ϕ is the representation
Ker ϕ = (Ker ϕ; Ker ϕe ∩ Ux | x ∈ P) ∈ rep P, where Ker ϕe is the kernel of the G-linear transformation ϕe : Uf0 ! Vf0. If ϕ is an epimor- phism, then 0 −! Ker ϕ −! U −! V −! 0 is an exact sequence of representations, since
0 −! Ker ϕe ∩ Ux −! Ux −! Vx −! 0 is an exact sequence of vector spaces for each x ∈ P ∪ {0}.
A representation morphism ϕ : U ! V is a pure morphism and ϕ(U) = (ϕ(U0), ϕe(Ux) | x ∈ P) is a pure subrepresentation of V if ϕe(Ux) = ϕ^(U0)∩Vx for each x ∈ P. For example, the kernel of a representation morphism is a pure subrepresentation. A pure morphism is called a proper morphism in [56].
If ϕ : U ! V is a pure morphism, then the cokernel of f is the representation
Coker ϕ = (V0/ϕ(U0); (Vx + ϕ^(U0))/ϕ^(U0) | x ∈ P) ∈ rep P.
The sequence 0 −! U −! V −! Coker ϕ −! 0 in rep P is exact if ϕ is a pure monomorphism, since
0 −! ϕ(U0) ∩ Vx −! Vx −! (Vx + ϕ^(U0))/ϕ^(U0) −! 0 is an exact sequence of vector spaces for each x.
A representation U ∈ rep P is projective if whenever ϕ : V ! W is a representation epimorphism and ψ : U ! W is a representation morphism with V,W ∈ rep P, then there is a morphism h : U ! V with ϕh = ψ. Dually, a representation U is injective if whenever ϕ : V ! W is a pure monomorphism and ψ : V ! U with V,W ∈ rep P, then there is a morphism h : W ! U with hϕ = ψ.
Henceforth, if P is an equipped poset then I(P) will denote an injective representation of the poset P such that I(P) = (F ; I(P)x | x ∈ P), where I(P)x = G, for each x ∈ P. CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 60
We recall that according to Zavadskij [67,68] to each equipped poset P, it can be attached its corresponding evolvent Pb, which is an ordinary poset whose theory of representation is very well known [3, 56]. Thus, the Auslander-Reiten quiver Γ(P), associated to the category rep P, can be obtained by using some “weak gluing” and “lifting” operations defined on objects and morphisms of the category rep Pb. We describe such operations below.
Zavadskij classified equipped posets of finite growth representation type in [68]. Our starting point will be focused in the description (via the lifting algorithm) of the Auslander- Reiten quiver of the sincere non-trivially equipped posets of finite type F13,...,F18. Such descriptions allow to give an explicit construction of the preprojective and preinjective component of the Auslander-Reiten quiver of the critical equipped posets K6 and K8.
Note that, the aforementioned equipped posets Fi and kj have as characteristic that x y whenever x ∈ P⊗ and y ∈ P◦ (viceversa). According to the Definition 24, each weak point 0 00 x ∈ P gives rise to two new incomparable points x and x in the ordinary poset Pb, which 0 00 also are incomparable with the point y = y = y in Pb.
Let P be a sincere non-trivially equipped poset of the form F13,...,F18, and an equipped poset A of the form K6, K8. We compute the Auslander-Reiten quiver Γ(P) of P and preprojective P(A) and preinjective I(A) components of the Auslander-Reiten quiver A of the equipped poset A via the lifting and weak gluing operations applied to the Auslander- Reiten quiver Γ(Pb) of Pb and of the preprojective and preinjective components of Γ(Ab) of Ab. Such operations constitute the following combinatorial lifting algorithm.
The lifting algorithm ./ The following algorithm is a way to compress the information (in terms of the dimension vectors of the indecomposable representations) given by the Auslander-Reiten quiver of the evolvent of an equipped poset. The different steps go as follows: CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 61
Algorithm 3.1.1: Input. An equipped poset of type P ∈ {F13,...,F18} or A ∈ {K6,K8}. Output. The Auslander-Reiten quiver Γ(P) or P(A), I(A).
1. choose one of the following equipped posets F13,...,F18, k6, k8, and denoted it by P,
2. compute the evolvent Pb of P,
3. compute the Auslander-Reiten quiver Γ(Pb) of the ordinary poset Pb, associating its corresponding dimension vectors,
4. fix a vertex U = (U0; Ux | x ∈ Pb) in Γ(Pb) with vector dimension d of the form dU = (d0; dx | x ∈ Pb),
5.( lifting; nΓ(Pb)) ⊗ 0 00 (a)( nU) for each x ∈ P with associated evolvent points x and x , do dx = dx0 + dx00 (b)( nf) for each arrow f : U −! V ∈ Γ(Pb) define the arrow nf : nU ! nV in Γ(P), keeping neighbors and arrows orientation of Γ(Pb).
6. fix two vertices U and V in nΓ(Pb),
7.( weak gluing; oΓ(Pb)) if dU = dV = d = (d0; dx | x ∈ P) then do 2d = (2d0; 2dx | x ∈ P) and identify U and V else keep dU and dV invariants,
8. fix a vertex W in oΓ(Pb) with vector dimension s = (s0; sx | x ∈ P) given in step 7,
9. define d = (d0; dx | x ∈ P) and do d0 = s0, ( sx if x ∈ P⊗, d = 2 x ◦ sx if x ∈ P .
10. assign vector d to a unique vertex U of Γ(P) keeping neighbors and arrows orientation of oΓ(Pb), 11. define the dimension vector l = (l = d ; l = dx | x ∈ P). 0 0 x dim rad Ux
Arrows in ./ Γ(Pb) Arrows in ./ Γ(Pb) are constructed as follows:
12. For each situation of the form
f P (x) −−! P (y)
for suitable vertices P (x),P (y) in Γ(Pb) with y associated to a fixed strong point y in the original equipped poset P and P (x) in the orbit of P (∅) or in the orbit of P (z), ./(f) with z a fixed strong point in P, is built an arrow of the form ./(P (x)) −−−! ./(P (y)) in ./ Γ(Pb). CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 62
13. For each situation of the form P (y0) f1
P (x) @ f2 @R P (y00)
for suitable vertices P (x),P (y0),P (y00) in Γ(Pb) with y0, y00 associated to a fixed weak point y in the original equipped poset P and P (x) in the orbit of P (∅), it is built 2 ./(f1,f2) an arrow of the form ./(P (x)) −−−−−−−! ./(P (y0) ⊕ P (y00)) in ./ Γ(Pb). 14. For each situation of the form P (y0) f1
f P (x) H2 Hj 00 A P (y ) f3A AU P (z) for suitable vertices P (x),P (y0),P (y00),P (z) in Γ(Pb) with y0, y00 (z) associated to a fixed weak (strong) point y (z) in the original equipped poset P and P (x) in the 2 ./(f1,f2) orbit of P (∅), two arrows are been built of the following form ./(P (x)) −−−−−−−! ./(f3) ./(P (y0) ⊕ P (y00)) and ./(P (x)) −−−! ./(P (z)) in ./ Γ(Pb). 15. For each situation of the form P (y0) f1
P (x00) @ g2 @R P (z00)
P (z0) g1
P (x0) @ f2 @R P (y00)
for suitable vertices P (x0),P (x00),P (y0),P (y00),P (z0),P (z00) in Γ(Pb) with x0, x00, y0, y00, z0, z00 associated to a fixed weak point x, y, z, respectively in the original equipped poset P, two arrows are been built of the following form ./(f ,f ) ./(g ,g ) ./(P (x0) ⊕ P (x00)) −−−−−!1 2 ./(P (y0) ⊕ P (y00)) and ./(P (x0) ⊕ P (x00)) −−−−−!1 2 ./(P (z0) ⊕ P (z00)) in ./ Γ(Pb). CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 63
16. For each situation of the form
P (y0) f1
P (x00) @ g2 @R P (z) g1
P (x0) @ f2 @R P (y00)
for suitable vertices P (x0),P (x00),P (y0),P (y00),P (z), in Γ(Pb) with x0, x00, y0, y00 as- sociated to a fixed weak point x, y, respectively in the original equipped poset P (and P (z) in the orbit of P (∅)), two arrows are been built of the following form ./(f ,f ) 2 ./(g ,g ) ./(P (x0) ⊕ P (x00)) −−−−−!1 2 ./(P (y0) ⊕ P (y00)) and ./(P (x0) ⊕ P (x00)) −−−−−−−!1 2 ./(P (z)) in ./ Γ(Pb).
We have contemplated the different situations for the source arrows, the situations for the sink arrows are analogous.
Meshes in ./ Γ(Pb) Meshes in ./ Γ(Pb) are constructed as follows:
17. For each situation of the form
P (x) f1 @f2 @R P (y) − − − − − τ −1P (y)
for suitable vertices P (x),P (y) in Γ(Pb) with y associated to a fixed strong point y in the original equipped poset P and P (x) in the orbit of P (∅) or in the orbit of P (z), with z a fixed strong point in P, is built a mesh of the form
./(P (x)) ./(f1) @./(f2) @R ./(P (y))− − − − − ./(τ −1P (y))
in ./ Γ(Pb). 18. For each situation of the form CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 64
P (y0) f1 @g1 @R P (x) − − − − − τ −1P (x) @ f2 @R g2 P (y00)
for suitable vertices P (x),P (y0),P (y00) in Γ(Pb) with y0, y00 associated to a fixed weak point y in the original equipped poset P and P (x) in the orbit of P (∅), it is built a mesh of the form
./(P (y0) ⊕ P (y00)) ./(f , f ) ./(g , g ) 1 2 22 @ 1 2 @R ./(P (x)) − − − − − − − − ./(τ −1P (x))
in ./ Γ(Pb). 19. For each situation of the form
P (y0) f1 @g1 @R P (x) − − − − − τ −1P (x) J@ f2 g2 J @R f3 g3 J P (y00) JJ^ P (z)
for suitable vertices P (x),P (y0),P (y00),P (z) in Γ(Pb) with y0, y00 (z) associated to a fixed weak (strong) point y (z) in the original equipped poset P and P (x) in the orbit of P (∅), it is built a mesh of the form
./(P (y0) ⊕ P (y00)) ./(f , f ) ./(g , g ) 1 2 22 @ 1 2 @R ./(P (x)) − − − − − − ./(τ −1P (x)) @ ./(f3) @R ./(g3) ./(P (z))
in ./ Γ(Pb). 20. For each situation of the form CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 65
P (y0) f1 @h1 @R P (x00)− − − − − τ −1P (x00) @ g2 @R l2 P (z00)
P (z0) g1 @l1 @R P (x0)− − − − − τ −1P (x0) @ f2 @R h2 P (y00)
for suitable vertices P (x0),P (x00),P (y0),P (y00),P (z0),P (z00) in Γ(Pb) with x0, x00, y0, y00, z0, z00 associated to a fixed weak point x, y, z, respectively in the original equipped poset P, it is built a mesh of the form
./(P (y0) ⊕ P (y00)) ./(f1, f2) @./(h1, h2) @R ./(P (x0) ⊕ P (x00)) − − − ./(τ −1P (x0) ⊕ τ −1P (x00)) @ ./(g1, g2) @R ./(l1, l2) ./(P (z0) ⊕ P (z00))
in ./ Γ(Pb). 21. For each situation of the form
P (y0) f1 @h1 @R P (x00)− − − − − τ −1P (x00) @ g2 @R l2 P (z) g1 @l1 @R P (x0)− − − − − τ −1P (x0) @ f2 @R h2 P (y00)
for suitable vertices P (x0),P (x00),P (y0),P (y00),P (z), in Γ(Pb) with x0, x00, y0, y00 asso- ciated to a fixed weak point x, y, respectively in the original equipped poset P (and P (z) in the orbit of P (∅)), it is built a mesh of the form CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 66
./(P (y0) ⊕ P (y00)) ./(f1, f2) @./(h1, h2) @R ./(P (x0) ⊕ P (x00)) − − − ./(τ −1P (x0) ⊕ τ −1P (x00))
@ 22 ./(g1, g2) @R ./(l1, l2) ./(P (z))
in ./ Γ(Pb).
Remark 3.1. The following facts must be considered in the development of the lifting algorithm in the finite representation type case:
1. Arrows ./(f): ./(U) ! ./(V ) correspond to irreducible morphisms, were f is a system of arrows in Γ(Pb) as we previously described. 0 00 0 00 2. nP (x ) = nP (x ). P (x ) and P (x ) denote the indecomposable projective repre- 0 00 −n 0 sentation of Pb associated to the points x , x ∈ Pb, respectively. And nτ P (x ) = −n 00 nτ P (x ) for the corresponding translated representations. −n 0 3. The gluing operation can be also applied to orbits by gluing nτ P (x ) with −n 00 nτ P (x ) for any positive integer n.
Henceforth, we let ./ denote the lifting algorithm. Note that, ./(P (∅b)) = P (∅) and ./(I(Pb)) = I(P).
The irreducibility of the morphisms in ./ Γ(Pb) arises from the irreducibility of the mor- phisms in Γ(Pb), this behavior is exposed in the following theorem. 0 00 Theorem 20. (i) If the morphism ./(f1, f2): ./(P (x ) ⊕ P (x )) −! ./(P (y)) in rep P, induced by a system of arrows of the form:
P (x0) @ f1 @R P (y) ∈ Γ(Pb) f2 P (x00)
is defined in such a way that ( f (z), if z ∈ X1 ./(f , f )(z) = 1 0 1 2 2 f2(z), if z ∈ X0
with
0 00 1 2 (./(P (x ) ⊕ P (x )))0 = X0 ⊕ X0 = X0 0 1 (P (x ))0 = X0 00 2 (P (x ))0 = X0 1 2 X0 ' X0 . CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 67
Then ./(f1, f2) is an irreducible morphism in rep P.
0 00 (ii) If the morphism ./(f1, f2): ./(P (y)) −! ./(P (x ) ⊕ P (x )) in rep P, induced by a system of arrows of the form (as in the case 13. in the lifting algorithm):
P (x0) f1
P (y) ∈ Γ(Pb) @ f2 @R P (x00)
is defined in such a way that ./(f1, f2)(z) = (f1(z), f2(z)). Then ./(f1, f2) is an irreducible morphism in rep P.
(iii) If the morphism ./(f): ./(P (x)) −! ./(P (y)) in rep P induced by the arrow (as in the case 12. in the lifting algorithm):
f P (x) −!− P (y) ∈ Γ(Pb)
is obtained by making ./(f) = f. Then ./(f) is an irreducible morphism in rep P.
0 00 0 00 (iv) If the morphisms ./(f1, f2): ./(P (x ) ⊕ P (x )) −! ./(P (y ) ⊕ P (y )) and ./(g1, g2): ./(P (x0) ⊕ P (x00)) −! ./(P (z0) ⊕ P (z00)) in rep P, induced by a system of arrows of the form (as in the case 15. in the lifting algorithm): P (y0) f1
P (x00) @ g2 @R P (z00)
P (z0) g1
P (x0) @ f2 @R P (y00)
are defined in such a way that f1 0 g1 0 ./(f1, f2) = ./(g1, g2) = . 0 f2 0 g2
Then ./(f1, f2) and ./(g1, g2) are irreducible morphisms in rep P.
Proof. CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 68
◦ (i) Firstly we note that by definition ./(f1, f2) is a linear map and that if j ∈ P 0 00 then ./^(f1, f2)(./(P (x ) ⊕ P (x ))j) ⊆ ./(P (y))j provided that for any z = (z1, z2) ∈ 0 00 ./(P (x ) ⊕ P (x ))j ∩ Xf0 it holds that
./^(f1, f2)(z) = (f1(z1), f2(z2)) ∈ ((P (y))j, (P (y))j) = (P (y))j.
Whereas if j ∈ P⊗ is a weak point associated to the points j0 and j00 in Pb then by definition 0 f1((P (x ))j0 ) ⊆ (P (y))j0 00 f2((P (x ))j00 ) ⊆ (P (y))j00 .
0 00 thus ./^(f1, f2)((P (x ) ⊕ P (x )j) ⊆ ./(P (y))j provided that
./(P (y))j = ((P (y))j0 , (P (y))j00 ) = (P (y))j.
Therefore ./(f1, f2) is a morphism in rep P.
Suppose that there exist an object Z ∈ rep P and radical morphisms h1 ∈ 0 00 rad (./(P (x ) ⊕ P (x )),Z) and h2 ∈ rad (Z, ./P (y)) in rep P such that
./(f1, f2) = h2h1.
That is, ./(f1, f2) is not irreducible. Thus, if π1 and π2 denote the corresponding projections, then
( 0 (h hx )(z), if z ∈ X1, ./(f , f )(z) = 2 1 0 1 2 x00 2 (h2h1 )(z), if z ∈ X0 ,
x0 x00 x0 x00 where h1 = h1π1 and h1 = h1π2. Since h1 , h1 and h2 are radical morphisms in x0 0 x00 00 x00 rep Pb, i.e., h1 ∈ rad (P (x ), Zb) and h1 ∈ rad (P (x ), Zb), and f2 = h2h1 are fac- torizations of the irreducible morphisms f1 and f2 in rep Pb which is a contradiction.
(ii) As above suppose that ./(f1, f2) is not an irreducible morphism, then there exist a representation Z ∈ rep P and radical morphisms h1 ∈ rad (./P (x),Z) and h2 ∈ rad (Z, ./(P (x0) ⊕ P (x00))) in rep P such that
./(f1, f2) = h2h1.
Thus, if g1 = π1h2 and g2 = π2h2 then g1, g2 and h1 are radical morphisms in rep Pb, 0 00 i.e., g1 ∈ rad (Z,Pb (y )) and g2 ∈ rad (Z,Pb (y )) in rep Pb such that f1 = g1h1 and f2 = g2h1 are factorizations of f1 and f2 in rep Pb which is a contradiction. (iii) The result follows from the definition of f.
0 00 (iv) Note that for (u, v) ∈ ((P (x ^) ⊕ P (x ))0 it holds that
./^(f1, f2)(u, v) = (f1(u), f2(v))
./^(g1, g2)(u, v) = (g1(u), g2(v)) CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 69
where
0 00 f1(u) = (u0, u1), u0 ∈ P (y )0, u1 ∈ P (y )0 0 00 f2(v) = (v0, v1), v0 ∈ P (y )0, v1 ∈ P (y )0 0 00 g1(u) = (s0, s1), s0 ∈ P (z )0, s1 ∈ P (z )0 0 00 g2(v) = (t0, t1), t0 ∈ P (z )0, t1 ∈ P (z )0,
in this case u0 = v1, u1 = −v0 and s0 = t1, s1 = −t0. Thus the result follows from items (i), (ii).
The remaining cases for irreducible morphisms are proved by considering similar arguments to the Theorem 20 and combining its items (i) − (iv).
3.2 The Auslander-Reiten quiver of the equipped posets F13 − F18
In this section, it is used the lifting algorithm to describe the Auslander-Reiten quivers of the sincere non-trivially equipped posets of finite representation type F13 − F18 (see Ap- pendix A.2 where we show the Auslander-Reiten quiver of some suitable ordinary posets). Below, we show explicitly the application of the lifting algorithm only for the equipped posets F13, F15, F17; and for the equipped poset F18. In Appendix A.2, we show The Auslander-Reiten quiver of the remaining equipped posets. To do that, we apply explic- itly the lifting algorithm. Worth noting that Simson [56] obtained the Auslander-Reiten quiver of the equipped posets F13 − F18 by using suitable valued quivers.
In section 1.6 it is given the definition of the objects P (x) and P (x, y), for an equipped poset P. Note that the representations P (x) are precisely the projective representations of the ordinary poset Pb.
We recall that according to Arnold [3] the indecomposable projective representations P (x) = (k; P (x)y | y ∈ P) (indecomposable injective representations I(x) = (k; I(x)y | y ∈ P)) of an ordinary poset P are given by the following formulas: ( ( O k if y ∈ x , 0 if y ∈ xM, P (x)y = I(x)y = 0 otherwise. k otherwise.
In order to compute the Auslander-Reiten quiver of equipped or ordinary posets, we draw the dimension vectors with the shape that the poset has, where the number described before in the vertical line is the dimension of the space U0, and the numbers that are after the vertical line, are the dimensions of the spaces Ux (see for example Table 3.1).
For the equipped poset F13 we have that Fd13 = F7 and ./ Γ(F7) = Γ(F13). The following table contains the indecomposable projective representations of the equipped poset F13 and its dimension vector. CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 70
P (∅) T (1)
1 | 0 2 | 1
Table 3.1. The indecomposable projective representations of the equipped poset F13 and its corresponding dimension vector.
- F13 : ⊗ b Fd13 = F7 : 0 00 1 1e1 e
1 | 1 0 1 | 1 2 | 1 ¡¡ A ¡¡ A ¡¡ A ¡ AAU ¡ AAU ¡ AAU - - Γ(F7) : 1 | 0 0 − − 1 | 1 1 nΓ(F7) : 1 | 0 − − − 1 | 2 oΓ(F7) :1 | 0 − − − 1 | 1 A ¡¡ A ¡¡ AAU ¡ AAU ¡ 1 | 0 1 1 | 1
2 | 1 2¡¡ 2 A ¡ AAU ./ Γ(F7) : 1 | 0 − − − 1 | 1
Figure 3.1. The Auslander-Reiten quiver Γ(F13) of the equipped poset F13 obtained from ./ Γ(F7) via the lifting algorithm ./.
We note that, all the arguments we are done until now have been given in the frame- mark of the theory of representation of posets. However arguments from the theory of representation of path algebras can be used accurately to solve some of these problems [3, 6, 56].
Now we consider the equipped poset F14 to apply the same procedure. Note that, Fd14 = F2. Therefore, after applying the lifting algorithm ./ to Γ(F2), we obtain Γ(F14).
P (∅) T (1) P (2)
1 | 0 0 2 | 1 0 1 | 0 1
Table 3.2. The indecomposable projective representations of the equipped poset F14 and its corresponding dimension vector. CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 71
F14 : ⊗ 1 2e
2 | 1 0 − − 2 | 1 2 ¡ A ¡ A 2¡ 22¡ 2 ¡ AAU ¡ AAU ./ Γ(F2) : 1 | 0 0 − −2 | 1 1 − − 1 | 1 1 A A ¡¡ ¡¡ AAU ¡ AAU ¡ 1 | 0 1 − − 1 | 1 0
Figure 3.2. The Auslander-Reiten quiver Γ(F14) of the equipped poset F14.
In the sequel, we describe all the Auslander-Reiten sequences in quiver Γ(F14) giving a detailed description of the behavior of the corresponding dimensions.
1 (i) 0 −! P (∅) −! T (1) ⊕ P (2) −! G1(1, 2) −! 0, 1 · (2 | 1 0) + 1 | 0 1 − 1 | 0 0 = 3 | 1 1 − 1 | 0 0 = 2 | 1 1, τ(G1(1, 2)) = P (∅). 2 (ii) 0 −! T (1) −! G1 (1, 2) −! G2(1, 2) −! 0, − 2 · (2 | 1 1) − 2 | 1 0 = 4 | 2 2 − 2 | 1 0 = 2 | 1 2, τ (T (1)) = G2(1, 2).
(iii) 0 −! P (2) −! G1(1, 2) −! P (1) −! 0, 2 | 1 1 − 1 | 0 1 = 1 | 1 0, τ −(P (2)) = P (1). 1 (iv) 0 −! G1(1, 2) −! G2 (1, 2) ⊕ P (1) −! P (1, 2) −! 0,
1 · (2 | 1 2) + 1 | 1 0 − 2 | 1 1 = 3 | 2 2 − 2 | 1 1 = 1 | 1 1, τ(P (1, 2)) = G1(1, 2).
Again, we show step by step the application of the lifting algorithm ./ to Γ(F3), since the ordinary poset F3 is the evolvent of the equipped poset F15, it holds that ./(F3) = Γ(F15).
P (∅) T (2) T (3) P (1)
0 0 1 0 1 0 0 2 0 1 2 0 0 1 1 0
Table 3.3. The indecomposable projective representations of the equipped poset F15 and its corresponding dimension vector.
0 00 3 3 3 ⊗ e e F15 : ⊗ −! F = F : b d15 3 0 00 1e 2 1e 2 e2 e CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 72
1 0 − − −0 1 − − −1 1 − − − 1 0 1 0 1 0 1 1 0 0 1 0 0 1 1 1 1 0 @ @ @ @ @ @ @ @ @R @R @R @R 1 0 −−−1 1 −−−1 2 −−−2 1 −−− 1 1 1 0 0 0 2 1 1 0 2 1 0 1 2 1 1 1 1 1 1 0 @ @ @ @ @ @ @ @ @ @ Γ(F3): @R @R @R @R @R 0 0 −−−1 1 −−−2 2 −−−2 2 −−−2 2 −−− 1 1 1 0 0 0 2 1 0 0 3 1 1 1 3 2 1 1 2 1 1 1 1 1 1 1 B@ ¢ B@ ¢ B@ ¢ B@ ¢ B@ ¢ ¢ ¢ ¢ ¢ ¢ B @ ¢ B @ ¢ B @ ¢ B @ ¢ B @ ¢ B @R B @R B @R B @R B @R B 0 0 −−−¢ B 1 1 −−−¢ B 1 1 −−−¢ B 1 1 −−−¢ B 1 1 ¢ 1 1 0 0 ¢ 1 0 0 0 ¢ 2 1 1 1 ¢ 1 1 0 0 ¢ 1 0 1 1 ¢ B ¢ B ¢ B ¢ B ¢ B ¢ B B B B B B ¢ B ¢ B ¢ B ¢ B ¢ BN ¢ BN ¢ BN ¢ BN ¢ BN ¢ 0 1 −−−1 1 −−−2 1 −−−1 2 −−− 1 1 1 0 0 0 2 1 0 1 2 1 1 0 2 1 1 1 1 1 0 1 @ @ @ @ @ @ @ @ @R @R @R @R 0 1 − − −1 0 − − −1 1 − − − 0 1 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1
1 − − −1 − − −2 − − − 1 1 0 1 1 1 0 1 0 1 1 1 1 @ @ @ @ @ @ @ @ @R @R @R @R 1 −−−2 −−−3 −−−3 −−− 2 1 0 0 2 1 1 2 1 1 2 1 2 1 1 1 @ @ @ @ @ @ @ @ @ @ @R @R @R @R @R 0 −−−2 −−−4 −−−4 −−−4 −−− 2 nΓ(F3) : 1 0 0 2 1 0 3 1 2 3 2 2 2 1 2 1 1 2 B@ ¢¢ B@ ¢¢ B@ ¢¢ B@ ¢¢ B@ ¢¢ B @ B @ B @ B @ B @ @R ¢ @R ¢ @R ¢ @R ¢ @R ¢ B 0 ¢ B 2 ¢ B 2 ¢ B 2 ¢ B 2 ¢ B 1 −−−B 1 −−−B 2 −−−B 1 −−− B 1 B 1 0 ¢ B 0 0 ¢ B 1 2 ¢ B 1 0 ¢ B 0 2 ¢ ¢ ¢ ¢ ¢ ¢ B ¢ B ¢ B ¢ B ¢ B ¢ BBN ¢ BBN ¢ BBN ¢ BBN ¢ BBN ¢ 1 −−−2 −−−3 −−−3 −−− 2 1 0 0 2 1 1 2 1 1 2 1 2 1 1 1 @ @ @ @ @ @ @ @ @R @R @R @R 1 − − −1 − − −2 − − − 1 1 0 1 1 1 0 1 0 1 1 1 1 CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 73
1 − − −1 − − −2 − − − 1 2 0 1 2 2 0 2 0 1 2 2 1 @ @ @ @ @ @ @ @ @R @R @R @R 1 −−−2 −−−3 −−−3 −−− 2 2 0 0 4 2 1 4 2 1 4 2 2 2 2 1 @ @ @ @ @ @ @ @ @ @ @R @R @R @R @R 0 −−−1 −−−2 −−−2 −−−2 −−− 1 oΓ(F3) : 1 0 0 2 1 0 3 1 1 3 2 1 2 1 1 1 1 1 @ @ @ @ @ @ @ @ @ @ @R @R @R @R @R 0 −−−1 −−−1 −−−1 −−− 1 1 1 0 1 0 0 2 1 1 1 1 0 1 0 1
0 − − −1 − − −1 − − − 0 2 0 1 2 2 0 2 0 1 2 2 1 @ @ @ @ @ @ @ @ @R @R @R @R 1 −−−1 −−−2 −−−1 −−− 1 2 0 0 4 2 1 4 2 1 4 2 2 2 2 1 @ @ @ @ @ 2 @ 2 @ 2 @ 2 @ 222 @ 2 2 2 @R @R @R @R @R 0 −−−1 −−−1 −−−1 −−−1 −−− 0 ./ Γ(F3): 1 0 0 2 1 0 3 1 1 3 2 1 2 1 1 1 1 1 @ @ @ @ @ @ @ @ @ @ @R @R @R @R @R 0 −−−1 −−−0 −−−1 −−− 0 1 1 0 1 0 0 2 1 1 1 1 0 1 0 1
Figure 3.3. The Auslander-Reiten quiver Γ(F15) of the equipped poset F15.
Now, for the equipped poset F16 such that Fd16 = F8, note that, to compute the corre- sponding Auslander-Reiten quiver Γ(F16), it is necessary to apply the lifting algorithm ./ to Γ(F8).
P (∅) T (1) P (2) P (3)
0 0 0 0 1 0 0 2 1 0 1 0 1 1 0 1
Table 3.4. The indecomposable projective representations of the equipped poset F16 and its corresponding dimension vector. 0
CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 74
3
e
F16 : ⊗ 1 2e
0 − − −2 − − − 0 2 1 0 2 1 0 2 1 2 @ @ @ 2 @ 2 @ 222 @ 2 @R @R @R 0 − − −1 − − −1 − − − 0 Γ(F16) : 1 0 0 2 1 0 2 1 1 1 1 1 @ @ @ @ @ @ @R @R @R 1 − − − − − − 1 1 0 0 2 1 1 1 1 0 @ @ @ @ @R @R 0 − − − 0 1 0 1 1 1 0
Figure 3.4. The Auslander-Reiten quiver Γ(F16) of the equipped poset F16.
In the sequel, we compute the Auslander-Reiten quiver Γ(F17) of F17 by applying the lifting algorithm to the ordinary poset (2, 2), which is the corresponding evolvent of F17.
P (∅) T (1) T (2)
0 1 0 1 0 2 0 2 1
Table 3.5. The indecomposable projective representations of the equipped poset F17 and its corresponding dimension vector.
2 ⊗
F17 : ⊗ 1 CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 75
1 0 1 1 1 0 1 1 @ @ @ @ @R @R 1 0 − − − − − 1 1 1 − − − − −− 2 1 0 0 1 1 0 1 0 1 1 @ @ @ @ @ @ @ @ @R @R @R @R n 0 0 − − − − −1 1 − − − − − 1 1 0 − − − − −−2 − − − − −− 2 Γ(2, 2) : 1 0 0 1 0 0 1 1 1 −! 1 0 1 0 1 2 @ @ @ @ @ @ @ @ @R @R @R @R 0 1 − − − − − 1 1 1 − − − − −− 2 1 0 0 1 0 1 1 0 1 1 @ @ @ @ @R @R 0 1 1 1 0 1 1 1
1 2 1 @ @ @R 1 − − − − −− 2 2 0 2 1 @ @ @ @ @R @R o 0 − − − − −−1 − − − − −− 1 −! 1 0 1 0 1 1
0 2 1 @ @ @R 1 − − − − −− 1 2 0 2 1 @ @ 2 @ 22 @ 2 @R @R 0 − − − − −−1 − − − − −− 0 ./(2, 2) : 1 0 1 0 1 1
Figure 3.5. The Auslander-Reiten quiver Γ(F17) of the equipped poset F17.
Below, we present the last sincere non-trivially equipped poset of finite representation type F18 and its corresponding Auslander-Reiten quiver Γ(F18). The computations are described in Appendix A.2, which are realized via the lifting algorithm ./ to the poset F, where Fd18 = F. CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 76
P (∅) T (2) T (3) T (4) P (1)
0 0 0 1 0 1 0 2 0 2 1 2 0 1 0 0 0 0 1 0 0 0 0 1 0
Table 3.6. The indecomposable projective representations of the equipped poset F18 and its corresponding dimension vector.
4 ⊗
⊗ 3 F18 =
⊗ 1e 2
0 0 2 0 − − − 2 1 0 1 2 0 @@R @@R 0 1 1 0 1 0 0 ./(F): 2 1 −−−2 0 −−−2 1 −−−4 1 −−−2 1 −−−2 1 −−− 2 0 0 0 2 0 0 0 2 1 2 0 0 1 2 1 @@R @@R @@R @@R @@R @@R @@R 1 1 2 1 1 0 0 0 2 0 −−−4 1 −−−4 1 −−−4 2 −−−4 1 −−−4 2 −−−4 1 −−− 2 1 0 0 2 0 2 0 2 0 2 1 2 1 2 2 2 1 2 2@@R 2 2@@R 2 2@@R 2 2@@R 2 2@@R 2 2@@R 2 2@@R 2 2@@R 0 1 1 1 1 0 0 0 0 1 0 −−−2 0 −−−3 1 −−−3 1 −−−2 1 −−−3 1 −−−3 1 −−−2 1 −−− 1 0 0 0 1 0 1 0 2 0 1 0 1 1 2 1 1 1 1 1 @@R @@R @@R @@R @@R @@R @@R @@R 0 1 0 1 0 0 0 0 1 0 −−−1 0 −−−2 1 −−−1 0 −−−1 1 −−−2 0 −−−1 1 −−− 1 0 1 0 0 0 1 0 1 0 0 0 1 1 1 0 0 1
Figure 3.6. The Auslander-Reiten quiver Γ(F18) of the equipped poset F18.
3.3 The Auslander-Reiten quiver of a completely weak chain Cn
In this section, we present an example of an equipped poset of finite growth representation type, which is linearly ordered and not sincere.
⊗ Let P be an equipped poset such that P = {1 ≺ 2 ≺ · · · ≺ n | 1, 2, . . . , n ∈ P } i.e. P = Cn is a completely weak chain. CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 77 ⊗ n
⊗ n − 1 . . Cn = ⊗ 3
⊗ 2
⊗ 1
Figure 3.7. The diagram associated to a completely weak chain Cn.
We recall that the complete list of indecomposable representations of a completely weak chain are P (i), T (i), T (i, j) with 1 ≤ i, j ≤ n, i < j and P (∅). They were obtained by Zavadskij in [68]. See [20] where it is described the Auslander-Reiten quiver of P = Cn. Note that C1 = F13 and C2 = F17, whose Auslander-Reiten quivers were already presented in Figures 3.1 and 3.5, respectively. Let Cn be a completely weak chain. Since Ccn = (n, n), we apply the lifting algorithm to Γ(n, n) and present here the final result ./ Γ(n, n) = Γ(Cn). Recall that in Figure A.7 it is described Γ(n, n).
T (1) T (2) T (3) ... T (n) P (∅)
0 0 0 1 0 0. 0. 0. 0. 0...... 2 0 2 0 2 1 ... 2 0 1 0 0 1 0 0 0 1 0 0 0 0
Table 3.7. The indecomposable projective representations of the equipped poset Cn and its cor- responding dimension vector.
For the sake of clarity, we will assume notations T (i), T (i, j), P (i) and P (∅) for vertices in the Auslander-Reiten quiver Γ(Cn) of the equipped poset Cn. CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 78 T (1) @ @R T (2) − − − T (1, n) @ @ @R @R T (3) − − −T (2, n) − − −T (1, n − 1) @ @ @ @R @R @R
T (4) − − −T (3, n) − − −T (2, n − 1) −−T (1, n − 2)
-
- -
- -
q - - q
T (n − 2) q - - Tq (1, 4)
q - - q
- -
- - @
- - @R
- -
T (n − 1) - - T (1, 3)
- - @ - - @ @R - @R T (n) −− T (n − 1, n) − − −−− −−− −−− −−− −−− −−− −−− − T (2, 3) − − − T (1, 2) @ @ @ 2 2 @R 2 2 22@R @R P (∅) − − − P (n) − − −−− −−− −−− −−− −−− −−− −−− −−− −−− −−− − P (2) − − − P (1)
Figure 3.8. The Auslander-Reiten quiver Γ(Cn) of the equipped poset Cn.
The following result is an immediate consequence of the procedures described above.
Theorem 21. Let P be an input poset of the Lifting algorithm, with P = P⊗ + P◦ and Pb its corresponding evolvent. Then, the following statements hold.
(i) If P (x) is an indecomposable projective representation of Pb then ./(P (x)) = P (x) is an indecomposable projective representation of P.
(ii) If P (x0) and P (x00) are indecomposable projective representations of Pb, then ./(P (x0) ⊕ P (x00)) = T (x) is an indecomposable projective representation of P.
Proof.
(i) This item follows immediately from the definition of the lifting algorithm.
(ii) If P (x0) and P (x00) are indecomposable projective representations of Pb, then 0 00 00 0 0 00 nP (x ) = nP (x ), but oP (x ) = oP (x ) = o(P (x ) ⊕ P (x )) and thus 0 00 ./(P (x ) ⊕ P (x )) = T (x) is an indecomposable projective representation of P.
3.4 The preprojective and preinjective components of the critical equipped posets K6 and K8
Let A be a critical equipped poset of finite growth representation type of kind K6 or K8 (which are one parametric). Since these posets have infinite number of representations, in this section, we present the preprojective and preinjective components of the Auslander- Reiten quiver Γ(A) of A. These components are denoted by P(A) and I(A), respectively. CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 79
Below, you can find the indecomposable projective (P (∅), T (1), T (2)), injective (I(1), I(2), P (1, 2) = I(K6)) representations, and the diagram associated to the sincere critical equipped poset K6.
P (∅) T (1) T (2) I(1) I(2) P (1, 2)
1 |0 0 2 |1 0 2 |0 1 2 |1 2 2 |2 1 1 |1 1
Table 3.8. The indecomposable projective and injective representations of the critical equipped poset K6 and its corresponding dimension vector [67].
⊗ ⊗ K6 = P = 1 2
In Remark A.2, we describe the orbits in the preprojective P(K1) and preinjective I(K1) components. The preprojective P(K6) and preinjective I(K6) components of the Auslander-Reiten quiver Γ(K6) of the critical equipped poset K6 have been obtained by applying the lifting algorithm ./ to these orbits, in this case, we assume that k, l, r ∈ Z with k ≥ 0, l = 2k and r = 2k + 1. The following steps are carried out to compute the vertices of the infinite components P(K6) and I(K6):
1. We start by applying n to the orbits in P(K1) and orbits in I(K1):
( −l 2l + 1 | 2l 2l if l = 2k, nτ P (∅) = 2r + 1 | 2r 2r if r = 2k + 1. ( −l 0 l + 1 | 2k + 1 2k if l = 2k, nτ P (1 ) = r + 1 | 2k + 1 2k + 2 if r = 2k + 1. ( −l 00 l + 1 | 2k + 1 2k if l = 2k, nτ P (1 ) = r + 1 | 2k + 1 2k + 2 if r = 2k + 1. ( −l 0 l + 1 | 2k 2k + 1 if l = 2k, nτ P (2 ) = r + 1 | 2k + 2 2k + 1 if r = 2k + 1. ( −l 00 l + 1 | 2k 2k + 1 if l = 2k, nτ P (2 ) = r + 1 | 2k + 2 2k + 1 if r = 2k + 1.
( l 2l + 1 | 2l + 2 2l + 2 if l = 2k, nτ I(K1) = 2r + 1 | 2r + 2 2r + 2 if r = 2k + 1. ( l 0 l + 1 | 2k + 1 2k + 2 if l = 2k, nτ I(1 ) = r + 1 | 2k + 3 2k + 2 if r = 2k + 1. CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 80 ( l 00 l + 1 | 2k + 1 2k + 2 if l = 2k, nτ I(1 ) = r + 1 | 2k + 3 2k + 2 if r = 2k + 1. ( l 0 l + 1 | 2k + 2 2k + 1 if l = 2k, nτ I(2 ) = r + 1 | 2k + 2 2k + 3 if r = 2k + 1. ( l 00 l + 1 | 2k + 2 2k + 1 if l = 2k, nτ I(2 ) = r + 1 | 2k + 2 2k + 3 if r = 2k + 1.
−l 0 −l 00 l 0 l 00 It is easy to see that nτ P (x ) = nτ P (x ) and nτ I(x ) = nτ I(x ), for each l. 2. We continue by applying o:
( −l 2l + 1 | l l if l = 2k, o(nτ P (∅)) = 2r + 1 | r r if r = 2k + 1. ( −l 0 00 2l + 2 | 2k + 1 2k if l = 2k, o(nτ (P (1 ) ⊕ P (1 ))) = 2r + 2 | 2k + 1 2k + 2 if r = 2k + 1. ( −l 0 00 2l + 2 | 2k 2k + 1 if l = 2k, o(nτ (P (2 ) ⊕ P (2 ))) = 2r + 2 | 2k + 2 2k + 1 if r = 2k + 1. ( l 2l + 1 | l + 1 l + 1 if l = 2k, o(nτ I(K1)) = 2r + 1 | r + 1 r + 1 if r = 2k + 1. ( l 0 00 2l + 2 | 2k + 1 2k + 2 if l = 2k, o(nτ (I(1 ) ⊕ I(1 ))) = 2r + 2 | 2k + 3 2k + 2 if r = 2k + 1. ( l 0 00 2l + 2 | 2k + 2 2k + 1 if l = 2k, o(nτ (I(2 ) ⊕ I(2 ))) = 2r + 2 | 2k + 2 2k + 3 if r = 2k + 1.
Since the points 1, 2 ∈ K6 are incomparable then the previous procedure allows us to conclude the following result.
Theorem 22. For the sincere critical equipped poset K6, the preprojective P(K6) and preinjective I(K6) components of its Auslander-Reiten quiver Γ(K6) can be computed by using the following identities, for which we assume that k, l, r ∈ Z with k ≥ 0, l = 2k and r = 2k + 1.
0 00 1. ./(P (x ) ⊕ P (x )) = T (x), P(K6) = ./ P(K1), ( −l 2l + 1 | l l if l = 2k, τ P (∅) = 2r + 1 | r r if r = 2k + 1. ( 2l + 2 | l + 1 l if l = 2k, τ −l(T (1)) = 2r + 2 | r r + 1 if r = 2k + 1. ( 2l + 2 | l l + 1 if l = 2k, τ −l(T (2)) = 2r + 2 | r + 1 r if r = 2k + 1. CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 81
τ −1 τ −2 τ −3 τ −l τ −r 2 | 1 0 − −4 | 1 2 − −6 | 3 2 − − 8 | 3 4 ··· 2l + 2 | l + 1 l − − 2r + 2 | r r + 1 2¡¡ 2A 2¡¡ 2A 2¡¡ 2A 2 ¡¡ 2¡¡ 2A 2 ¡¡ ¡ AAU ¡ AAU ¡ AAU ¡ ¡ AAU ¡ 1 | 0 0 − −3 | 1 1 − −5 | 2 2 − − 7 | 3 3 ··· 2l + 1 | l l − − − 2r + 1 | r r ··· A A A A A A 22¡¡ 2 2¡¡ 2 2 ¡¡ 2 22 ¡¡ 2 AAU ¡ AAU ¡ AAU ¡ AAU AAU ¡ AAU 2 | 0 1 − −4 | 2 1 − −6 | 2 3 − − 8 | 4 3 ··· 2l + 2 | l l + 1 − − 2r + 2 | r + 1 r
Figure 3.9. The preprojective component P(K6) associated to the critical equipped poset K6.
0 00 2. ./(I(x ) ⊕ I(x )) = I(x), I(K6) = ./ I(K1), ( l 2l + 1 | l + 1 l + 1 if l = 2k, τ I(K6) = 2r + 1 | r + 1 r + 1 if r = 2k + 1. ( 2l + 2 | l + 1 l + 2 if l = 2k, τ lI(1) = 2r + 2 | r + 2 r + 1 if r = 2k + 1. ( 2l + 2 | l + 2 l + 1 if l = 2k, τ lI(2) = 2r + 2 | r + 1 r + 2 if r = 2k + 1.
τ r τ l τ 3 τ 2 τ 1 2r + 2 | r + 2 r + 1 − 2l + 2 | l + 1 l + 2 ··· 8 | 5 4 − −6 | 3 4 − −4 | 3 2 − − 2 | 1 2 2A 2 ¡¡ 2A 2A 2¡¡ 2A 2¡¡ 2A 2 ¡¡ 2A AAU ¡ AAU AAU ¡ AAU ¡ AAU ¡ AAU ··· 2r + 1 | r + 1 r + 1 − − 2l + 1 | l + 1 l + 1 ··· 7 | 4 4 − −5 | 3 3 − −3 | 2 2 − − 1 | 1 1 A A A A 2¡¡ 2 2 ¡¡ 2¡¡ 2 2¡¡ 2 2¡¡ 2 2 ¡¡ ¡ AAU ¡ ¡ AAU ¡ AAU ¡ AAU ¡ 2r + 2 | r + 1 r + 2 − 2l + 2 | l + 2 l + 1 ··· 8 | 4 5 − −6 | 4 3 − −4 | 2 3 − −2 | 2 1
Figure 3.10. The preinjective component I(K6) associated to the critical equipped poset K6.
From now on, we state a similar result to the Theorem 22, for the critical equipped posets K8. We do that without a proof, provide that it is enough to adapt the arguments to the poset K8.
In Table 3.9, we present the indecomposable projective (P (∅), T (1), P (2), P (3)), injective (I(1), P (1, 2) = I(2), P (1, 3) = I(3), P (1, 2, 3) = I(K8)) representations and the dimen- sion vectors of these representations and the diagram associated to the sincere critical equipped poset K8 is presented. CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 82
P (∅) T (1) P (2) P (3) I(1) P (1, 2) P (1, 3) P (1, 2, 3)
1 |0 0 0 2 |1 0 0 1 |0 1 0 1 |0 0 1 2 |1 2 2 1 |1 1 0 1 |1 0 1 1 |1 1 1
Table 3.9. The indecomposable projective and injective representations of the critical equipped poset K8 and its corresponding dimension vector [67].
⊗ K8 = P = 1 2e 3 e
The following result describes the components P(K8) and I(K8) of Γ(K8).
Theorem 23. For the sincere critical equipped poset K8, the preprojective P(K8) and preinjective I(K8) components of its Auslander-Reiten quiver Γ(K8) can be computed by using the following identities, for which we assume that k, l, r ∈ Z with k ≥ 0, l = 2k and r = 2k + 1.
0 00 1. ./(P (1 ) ⊕ P (1 )) = T (1), ./(P (2)) = P (2), ./(P (3)) = P (3), P(K8) = ./ P(K1), ( −l 2l + 1 | l l l if l = 2k, τ P (∅) = 2r + 1 | r r r if r = 2k + 1. ( 2l + 2 | l + 1 l l if l = 2k, τ −l(T (1)) = 2r + 2 | r r + 1 r + 1 if r = 2k + 1. ( l + 1 | k k + 1 k if l = 2k, τ −l(P (2)) = r + 1 | k + 1 k k + 1 if r = 2k + 1. ( l + 1 | k k k + 1 if l = 2k, τ −l(P (3)) = r + 1 | k + 1 k + 1 r if r = 2k + 1.
τ −1 τ −2 τ −3 τ −l τ −r 2 | 1 0 0 − −4 | 1 2 2 − −6 | 3 2 2 − −8 | 3 4 4 ··· 2l + 2 | l + 1 l l − − 2r + 2 | r r + 1 r + 1 ¡ A ¡ A ¡ A ¡ ¡ A ¡ 2¡ 2¡ 2¡ 2 222 ¡ 2¡ 22 ¡ ¡ AAU ¡ AAU ¡ AAU ¡ ¡ AAU ¡ 1 | 0 0 0 − −3 | 1 1 1 − −5 | 2 2 2 − −7 | 3 3 3 ··· 2l + 1 | l l l − − 2r + 1 | r r r ··· CA ¡¡£ CA ¡¡£ CA ¡¡£ CA CA ¡¡£ CA C AAU ¡ £ C AAU ¡ £ C AAU ¡ £ C AAU C AAU ¡ £ C AAU £ £ £ £ C 1 | 0 1 0 − −C 2 | 1 0 1 − −C 3 | 1 2 1 − −C 4 | 2 1 2 ··· C l + 1 | k k + 1 k − −C r + 1 | k + 1 k k + 1 C £ C £ C £ C C £ C C £ C £ C £ C C £ C CW £ CW £ CW £ CW CW £ CW 1 | 0 0 1 − −2 | 1 1 0 − −3 | 1 1 2 − −4 | 2 2 1 ··· l + 1 | k k k + 1 − − r + 1 | k + 1 k + 1 k
Figure 3.11. The preprojective component P(K8) of the critical equipped poset K8. CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 83
0 00 2. ./(I(1 ) ⊕ I(1 )) = I(1), ./(I(2)) = I(2), ./(I(3)) = I(3), I(K8) = ./ I(K1), ( l 2l + 1 | l + 1 l + 1 l + 1 if l = 2k, τ I(K8) = 2r + 1 | r + 1 r + 1 r + 1 if r = 2k + 1. ( 2l + 2 | l + 1 l + 2 l + 2 if l = 2k, τ lI(1) = 2r + 2 | r + 2 r + 1 r + 1 if r = 2k + 1. ( l + 1 | k + 1 k k + 1 if l = 2k, τ lI(2) = r + 1 | k + 1 k + 2 k + 1 if r = 2k + 1. ( l + 1 | k + 1 k + 1 k if l = 2k, τ lI(3) = r + 1 | k + 1 k + 1 k + 2 if r = 2k + 1.
τ r τ l τ 3 τ 2 τ 1 2r + 2 | r + 2 r + 1 r + 1 − 2l + 2 | l + 1 l + 2 l + 2 ··· 8 | 5 4 4 − −6 | 3 4 4 − −4 | 3 2 2 − − 2 | 1 2 2 A ¡ A A ¡ A ¡ A ¡ A 2¡ 22 2¡ 2¡ 2¡ 2 222 AAU ¡ AAU AAU ¡ AAU ¡ AAU ¡ AAU ··· 2r + 1 | r + 1 r + 1 r + 1 − 2l + 1 | l + 1 l + 1 l + 1 ··· 7 | 4 4 4 − −5 | 3 3 3 − −3 | 2 2 2 − − 1 | 1 1 1 ¡¡ £ C A ¡¡ £ ¡¡ £ CA ¡¡ £ CA ¡¡ £ CA ¡¡ £ ¡ £ C AAU ¡ £ ¡ £ C AAU ¡ £ C AAU ¡ £ C AAU ¡ £ £ £ £ £ £ £ r + 1 | k + 1 k + 2 k + 1 C l + 1 | k + 1 k k + 1 ··· 4 | 2 3 2 − −C 3 | 2 1 2 − −C 2 | 1 2 1 − −− C 1 | 1 0 1 £ C £ £ C £ C £ C £ £ C £ £ C £ C £ C £ £ CW £ £ CW £ CW £ CW £ r + 1 | k + 1 k + 1 k + 2 l + 1 | k + 1 k + 1 k ··· 4 | 2 2 3 − −3 | 2 2 1 − −2 | 1 1 2 − −− 1 | 1 1 0
Figure 3.12. The preinjective component I(K8) of the critical equipped poset K8.
3.5 The Auslander-Reiten quiver of the equipped poset Yt in the subcategory addN
In this section, we describe the Auslander-Reiten quiver of subcategories of representations of equipped posets of type Yt.
Definition 26. An equipped poset of type Yt is an equipped poset P with 3t − 2 points, t ≥ 2, t ∈ N such that: P = {1 ≺ 2 ≺ · · · ≺ t − 1} + {t ≺ t + 1 ≺ · · · ≺ 2t − 1} + {2t ¡ 2t + 1 ¡ ··· ¡ 3t − 2} with t − 1 ≺ t, t ¡ 2t, {1, 2,..., 2t − 1} ⊂ P⊗, {2t, 2t + 1,..., 3t − 2} ⊂ P◦.
It should be noted that these posets can also be seen in the following way, P = tf\{t} + g 5 g 5 t + t where | tf\{t} |=| t \{t} |=| t |. The following is the diagram of the equipped posets of type Yt. CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 84 ⊗ 2t − 1 3t − 2
@@ e @ ⊗ 2t − 2 3t − 3 · ·· e ·· · ⊗ t + 1 2t @ @ e @ P = ⊗ t
⊗ t − 1 . . . ⊗ 2
⊗ 1
Figure 3.13. The diagram associate with an equipped poset of type Yt.
As before, we assume the notations P (∅), P (i), P (j, k), T (j), T (j, k) and Gh(j, k) (with h ∈ {1, 2}), for indecomposable representations of posets of type Yt.
Henceforth, we let add Yt denote the class of objects in the category of representations rep P, of an equipped poset P of type Yt, as follows:
add N = add Yt = hP (k),T (i),T (i, j),P (i, l),Gh(i, l)i, where : h ∈ {1, 2}, 1 ≤ i ≤ 2t − 1, (3.1) 1 < j ≤ 2t − 1, 1 ≤ k ≤ 4t − 3, 2t ≤ l ≤ 3t − 2 if t + 1 ≤ i ≤ 2t − 1.
Theorem 24. The following statements hold, for the equipped posets of type Yt = P.
i) If t = 2 then P is of finite representation type.
ii) If t = 3 then P is of finite growth representation type.
iii) If t ≥ 4 then P is of wild representation type.
Proof. The proof is based on the definition of the evolvent of an equipped poset, as follows:
i) If t = 2 then the diagrams of P and its corresponding evolvent Pb have the following forms:
⊗ 3 4 30 4 300 @ @ @ @ e e@ e@ e @ ⊗ 2 @ 0 @ 00 P = Pb = 2 2 e e ⊗ 1 10 100 e e CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 85
P is of finite representation type provided that Pb does not contain Kleiner's critical posets.
ii) If t = 3 then the diagram of P, and its corresponding evolvent Pb, have the following forms:
⊗ 5 7 50 7 500 @ @ @ e u@ u u @ ⊗ 4 6 @ 40 6 400 @ @ @ @ e u@ u@ u @ ⊗ 3 @ 0 @ 00 P = Pb = 3 3 e e ⊗ 2 20 200 e e ⊗ 1 10 100 e e
In this case, Pb contains the subposet {40 < 50, 6 < 7, 400 < 500} = (2, 2, 2). Thus P is of infinite representation type. Moreover, since Pb does not contains Nazarova's hypercritical posets N1,...,N6, P is of tame representation type. Thus P is of finite growth representation type provided that Pb does not contain the critical posets of type G1,...,G7.
iii) If t ≥ 4 then the diagrams of P, and its corresponding evolvent Pb, have the following forms:
3t − 2 ⊗2t − 1 (2t − 1)0 3t − 2 (2t − 1)00 @@ b @ ⊗2t − 2 3t − 3 b@ (2t − 2)0 b3t − 3 (2t − b 2)00 · ·· · . · ·· · b ·· . ·· ⊗t + 3 2t + 2 0 00 b(t + 3) b2t + 2 (t + b 3) @@ b @ ⊗t + 2 2t + 1 b@ (t + 2)0 b2t + 1 (t + br 2)00
@@ @ ⊗t + 1 2t b r@ (t + 1)0 r2t (t + r 1)00 @ P : @ @@ @@ ⊗t b Pb : rt0 rt00 r ⊗t − 1 b(t − 1)0 b(t − 1)00 ...... ⊗4 b40 b400 ⊗3 b30 b300 ⊗2 b20 b200 ⊗1 b10 b100 b b As a consequence of Theorem 6, we conclude that P is of wild representation type provided that Pb contains the subposet {(t + 1)0 < (t + 2)0, 2t < 2t + 1, (t + 1)00 < 00 00 (t + 2) < (t + 3) } = (2, 2, 3) = N3.
The following theorem characterizes monomorphisms and epimorphisms in subcategories of type add Yt. CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 86
Theorem 25. Let ϕ : U −! V be a non zero morphism in add Yt, with U, V ∈ add Yt. Then, the following statements hold.
1. ϕ is a monomorphism if and only if it satisfies one of the following conditions:
(i) U = P (i) and V = P (j), where i, j ∈ 1g or i, j ∈ 1O; U = P (i) and V = T (j, k), where i ∈ kg; U = P (i) and V = P (j, k), where i ∈ kg + jO; U = P (i) and V = Gh(k, j), where i ≥ j ∈ tO. (ii) U = T (i) and V = T (j), where i ≥ j; U = T (i) and V = T (k, j), where i ≥ k; U = T (i) and V = Gh(k, j), where i ≥ k. (iii) U = T (i, j) and V = T (k, l), where i ∈ kg and j ∈ lg. (iv) U = P (i, j) and V = P (k, l), where i ∈ kg and j ∈ lO;
U = P (i, j) and V = P (k), where k ∈ tf. 0 0 (v) U = Gh(i, j) and V = Gh(k, l), where h ≤ h , i > k and j ≤ l. 2. ϕ is an epimorphism if and only if it satisfies one of the following conditions:
(a) U = V = P (i); (b) U = T (i) and V ∈ {P (i),T (i)}; (c) U = T (i, j) and V ∈ {P (i),T (i, j)}; (d) U = V = P (i, j);
(e) U = Gh(i, j) and V ∈ {P (i, j),Gh(i, j)};
Proof.
1. (i) If ϕ : U ! V is a morphism, with U = P (i), then dimF U0 = 1. Thus, if V = P (j) with i, j ∈ 1g, i ∈ jg, then dimF V0 = 1. Therefore if ϕ 6= 0, it is a monomorphism. Worth noting, that if j > i then ϕ = 0.
If U = P (i) and V = T (j, k), then ϕ = 0 if i ∈ kf\{k}. On the other hand, if i ∈ kg and ϕ 6= 0, then dimF U0 = 1, dimF V0 = 2 and ϕ is a monomorphism.
If U = P (i), V = Gh(k, j), with i ∈ tf, then it follows that ϕ = 0. In case ϕ 6= 0, we get i ∈ jO,dimF U0 = 1, dimF V0 = 2 and so ϕ is a monomorphism. If U = P (i), V = P (k, j) and ϕ 6= 0, then i ∈ kg + jO, dimF U0 = 1, dimF V0 = 1 and thus ϕ is a monomorphism.
(ii) If U = T (i) and either V = P (j) or V = P (k, j), then we have that dimF U0 = 2, dimF V0 = 1 and ϕ is not a monomorphism. On the other hand, if ϕ 6= 0, U = T (i) and V = T (k, j), then i ∈ kg, dimF U0 = dimF V0 = 2 and the matrix representation of ϕ has the form:
−1 α0 −λ α1 M(ϕ) = −1 . (3.2) α1 α0 + µλ α1
Therefore ϕ is a monomorphism.
If U = T (i), V = Gh(k, j) and ϕ 6= 0, then i ∈ kg, dimF U0 = dimF V0 = 2 and M(ϕ) is the form of the matrix representation of ϕ. Hence ϕ is a monomorphism. CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 87
(iii) If U = T (i, j), V = T (k, l) and ϕ 6= 0, we get i ∈ kg, j ∈ lg, dimF U0 = dimF V0 = 2 and the matrix representation of ϕ has the form M(ϕ). Thus ϕ is a monomorphism. Besides, if U = T (i, j) and V = P (k) or V = P (k, l), then ϕ is not a monomorphism.
If U = P (i, j), V = P (k) and ϕ 6= 0, it follows that k ∈ tf, dimF U0 = dimF V0 = 1 and ϕ is a monomorphism.
(iv) If U = P (i, j), V = P (k, l) and ϕ 6= 0, then i ∈ kg, j ∈ lO and dimF U0 = dimF V0 = 1, and thus ϕ is a monomorphism.
(v) The morphisms ϕ : U ! V 6= 0, with U = Gh(i, j) and V = P (k) or V = P (k, l), are not monomorphisms. Whereas if V = T (j) or V = T (j, l), ϕ = 0. 0 Finally, whether ϕ 6= 0, h ≤ h , i ≥ k, j ≤ l and V = Gh0 (k, l), then M(ϕ) is the form of the matrix representation of ϕ, and thus ϕ is a monomorphism.
2. (a) If ϕ : U ! V is a morphism, with U = P (i) and V = T (i) or V = T (i, j), we have ϕ = 0. Whereas whether ϕ 6= 0, V = P (j) or V = Gh(k, l), then j = i or i ∈ lO thus V = P (i) in the first case, whereas in the second case it holds that Uk = 0 and Vk 6= 0, and hence ϕ is not an epimorphism. (b) If ϕ : U ! V 6= 0 is a morphism, with U = T (i) and V = T (j), then i ≤ j. In the case that i = j, we have that ϕ is an epimorphism; whether i > j it holds that Uj = 0 and Vj 6= 0 and therefore ϕ is not an epimorphism. If U = T (i), V = P (j) and ϕ 6= 0, then i ≥ j. Whether i > j, it holds that Uj = 0 and Vj 6= 0 and thus ϕ is not an epimorphism. Therefore if ϕ 6= 0, U = T (i) and V = P (i), then ϕ is an epimorphism. We note that if ϕ 6= 0 with U = T (i), V = T (k, l) is not an epimorphism due that dimG Ul = 1 and dimG Vl = 2. Also, the morphisms T (i) ! P (k, l) or T (i) ! Gh(k, l) are not epimorphisms. (c) If ϕ : U ! V 6= 0 is a morphism, with U = T (i, j) and V = P (k), then i ∈ kg. If i > k, we get Uk = 0 and Vk 6= 0 and thus ϕ is not an epimorphism. In the case that k = i, we have that ϕ is an epimorphism. We note that, if V = T (i, j), ϕ is an epimorphism. On the other hand, if U = T (i, j) and V = P (k, l) or V = Gh(k, l), then ϕ is not an epimorphism since Ul = 0 and Vl 6= 0. (d) If ϕ : U ! V 6= 0 is a morphism with U = P (i, j) and V = P (k, l), then i ∈ kg and j ∈ lO. If i > k or j > l, we have that ϕ is not an epimorphism due that Uk = 0, Vk 6= 0 or Ul = 0, Vl 6= 0, and thus ϕ is an epimorphism if and only if U = V = P (i, j).
(e) If ϕ : U ! V 6= 0 is a morphism with U = Gh(i, j) and V = Gh0 (k, l), then 0 0 h = h , due that dimG Ul < dimG Vl if h < h . Moreover, i = k and j = l due that otherwise Uk = 0 and Vk 6= 0 or Ul = 0 and Vl 6= 0 and under this circumstances ϕ is not an epimorphism.
If ϕ : U ! V 6= 0 is a morphism with U = Gh(i, j) and V = P (k, l), then i ≥ k and j ≥ l, since i > k or j > l. Hence ϕ is not an epimorphism and thus i = k and j = l.
If ϕ : U ! V 6= 0 is a morphism with U = Gh(i, j) and V = Gh(k, l), then i ≥ k and j ≥ l. Since i > k or j > l, we have that ϕ is not an epimorphism and thus i = k and j = l.
Corollary 6. For the equipped posets P of type Yt, CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 88 ⊗ 2t − 1 3t − 2
@@ e @ ⊗ 2t − 2 3t − 3 · ·· e ·· · ⊗ t + 1 2t @ @ e @ P = ⊗ t
⊗ t − 1 . . . ⊗ 2
⊗ 1 the following statements hold.
(i) The only indecomposable projective representations of add Yt are: P (j) for j ∈ tO, and T (i) for i ∈ (2t − 1)f.
(ii) P (1) is the only indecomposable injective representation of add Yt .
Proof.
(i) For j ∈ tO, we have the following cases:
1. Any morphism ϕ : P (j) ! V 6= 0, with V = P (i), i ∈ tf can be factorized in the ϕ ϕ form P (j) −!1 U −!2 P (i), where U ∈ {P (i),T (i),T (i, k)}. If U = P (i), then ϕ1 = ϕ and ϕ2 is the identity. If U = T (i) then ϕ1 (with ϕ1(t) = (t, 0), for any t) is an inclusion and ϕ2(t, 0) = ϕ(t). The same can be done if U = T (i, j), and for all these cases ϕ2 is an epimorphism. 2. If V ∈ {T (k),T (k, l)} then ϕ : P (j) ! V = 0. 3. If ϕ : P (j) ! V 6= 0, where V is an indecomposable representation of the form ϕ1 ϕ2 Gh(k, l), then j ∈ lO and thus ϕ can be factorized in the form P (j) −! U −! Gh(k, l) with U = Gh(k, l), ϕ1 = ϕ and ϕ2 is the identity. 4. If ϕ : P (j) ! V 6= 0, where V is an indecomposable representation of the form ϕ ϕ P (k, l), then ϕ can be factorized in the form P (j) −!1 U −!2 P (k, l). If U = P (k, l), then ϕ1 = ϕ and ϕ is the identity.
Since the cases 1. − 4. cover all morphisms ϕ : P (j) ! V , with V ∈ add Yt, we conclude that P (j), j ∈ tO is projective. For i ∈ tg, we have the following cases:
(a) If ϕ : T (j) ! P (i) 6= 0 is a morphism and i ∈ (2t − 1)f, then ϕ can be factorized ϕ ϕ in the form T (j) −!1 U −!2 P (i), with U ∈ {P (i),T (i),T (i, k)}. If U = P (i) then ϕ1 = ϕ and ϕ2 is the identity, if U = T (i), then ϕ is an epimorphism. (b) If ϕ : T (i) ! T (j) 6= 0 is a morphism with i ≥ j, then ϕ can be factorized in ϕ1 ϕ2 the form T (i) −! U −! T (j). If ϕ2 is an epimorphism, then U = T (j), ϕ1 = ϕ and ϕ2 is the identity. CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 89
(c) If ϕ : T (i) ! T (k, l) 6= 0 is a morphism, with i ≥ k, then ϕ can be factorized ϕ1 ϕ2 in the form T (i) −! U −! T (k, l). If ϕ2 is an epimorphism, then U = T (k, l), ϕ1 = ϕ and ϕ2 is the identity. (d) If ϕ : T (i) ! P (k, l) 6= 0 is a morphism, with i ∈ kg, then ϕ can be fac- ϕ1 ϕ2 torized in the form T (i) −! U −! P (k, l). If ϕ2 is an epimorphism, then U ∈ {Gh(k, l),P (k, l)}. In the first case, ϕ1 has the form (3.2) and ϕ2 = ϕ, in the second case, ϕ1 = ϕ and ϕ2 is the identity.
(e) If ϕ : T (i) ! Gh(k, l) 6= 0 is a morphism, with i ∈ kg, then ϕ can be factorized in ϕ1 ϕ2 the form T (i) −! U −! Gh(k, l). If ϕ2 is an epimorphism, then U = Gh(k, l), ϕ1 = ϕ and ϕ2 is the identity.
The cases (a)-(e) permit us to conclude that indecomposable representations of the form T (i) are projective.
Finally, we note that HomF (P (i),T (j)) = 0, for any i, j. Moreover 0 O 0 HomF (P (i),T (j, j )) = 0 if i ∈ jf + t , and HomF (P (i),Gh(j, j )) = 0, for any i, j. Therefore, no other indecomposable projective representation exists in add Yt.
(ii) For the injective case, we consider the following cases:
(a) If ϕ : P (i) ! P (1) 6= 0 is a morphism, then ϕ can be factorized in ϕ1 ϕ2 the form P (j) −! U −! P (1). If ϕ1 is a monomorphism, then U ∈ {P (j),T (j, k),P (j, k), Gh(k, j)}. In the first case i, j ∈ 1g or i, j ∈ 1O, ϕ is an inclusion and ϕ2 = ϕ; in the second case, i ∈ kg, for any t ∈ F , ϕ1(t) = (ϕ(t), 0) and ϕ(t1, t2) = t1, for 2 any (t1, t2) ∈ F ; in the third case, i ∈ kg + jO, ϕ1 = ϕ and ϕ2 is an inclusion; in the fourth case, i ≥ j ∈ tO, ϕ1(t) = (ϕ(t), 0) and ϕ2(t1, t2) = t1. (b) If ϕ : T (i) ! P (1) 6= 0 is a morphism, then ϕ can be factorized in ϕ1 ϕ2 the form T (i) −! U −! P (1). If ϕ1 is a monomorphism, then U ∈ {T (j),T (k, j),Gh(k, j)}. In the first case i ≥ j, ϕ1 has the form (3.2) and ϕ2 = ϕ; in the second case, i ≥ k, ϕ1 has the form (3.2) and ϕ2 = ϕ; in the third case, i ≥ k, ϕ1 has the form (3.2) and ϕ2 = ϕ. (c) If ϕ : T (i, j) ! P (1) 6= 0 is a morphism, then ϕ can be factorized in the form ϕ1 ϕ2 T (i, j) −! U −! P (1). If ϕ1 is a monomorphism, then U ∈ {T (k, l)}, i ∈ kg, j ∈ lg, ϕ1 with the form (3.2) and ϕ2 = ϕ. (d) If ϕ : P (i, j) ! P (1) 6= 0 is a morphism, then ϕ can be factorized in the form ϕ1 ϕ2 P (i, j) −! U −! P (1). If ϕ1 is a monomorphism, then U ∈ {P (k),P (k, l)}. In g the first case, k ∈ tf, ϕ1 = ϕ and ϕ2 is an inclusion; in the second case, i ∈ k , j ∈ lO, ϕ2 = ϕ and ϕ1 is an inclusion (ϕ1(t) = t for any t ∈ F ).
(e) If ϕ : Gh(i, j) ! P (1) 6= 0 is a morphism, then ϕ can be factorized in the ϕ1 ϕ2 0 form Gh(i, j) −! U −! P (1). If ϕ1 is a monomorphism, then U = Gh(k, l), 0 2 h ≤ h , i > k, j ≤ l, ϕ1(z) = z for any z ∈ F and ϕ2 = ϕ.
Cases (a)-(e) allow us to conclude that P (1) is injective. In fact, the arguments de- scribed above permit us to conclude that there is no other indecomposable injective.
Theorem 25 and Corollary 6 allows us to define the irreducible morphisms as follows. CHAPTER 3. THE AUSLANDER-REITEN QUIVER OF SOME EQUIPPED POSETS 90
Corollary 7. For a morphism ϕ : U ! V ∈ add Yt(U, V ), the following statements hold. 1. ϕ : P (i) ! P (j) is irreducible if and only if i = j + 1, i, j ∈ tO. 2. ϕ : T (i, j) ! P (i), ϕ : T (i + 1, j) ! T (i, j) and ϕ : T (i + 1, j) ! T (i, j − 1) are irreducible morphisms. 3. ϕ : T (i) ! T (i, 2t − 1) and ϕ : T (i) ! T (i, i − 1) are irreducible morphisms. 4. ϕ : P (k, l) ! T (k − 1, l − 1), k ∈ tg, l ∈ tO, P (k, l) ! P (k − 1, l) and P (k, l) ! P (k, l − 1), are irreducible morphisms.
5. ϕ : G1(k, l) ! G1(k, l − 1), ϕ : G1(k, l) ! G1(k − 1, l) and ϕ : G1(k, l) ! G2(k, l) are irreducible morphisms.
6. ϕ : G2(k, l) ! G2(k, l − 1), ϕ : G2(k, l) ! G2(k − 1, l), ϕ : G2(k, l) ! P (k, l) and ϕ : P (i) ! G1(2t − 1, i), for i ∈ tO, are irreducible morphisms. 7. ϕ : P (2t − 1) ! P (2t − 1, 3t − 2) is an irreducible morphism.
Proof. By Theorem 25, it suffices to see that:
g O 1. HomF (P (i),P (j)) = 0, if i ∈ 1 and j ∈ (if\{i}) + t ,
2. HomF (P (i),T (i)) = 0, for i ∈ 1g, 0 0 0 0 0 0 3. HomF (P (k, l),Gh(k , l )) = 0, if k, k ∈ 1g, k < k , l, l ∈ tO, l < l , 0 0 4. HomF (G2(k, l),G1(k , l )) = 0,
5. HomF (T (i, j),T (k)) = 0, if j ≤ k.
XPX(5, 10) ¨* XXz P (5, 9) ¨ J] ¨* XX ¨ r J¨ XXz P (5, 8) X¨P (6, 10) J] ¨* ¨* XXXz ¨¨J r J¨ J] ¨ J] ¨*XPX(6, 9)J ¨J r J ¨ r J ¨ J] XXz ¨P (6, 8) X¨P (7, 10) ¨* J J X ¨J r J ¨ JJ]XG2(5, 10) 6J] XXz X¨P (7, 9) ¨* XX J X J ¨J¨ r J XzJXG2(5, 9) rJ J] XXz ¨P (7¨, 8) 6 ¨* XX J J r J¨ XzJ G2(5, 8) J r JJ]¨XG2(6, 10) 6 ¨* ¨* J XX ¨ J r ¨ J r XzJ¨XG2(6, 9) 6 J ¨¨ 6 ¨* XX ¨ r J r J¨ XzJ¨G2(6, 8) J¨G2(7, 10) 6G1(5¨*, 10) XXX ¨ J ¨*r X¨XX XzJ¨XG2(7, 9)¨ Xz6XG1(5, 9) 6 X¨X ¨ ¨*r XX r XzJ¨G2(7r , 8)¨ Xz G1(5, 8) 6X¨G1(6, 10) ¨* ¨*r XXXz6 ¨ r ¨ ¨ X¨G1(6, 9) ¨ ¨ ¨*r XXXz ¨ r ¨* HT (1) ¨ r ¨ G1(6, 8) ¨ H P (7) XG1(7, 10) ¨ ¨* ¨ Hj T (1, 7) XXXz ¨ r ¨ ¨* HT (2) r ¨* H @I @I@ XG1X(7, 9) ¨ ¨ H ¨ H XXz ¨ r ¨T (3) Hj ¨T (2, 7) Hj T (1, 6) r @ r @I G1(7, 8) ¨* H r ¨* H r ¨* H @ @ ¨ H ¨ HHj ¨ HHj @ r @I ¨T (4) Hj ¨T (3, 7) ¨T (2, 6) T (1, 5) @ @ r @ ¨¨* H r ¨¨* H r ¨¨* H r ¨¨* H P (8) HHj ¨ HHj ¨ HHj ¨ HHj @ @ @ ¨T (5) HT (4, 7) HT (3, 6) HT (2, 5) HT (1, 4) @ 6 ¨¨* HH r ¨¨* H r ¨¨* H r ¨¨* H r ¨¨* H @ @ ¨T (6) Hj ¨T (5, 7) Hj ¨T (4, 6) Hj ¨T (3, 5) Hj ¨T (2, 4) Hj T (1, 3) @ r ¨* H r ¨¨* H r ¨¨* H r ¨¨* H r ¨¨* H r ¨¨* H @ @ ¨¨ HHj ¨ HHj ¨ HHj ¨ HHj ¨ HHj ¨ HHj @ @ P (9) HT (7) r ¨¨* HT (6, 7)r ¨¨* HT (5, 6)r ¨¨* HT (4, 5)r ¨¨* HT (3, 4)r ¨¨* HT (2, 3)r ¨¨* HT (1, 2) 6 ¢ HHj ¨ HHj ¨ HHj ¨ HHj ¨ HHj ¨ HHj ¨ HHj @ @ r ¢ r P (7) r P (6) r P (5) r P (4) r P (3) r P (2) r P (1) @ @ ¢ @ P (10) r r r r r r r 6¢ @ r¢ @¢ P (∅) r Figure 3.14. The Auslander-Reiten quiver Γ(Y4) of the equipped poset Y4 in the subcategory add N. CHAPTER 4
Applications of equipped posets to number theory and differential equations
In this chapter, we give some applications of the theory of representation of equipped posets. In particular it is given a solution to a problem posed by Andrews in 1987 re- garding partitions [1]. We also give a categorification of Delannoy numbers by using some suitable lattice representations. Finally explicit solutions of non-linear systems of differen- tial equations of the form X0(t) − X2(t)A = A, (where A is a suitable l × l-integer matrix) are given. Such solutions arise from the introduction of an interpretation of the classical calculus which we have called Zavadskij calculus [17, 28, 35, 59, 60].
4.1 Representation of equipped posets to generate Delan- noy numbers
In this section, Delannoy numbers are interpreted as dimensions of suitable representations of some equipped posets induced by compositions of integer numbers.
Delannoy numbers were introduced by Henri-Auguste Delannoy (1833-1915). He inves- tigated the different moves on a chessboard and observed that such numbers arise by investigating the queen movement la marche de la Reine [35, 59, 60].
For integer numbers i and j, Delannoy numbers satisfy the recurrence relation:
d(i, j) = d(i − 1, j) + d(i, j − 1) + d(i − 1, j − 1), (4.1) d(0, 0) = 0, d(i, j) = 0, if i < 0 and j < 0.
The central Delannoy numbers d(i, i) = {1, 3, 13, 63, 321, 1683, 8989,... } appears as the sequence A001850 in the On-Line Encyclopedia of Integer Sequences (OEIS).
91 CHAPTER 4. APPLICATIONS OF EQUIPPED POSETS TO NUMBER THEORY AND DIFFERENTIAL EQUATIONS92
According to Sulanke, very few combinatorial elements are known to be counted by these numbers. Actually, he describes in [60] 29 configurations which are counted by central Delannoy numbers.
4.1.1 Integer partitions
A partition λ of a positive integer n is a nonincreasing sequence of positive integers λ1 ≥ λ2 ≥ · · · ≥ λm such that: m X n = λi. i=1 A composition is a partition for which the order of its parts matters [2].
For instance, {1, 1, 1}, {2, 1}, and {3} are the three partitions of 3 whereas {1, 1, 1}, {2, 1}, {1, 2} and {3} are the four compositions of 3.
Regarding partitions and compositions, there are numerous open problems. For instance, in 1987 G.E. Andrews proposed the following problems [1].
(1) For what sets, of positive integers S and T , is P (S, n) = P (T, n − a) for n ≥ a and a fixed?
(2) For each pair S and T , which answer question (1), can a bijection be found between the partitions of n into elements of S and the partitions of n − a into elements of T ?
For a = 1, some identities introduced by Gessel and Stanton imply the solutions:
S = {n | n odd or n ≡ ±4, ±6, ±8, ±10 mod 32}, (4.2) T = {n | n odd or n ≡ ±2, ±8, ±12, ±14 mod 32}.
S = {n | n ≡ ±1, ±4, ±5, ±6, ±7, ±9, ±10, ±11, ±13, ±15, ±16, ±19 mod 40}, (4.3) T = {n | n ≡ ±1, ±3, ±4, ±5, ±9, ±10, ±11, ±14, ±15, ±16, ±17, ±19 mod 40}.
The above problem is still open if we consider integer compositions.
4.1.2 An advance to the Andrews problem
In this section, we define ordered compositions whose structure allows to give an advance to the problem posed by Andrews.
Let (D, ¢) be a partially ordered set of integer compositions {x1, x2, x3, x4} such that:
1. xi ≥ 0, 1 ≤ i ≤ 4, 2. At least two of its elements are positive,
3. x2 = x4, and the difference x3 − x1 ≥ 0. CHAPTER 4. APPLICATIONS OF EQUIPPED POSETS TO NUMBER THEORY AND DIFFERENTIAL EQUATIONS93
0 0 0 0 0 0 0 Besides, {x1, x2, x3, x4} ¢ {x1, x2, x3, x4} if and only if x1 ≤ x1, x3 ≤ x3, x2 ≤ x2 and 0 x4 ≤ x4.
It is clear that if Dn denotes the set of compositions of type D of a fixed integer n ≥ 2, then: [ D = Dn. n≥2
Theorem 26. The poset of compositions Dn of type D, of a fixed integer n ≥ 2, is a sum n of b 2 c chains.
Proof. The set of minimal points of Dn consists of all compositions {x, y, z, w} such that n y = w = 0, x + z = n. Thus x ∈ {1, 2,..., b 2 c}.
As an example, we note that {2, 0, 4, 0}¢{1, 1, 3, 1}¢{0, 2, 2, 2} is a chain of compositions of type D of 6. In the following Figure 4.1, we show examples of compositions of type Dn. We let Dn(k0) denotes the set of all compositions {x1, x2, x3, x4} with fixed difference x3 − x1 = k0. n Remark 4.1. Note that, each one of the b 2 c chains of Theorem 26, which we denote by Dn(k0), can be build in the following form: We begin by finding all possible pairs x, y such that x ≤ y and x + y = n (of course, y − x = k0). Then {x, 0, y, 0} is a composition of the number n of type D. Where Dn(k0) = {{x, 0, y, 0} ¡ {x − 1, 0 + 1, y − 1, 0 + 1} ¡ ··· ¡ {x − l, 0+l, y −l, 0+l}}. These chains attains maximal points whenever x−l = 0 or y −l = k0. Actually, by definition it is easy to see that to each composition {x1, x2, x3, x4} of type D of an integer n = 2k + 1, k ≥ 2 corresponds a unique composition {x1, x2, x3 − 1, x4} of n = 2k. ⊗
⊗ ⊗ ⊗ @ @ @ D5 = D4 ⊗ D7 = D6 ⊗ ⊗ D9 = D8 ⊗ ⊗ ⊗ @ @ @ @ @ @ ⊗ ⊗ ⊗ ⊗ @⊗ ⊗ ⊗ @⊗ @⊗ @ @ @ @ @ @ @ @ @ @ @ @ ⊗ @⊗ ⊗ @⊗ @⊗ ⊗ @⊗ @⊗ @⊗ D5(k0) D5(k1) D7(k0) D7(k1) D7(k2) D9(k0) D9(k1) D9(k2) D9(k3)
Figure 4.1. The diagram of compositions of type Dn with 4 ≤ n ≤ 9.
Regarding the number of antichains in Dn, we have the following result, where the symbol ti denotes the ith triangular number.
2 Theorem 27. The number Dn of two-point antichains contained in Dn is given by the formula: i n b c XX2 hi,j(ti − 2tj), i=1 j=0 CHAPTER 4. APPLICATIONS OF EQUIPPED POSETS TO NUMBER THEORY AND DIFFERENTIAL EQUATIONS94 where 0 if i = n and j = 0, n hi,j = n + 1 − i if i = 2j and 1 ≤ j ≤ b 2 c, 1 otherwise.
2 Dn = {2, 10, 29, 66, 129, 228, 374,... }.
Proof. We proceed by induction on n. If n = 2, the Hasse diagram of the poset D2 has the following form
⊗ a32
a21⊗ ⊗ a22 @ @ @ a11⊗ ⊗ a12
then, the pairs (a11, a12) and (a21, a22) are the only two-point antichains in D2, so
2 D2 = 1 + 1 = t1 − 2t0 + t2 − 2t1 = h1,0(t1 − 2t0) + h2,0(t2 − 2t0) + h2,1(t2 − 2t1), with h2,0 = 0, as required. Assume that the theorem holds for n = k > 2, i.e.
i k b 2 c 2 XX Dk = hi,j(ti − 2tj), i=1 j=0 where 0 if i = k and j = 0, k hi,j = k + 1 − i if i = 2j and 1 ≤ j ≤ b 2 c, (4.4) 1 otherwise. Now, we are going to prove that the theorem holds for n = k + 1. The Hasse diagram of the poset Dk+1 has the following form
⊗ ak+2,k+1
⊗ak+1,k⊗ ak+1,k+1 ...... a32⊗ ··· ⊗ a3k ⊗ a3,k+1 @ @ @ a21 ⊗ a22⊗ ··· ⊗ a2k ⊗ a4,k+1 @ @ @ @ @ @ a11 ⊗ a12⊗ ··· ⊗ a1k ⊗ a1,k+1 | {z } Dk CHAPTER 4. APPLICATIONS OF EQUIPPED POSETS TO NUMBER THEORY AND DIFFERENTIAL EQUATIONS95
As we can see in the previous diagram, Dk is a subposet of Dk+1, then we can obtain 2 2 Dk+1 from Dk as follows:
k+1 k b 2 c b 2 c 2 2 X X Dk+1 = Dk + hk+1,j(tk+1 − 2tj) + tk − 2t0 + (t2l − 2tl). (4.5) j=0 l=1
New summands appear by enumerating the number of two-point antichains formed by each point of the chain {a11 < a21 < a32 < ··· < ak+1,k} in the poset Dk+1. Note that k+1 2 hk+1,0 = 0 and hk+1,j = 1 if 1 ≤ j ≤ b 2 c in (4.5). From the equality for Dk we have that the equation (4.5) can be written in the following form:
i k+1 k k b 2 c b 2 c b 2 c 2 XX X X Dk+1 = hi,j(ti − 2tj) + hk+1,j(tk+1 − 2tj) + tk − 2t0 + (t2l − 2tl) i=1 j=0 j=0 l=1 i k k+1 b c b c XX2 X2 = hi,j(ti − 2tj) + tk − 2t0 + (t2l − 2tl) i=1 j=0 l=1 i k+1 b c XX2 = hi,j(ti − 2tj), i=1 j=0 where 0 if i = k + 1 and j = 0, k+1 hi,j = (k + 1) + 1 − i if i = 2j and 1 ≤ j ≤ b 2 c, (4.6) 1 otherwise.
Note that in (4.4) hk,0 = 0, but with new summand tk − 2t0 = hk,0(tk − 2t0) it changes to hk,0 = 1 in (4.6). Now the summands t2l − 2tl add one unit to the expression k + 1 − i, if k i = 2j, 1 ≤ j ≤ b 2 c from the equation (4.5), changing it by the new expression (k+1)+1−i, k+1 if i = 2j, 1 ≤ j ≤ b 2 c in the equation (4.6).
2 The sequence Dn does not appear in the OEIS.
Example 3. In this example we consider the particular case when n = 6 in the Theorem 2 27. Then, D6 is obtained as follows:
i 6 b 2 c 2 XX D6 = hi,j(ti − 2tj), i=1 j=0 where 0 if i = k and j = 0, hi,j = 7 − i if i = 2j and 1 ≤ j ≤ 3, (4.7) 1 otherwise.
2 D6 = h1,0(t1 − 2t0) + h2,0(t2 − 2t0) + h2,1(t2 − 2t1) + h3,0(t3 − 2t0) + h3,1(t3 − 2t1) + h4,0(t4 − 2t0) + h4,1(t4 − 2t1) + h4,2(t4 − 2t2) + h5,0(t5 − 2t0) + h5,1(t5 − 2t1)
+ h5,2(t5 − 2t2) + h6,0(t6 − 2t0) + h6,1(t6 − 2t1) + h6,2(t6 − 2t2) + h6,3(t6 − 2t3) CHAPTER 4. APPLICATIONS OF EQUIPPED POSETS TO NUMBER THEORY AND DIFFERENTIAL EQUATIONS96
= 1 · 1 + 1 · 3 + 5 · 1 + 1 · 6 + 1 · 4 + 1 · 10 + 1 · 8 + 3 · 4 + 1 · 15 + 1 · 13 + 1 · 9 + 0 · 21 + 1 · 19 + 1 · 15 + 1 · 9 = 129
The structure of the posets Dn allows us to give the following result regarding the An- drews's problems:
Corollary 8. Let C(n, D) be the number of compositions of type D of the positive integer n, then C(2n + 1, D) = C(2n, D), for any n ≥ 1.
2n+1 Proof. For each 1 ≤ i ≤ b 2 c, there are i + 1 compositions {x, y, z, y} of type D with x + y = i, see Remark 4.1. Then
C(2n + 1, D) = t 1 − 1 = C(2n, D). bn+ 2 c+1 As an example, we present in what follows the compositions in the particular cases when n = 4 and n = 5. It proves that the diagrams associated to D4 and D5 of type D are isomorphic, the diagrams of D4 and D5 correspond with the first diagram from the left to the right of the Figure 4.1.
{0, 0, 0, 4}{0, 0, 4, 0}{0, 4, 0, 0}{4, 0, 0, 0} {0, 0, 3, 1}{0, 3, 0, 1}{0, 3, 1, 0}{3, 0, 1, 0}{3, 0, 0, 1}{3, 1, 0, 0} {0, 0, 1, 3}{0, 1, 0, 3}{0, 1, 3, 0}{1, 0, 3, 0}{1, 0, 0, 3}{1, 3, 0, 0} {0, 0, 2, 2}{0, 2, 0, 2}{0, 2, 2, 0}{2, 0, 2, 0}{2, 0, 0, 2}{2, 2, 0, 0} {0, 1, 1, 2}{1, 0, 1, 2}{1, 1, 0, 2}{1, 1, 2, 0} {0, 1, 2, 1}{1, 0, 2, 1}{1, 2, 0, 1}{1, 2, 1, 0} {0, 2, 1, 1}{2, 0, 1, 1}{2, 1, 0, 1}{2, 1, 1, 0} {1, 1, 1, 1} D4 = {{1, 0, 3, 0}, {0, 2, 0, 2}, {2, 0, 2, 0}, {0, 1, 2, 1}, {1, 1, 1, 1}}
Table 4.1. Compositions of the number n = 4 and D4.
{0, 0, 0, 5}{0, 0, 5, 0}{0, 5, 0, 0}{5, 0, 0, 0} {0, 0, 4, 1}{0, 4, 0, 1}{0, 4, 1, 0}{4, 0, 1, 0}{4, 0, 0, 1}{4, 1, 0, 0} {0, 0, 1, 4}{0, 1, 0, 4}{0, 1, 4, 0}{1, 0, 4, 0}{1, 0, 0, 4}{1, 4, 0, 0} {0, 0, 3, 2}{0, 3, 0, 2}{0, 3, 2, 0}{3, 0, 2, 0}{3, 0, 0, 2}{3, 2, 0, 0} {0, 0, 2, 3}{0, 2, 0, 3}{0, 2, 3, 0}{2, 0, 3, 0}{2, 0, 0, 3}{2, 3, 0, 0} {0, 1, 1, 3}{1, 0, 1, 3}{1, 1, 0, 3}{1, 1, 3, 0}{0, 1, 3, 1}{1, 0, 3, 1} {1, 3, 0, 1}{1, 3, 1, 0}{0, 3, 1, 1}{3, 0, 1, 1}{3, 1, 0, 1}{3, 1, 1, 0} {0, 1, 2, 2}{1, 0, 2, 2}{1, 2, 0, 2}{1, 2, 2, 0}{0, 2, 1, 2}{2, 0, 1, 2} {2, 1, 0, 2}{2, 1, 2, 0}{0, 2, 2, 1}{2, 0, 2, 1}{2, 2, 0, 1}{2, 2, 1, 0} {1, 1, 1, 2}{1, 1, 2, 1}{1, 2, 1, 1}{2, 1, 1, 1} D5 = {{1, 0, 4, 0}, {2, 0, 3, 0}, {0, 1, 3, 1}, {0, 2, 1, 2}, {1, 1, 2, 1}}
Table 4.2. Compositions of the number n = 5 and D5. CHAPTER 4. APPLICATIONS OF EQUIPPED POSETS TO NUMBER THEORY AND DIFFERENTIAL EQUATIONS97
D4 {0, 2, 0, 2} 5 {0, 2, 1, 2}D ¨ ¨ ¨ ¨ ¨¨ ¨¨ ¨ ¨ ¨¨ ¨¨ ¨ ¨ {0, 1, 2, 1}{1, 1, 1, 1} {0, 1, 3, 1}{1, 1, 2, 1} H ¨ H ¨ H ¨ H ¨ HH¨¨ HH¨¨ ¨ H ¨ H ¨¨ HH ¨¨ HH ¨ H ¨ H {1, 0, 3, 0}{2, 0, 2, 0} {1, 0, 4, 0}{2, 0, 3, 0}
Figure 4.2. The diagram of the compositions of D4 and D5 of type D.
4.1.3 The category of lattice representations
In this section, we associate to each composition {x1, x2, x3, x4} of type D a pair of points 2 (x1, x2) and (x3, x4) in the usual lattice N .
A weak lattice path (or ascending weak lattice path) L ⊂ D from {x, 0, y, 0} to {0, x, k0, x} containing all points in Dn(k0) is defined in such a way that two different adjacent vertices have the form: {x, y, z, y}, {x − 1, y + 1, z − 1, y + 1}.
Thus, for each vertex in L there are two directions to attain the next vertex in Dn(k0), {(−1, 0), (0, 1)} or {(0, 1), (−1, 0)} [59]. Henceforth, we let Ln(k0) denote the set of all weak lattice paths linking out all the points in Dn(k0).
2 A segment p0p1 in a subset U0 ⊂ N is a two-point set whose elements have the form: p0 = (x0, y0) and p1 = (x1, y0), x0 ≤ x1.
Note that a more general definition of a weak lattice path connecting two segments (for instance, a descending weak lattice path) can be given as follows:
0 z0z0 0 0 If W 0 is a descending weak lattice path from a segment zkzk to a segment z0z0, then zkzk 0 z0z0 the associated adjacent segments to W 0 can be obtained in the following fashion: zkzk
{(xk, yk), (xk + t, yk), (xk + x,k, yk + y,k), (xk + t + x,k, yk + y,k),..., (x0, y0), (x0 + l, y0)},
0 0 where zk = (xk, yk), zk = (xk +t, yk), z0 = (x0, y0), z0 = (x0 +t, y0), t ≥ 0, xk ≤ x0, y0 ≤ yk, l ≥ 0, y,s = −1, x,s = 1, bearing in mind that new orientations {(1, 0), (0, −1)} and {(0, −1), (1, 0)} are considered to connect two consecutive segments. The following Figure 4.3 shows an example of a weak lattice path.
Since definitions of weak lattice paths (ascending or descending) are in general equivalent henceforth we will not specify the class of lattice paths we are dealing with. CHAPTER 4. APPLICATIONS OF EQUIPPED POSETS TO NUMBER THEORY AND DIFFERENTIAL EQUATIONS98
••••••
••••••- @ @ @ ?@ ? ••••••@ @ @ @ ••••••? -@? @ - @ @ @ @ ••••••@? @?
Figure 4.3. A weak lattice path.
Strong lattice paths belong to one of the following classes:
pk (I) Lattice paths Sp0 = {(x0, y0), (x1, y1),..., (xk, yk)} from p0 = (x0, y0) to pk = (xk, yk), where for a given point (xi, yi) it holds that either xi = xi−1 and yi = yi−1 + 1 or xi = xi−1 + 1 and yi = yi−1.
p0 (II) Lattice paths Spk = {(xk, yk), (xk−1, yk−1),..., (x0, y0)} from pk = (xk, yk) to p0 = (x0, y0), where for a given point (xj, yj) it holds that either xj = xj +1 and yj = yj−1 or xj = xj−1 and yj = yj − 1. (III) Products (P )(Q), where P is a lattice path of type (I) and Q is a lattice path of type (II).
The following is an example of a product of strong lattice paths.
••••••
••••••
••••••
••••••- 6 ••••••- ?
Figure 4.4. Product of strong lattice paths.
Regarding the number of weak lattice paths, we have the following result:
Theorem 28. For a fixed k0 ≥ 0, the number of weak lattice paths from {x, 0, x + k0, 0} x to {0, x, k0, x} containing all points in Dn(k0) equals 2 .
Proof. For each y, 0 ≤ y ≤ x, the ways to connect two adjacent vertices {{x, y, z, y}, {x− 1, y + 1, z − 1, y + 1}} are {{x, y, z, y}, {x − 1, y, z − 1, y}, {x − 1, y + 1, z − 1, y + 1}} and {{x, y, z, y}, {x, y + 1, z, y + 1}, {x − 1, y + 1, z − 1, y + 1}}. And the sequences of points in this case consists of the points, {{x, 0, k0 + x, 0}, {x − 1, 1, k0 + x − 1, 1},..., {0, x, k0, x}}. CHAPTER 4. APPLICATIONS OF EQUIPPED POSETS TO NUMBER THEORY AND DIFFERENTIAL EQUATIONS99
Lattice path products
Products of lattice paths (strong or weak) are defined as follows:
2 0 A weak product Pw in the sublattice U0 ⊂ N is defined in such a way that if zkzk is a z0 z z0 zk k 0 0 segment, Sp0 and Sp0 are strong lattice paths and W 0 is a weak lattice path, then zkzk
z0 z z0 z0 z0 P = (Szk ,S k )W 0 0 = (Szk W z0 ,S k W 0 ) w p0 p0 0 p0 zk p0 z0 zkzk k
0 In such a case, we write zkzk ¡ Pw.
A strong product Ps is defined in such a way that:
z z0 z0 P = (Szk , )W 0 0 = (Szk W z0 ,W 0 ) or s p0 ∅ 0 p0 zk z0 zkzk k
z z z0 z0 z0 P = ( ,S k0 )W 0 0 = (W z0 ,S k W 0 ) s ∅ p0 0 zk p0 z0 zkzk k
For z = (x0, y0), Pz,k denotes the set of all the products passing by the segment (x0, y0), (x0 + k, y0). The derivative δ(P ) of a given product P is defined as follows:
[ z0 z0 δ(P ) = {(Szk W z0 ,W zk )} {(W z0 ,S k W 0 )}, w p0 zk z0 zk p0 z0 k (4.8) [ δ(P ) = {(Szk W z0 ,W zk )} {(W z0 , )}. s p0 zk z0 zk ∅
The following Figure 4.5 shows examples of these kind of products.
Weak product Strong product •••••••• ••••••••
•••••••• ••••••••
•••••••• ••••••••
••••••••- - ••••••••- • - 6 6 6 d ••••••••- - ? ? ••••••••- ? •? 6 6 6 d ••••••••- ? -? - ••••••••? -? • - 6 6 6 d ••••••••- - ? ? ••••••••- ? •? d Figure 4.5. Weak and strong products.
2 For a given fixed points (x0, y0), (x0 + m, y0) ∈ N with m > 1, a lattice representation U of an equipped poset P is a system of the form:
U = (U0; Ux | x ∈ P), CHAPTER 4. APPLICATIONS OF EQUIPPED POSETS TO NUMBER THEORY AND DIFFERENTIAL EQUATIONS100
2 where U0 is an m × n-order sublattice of N and (x0, y0) is the minimum of U0. For each x ∈ P, Ux is a system of the form:
(Du; P u ,P u ,...,P u ), x x1,k1 x2,k2 xJ ,kj
u where Dx ⊂ U0 is an m × nx-order sublattice of D0 containing all the products P u ,P u ,...,P u from (x , y ) to (x + k , y ) with x = x + r, r, k ≥ 0, r + k ≤ m. x1,k1 x2,k2 xJ ,kj t t t t t t 0 t
u u u u + Dx ⊆ Dy if x ≤ y (Dy covers Dx and Ax < Ay if y ∈ x ).
S Moreover, x is a weak point if and only if the products in Ux up to the radical Uy are y∈xN all weak. And:
Dx ⊂ Dy if x ≤ y,
Ux ⊂ Uy if x y, (4.9) u u δ(Px ) ⊂ Py if x ¢ y. For z ∈ P
0 S Az = {t ∈ N | (x, t), (x , t) ¡ P }P ∈Uz − Uw. w∈zN
Remark 4.2. P u = SP u is the set of products associated to a given point x in a lattice x xj ,kj j representation U of a poset P.
The presentation ∅ with U0 = ∅ is the only lattice representation with no associated products in repL P.
A morphism ϕ : U ! V between two lattice representations is an order lattice homomor- phism ϕ : U0 ! V0 such that ϕ(Ux) ⊆ Vx for each x ∈ P.
A morphism ϕ : U ! V is an isomorphism of lattice representations if and only if ϕ :
U0 ! V0 is an isomorphism such that, for each x ∈ P, the restriction ϕx = ϕ|x : Ux ! Vx is an isomorphism as well.
The usual meet ∧ and join ∨ operations on sets allow to define the sum and intersection in repL P as follows:
U ∩ V = (U0 ∧ V0; Ux ∧ Vx | x ∈ P),
U ⊕ V = (U0 ⊕ V0; Ux ⊕ Vx | x ∈ P), where
U0 ⊕ V0 = U0 ∨ V0, u v u v Dx ⊕ Dx = Dx ∨ Dx, (4.10) u v u [ v Ux ⊕ Vx = (Dx ∨ Dx; Px Px ). CHAPTER 4. APPLICATIONS OF EQUIPPED POSETS TO NUMBER THEORY AND DIFFERENTIAL EQUATIONS101
A lattice representation U is decomposable if there exist not empty lattice representations U1 and U2 such that U = U1 ⊕ U2.
The following Figure 4.6 have examples of indecomposable lattice representations:
••••- 6 ••••- ? -? ••••- - -
Figure 4.6. Examples of lattice representations.
If x is a weak point, then an indecomposable representation U of x has the form:
u U = (Dx;(Dx; Pz,k)), where z = (xt, yt), and
2 Dx = {(x, y) ∈ N | 0 ≤ x ≤ (yt + 1)k + xt, 0 ≤ y ≤ yt}.
Henceforth, we let repL P be the category of lattice representations of a given equipped 2 poset P attached to the sublattice L ⊂ N .
The size kUk or dimension of a lattice representation U ∈ P is a sequence of nonnegative integers: (dx | x ∈ P), u u where, for each x ∈ P, dx =| Px − Σ Pz |. z∈xN
Points and relations in (D, ¢) are either weak or strong. We say that a point x ∈ D is weak if and only if its lattice representation only has associated weak products. Moreover, a chain C ⊂ D is weak if all of its points are weak. Further, relations between points in D with strong points are also strong.
Now, we consider the lattice representation
Un(k0) = (U0; Uci | x ∈ Dn(k0)) of the weak chain Dn(k0). The dimension dp, of the subset Up, equals to the number of all the weak products from (0, 0) to the weak lattice path starting in p and finishing in p0 = {x0, 0, x0 + k0, 0}, for some fixed x0 > 0.
We note that, |Dn(k0)| = 2 + k0 and; n + k n − k U = {(x, y) | 0 ≤ x ≤ 0 , 0 ≤ y ≤ 0 }, 0 2 2 n − k D = {(x, y) | 0 ≤ x ≤ n − 1, 0 ≤ y ≤ i}, 0 ≤ i ≤ 0 , (4.11) ci 2
Pci,k0 = P n−k0 . ( 2 −i,i),k0 CHAPTER 4. APPLICATIONS OF EQUIPPED POSETS TO NUMBER THEORY AND DIFFERENTIAL EQUATIONS102
The following Figure 4.7 illustrates the lattice representation of D5(1) as defined above:
(0, 2)⊗ (1, 2) (2, 2) (3, 2) • • •• • @@ @ •••••••⊗ (0, 1) (1, 1)@@ (2, 1) (3, 1) @ •••••••weak ⊗ D5(1) (0, 0) (1, 0) (2, 0) (3, 0)
Figure 4.7. Lattice representation of D5(1).
4.1.4 Categorification of Delannoy numbers
In this part, we interpret Delannoy numbers as dimensions of lattice representations of weak chains of type Dn(k0).
Theorem 29. For k0 ≥ 0, the dimension vector kUn(k0)k = (dp | p ∈ Dn(k0)), where for p = {x, y, x + k0, y} ∈ Dn(k0);
x dp = 2 c(x − y, y)c(x + k0 − y, y),
x+y and c(x, y) = x .
Proof. For each p = {x, y, x + k0, y} ∈ Dn(k0), dp is the number of weak products from (0, 0) to {x0, 0, x0 + k0, 0}, where the corresponding weak chain has as starting vertex p. x Since the number of weak lattice paths from p to {x0, 0, x0 + k0, 0} to p is 2 , and the number of lattice paths from (0, 0) to the segment ((x, y), (x + k0, y)) is c(x + y, x)c(x + k0 + y, y), we are done.
x0 P P y Corollary 9. For x0, k0 ≥ 0 fixed, D(x0, k0) = dp = 2 c(x0 − y, y)c(x0 + k0 − p∈Dn(k0) y=0 y, y).
Theorem 29 and Corollary 9 allow to obtain a categorification (in the sense of Ringel and Fahr [26,27]) of numbers D(x0, k0) which can be seen as Delannoy numbers d(x, y). As we described before, such numbers are also obtained by counting the number of lattice paths 2 in N from (0, 0) to (x, y) considering directions (1, 0), (0, 1) and (1, 1). That is: d(x, y) = d(x − 1, y) + d(x, y − 1) + d(x − 1, y − 1) (4.12) d(0, 0) = 1.
The next Figure 4.8 shows some of these numbers. CHAPTER 4. APPLICATIONS OF EQUIPPED POSETS TO NUMBER THEORY AND DIFFERENTIAL EQUATIONS103
1•••••••- 13- 85- 377- 1289- 3653- 8989 6 6 6 6 6 6 6
1••••••• - 11- 61- 231- 681- 1683- 3653 6 6 6 6 6 6 6
1••••••• - 9- 41- 129- 321- 681- 1289 6 6 6 6 6 6 6
1••••••• - 7- 25- 63- 129- 231- 377 6 6 6 6 6 6 6
1••••••• - 5- 13- 25- 41- 61- 85 6 6 6 6 6 6 6
1••••••• - 3- 5- 7- 9- 11- 13 6 6 6 6 6 6 6
1••••••• - 1- 1- 1- 1- 1- 1
Figure 4.8. Delannoy numbers and its reticular trajectories.
4.2 The Zavadskij calculus
In this section, we use Krawtchouk-Zavadskij matrices or K Z-matrices in order to give explicit solutions to some linear and non-linear differential equations. In order to do that, we introduce the Zavadskij calculus based on K Z-matrices, and some of its properties proved by Zavadskij in [18, 70].
Randomly equipped posets
As a motivation, let us start by introducing an equipped version of the Ehrenfest urn model, which is defined as follows: suppose that an n-point chain (ordinary) will be equipped randomly, to do that, a point xi of the chain is selected at random to change its nature (weak !strong), ordinary points are considered strong. Then a point is selected from this new poset in order to change its nature. The procedure follows at infinitum.
Several questions arise from this probabilistic experiment. For instance:
(i) What is the distribution of the weak points?
(ii) Is weak the relations between two consecutive weak points? Otherwise, if the relation between points is strong, what is the distribution of the indecomposable representa- tion T (xi, xj), i < j?
The first question can be solved by invoking the Ehrenfest urn model [28]. For example, if N = 3, we have the following (s0, s1, s2, s3) four states: CHAPTER 4. APPLICATIONS OF EQUIPPED POSETS TO NUMBER THEORY AND DIFFERENTIAL EQUATIONS104
⊗ ⊗ e e ⊗ ⊗ ⊗ e ⊗ s0 s1 s2 s3 noe weak onee weak twoe weak three weak points point points points Figure 4.9. Ehrenfest urn model.
The corresponding matrix of changes has the following form S(3):
1 s0,0 s0,1 s0,2 s0,3 0 3 0 0 0 1 0 0 2 (3) s1,0 s1,1 s1,2 s1,3 1 0 3 0 1 3 0 2 0 (3) 1 (3) S = = 2 = : S = A s2,0 s2,1 s2,2 s2,3 0 3 0 1 3 0 2 0 3 3 1 s3,0 s3,1 s3,2 s3,3 0 0 3 0 0 0 1 0 entries above and below of the main diagonal in matrices of type A(N) are descending- ascending from 1 to N, respectively.
It is easy to see that eigenvectors of a given matrix of type A(N) are columns of a matrix (N) (N) K whose entries Ki,j are given by the following formula: (N) X j − 1 N − j K = (−1)K i,j k i − k − 1 k X jN − j = (−1)K . k i − k k
We call the matrices K(N) Krawtchouk-Zavadskij matrices or K Z-matrices. These ma- trices were introduced in the late 1920’s by M. Krawtchouk as coefficients of some Krawtchouk polynomials with the form:
N N−j j X i (N) (1 + x) (1 − x) = x Ki,j i=0 According to Feinsilver [28], the idea of setting Krawtchouk polynomials in matrix form was given by Bose in 1985. However, we recall that these matrices were rediscovered by Zavadskij in 2007 in his work regarding generalizations of the Kronecker matrix problem. In such a work, Zavadskij defined the previously described Krawtchouk matrices given many additional properties. He used such matrices to give a simple description of the solution of the semilinear Kronecker problem obtained by Djokovi´cin 1978.
K Z-matrices are ubiquitous in Mathematics, for instance, they appeared in areas as har- monic analysis, statistics, combinatorics, coding theory, Lie algebras representation theory, Quantum theory, probability theory, etc. CHAPTER 4. APPLICATIONS OF EQUIPPED POSETS TO NUMBER THEORY AND DIFFERENTIAL EQUATIONS105
Remark 4.3. We note that if S = {v1, v2, . . . , vn} is a set of eigenvalues of a matrix of type A(n) (Kac matrix) then S corresponds to a unique irreducible representation of the Lie algebra Sl(n, C).
(N) The correspondence between the Krawtchouk matrices K and the Θn matrices redis- covered by Zavadskij is given by the identity
(N) K = ΘN+1. (4.13)
4.2.1 Matrix problems and the solution of some system of differential equations
Often, matrix problems are useful to obtain applications in different fields of mathematics. Perhaps one of the most remarkable matrix problems with applications in analysis is the Kronecker problem which was solved by Weierstrass in 1867, for some particular cases, and by Kronecker in 1890 for the complex number field case. This problem is equivalent to the problem of finding canonical Jordan form of pairs (A, B) of matrices with respect to the following elementary transformations [56]:
(i) All elementary transformations on rows of the block matrix (A, B).
(ii) All elementary transformations made simultaneously on columns of A and B having the same index number.
This classification problem allows to describe solutions of systems of differential equations dX(t) of the form A dt + BX(t) = f(t), where A and B are m × n-matrices. Actually, the following is the theorem stated by Kronecker and Weierstrass regarding this kind of systems of differential equations [54]:
Theorem 30. Let A and B be a pair of m × n-matrices (with entries in a field k). Consider the system: dX(t) A + BX(t) = f(t). (4.14) dt Then, there exists an m × m-matrix P and an n × n-matrix Q such that the system: dz A0 + B0z(t) = f 0(t), (4.15) dt with A0 = P AQ and B0 = PBQ and f 0(t) = P f(t) is a sum of systems of one of the following three types:
dzj 0 1. (s + 1) × (s); dt + zj−1 = fj(t); 1 ≤ j ≤ s; z0 = 0.
dzi 0 0 2. (s) × (s); dt + λzi + zi+1 = fi (t); 1 ≤ i ≤ s; zs+1 = 0, or zj = fj(t), λ ∈ k.
dzi 0 3. (s) × (s + 1); dt + zi+1 = fi (t); 1 ≤ i ≤ s; zs+1 = 0.
We recall that, if k is an algebraically closed field, then up to isomorphism every indecom- posable Kronecker module belongs to one of the following three classes [54, 56, 70]: CHAPTER 4. APPLICATIONS OF EQUIPPED POSETS TO NUMBER THEORY AND DIFFERENTIAL EQUATIONS106
0: In C(p) , where C(p) is a Frobenius matrix or companion matrix of a minimal polynomial ps(t), with n = s∂p(t), ∂p(t) denotes the degree of the polynomial p(t).
∗ I = I : (a) In Jn(0) ,
(b) Jn(0) In ,
+ − ± where Jn(0) ∈ {Jn (0),Jn (0)} and Jn (0) denotes a corresponding upper or lower Jordan block. Whereas, I∗ denotes the dual case defined by the classification problem.
! ∗ II = III : In In ,
∗ " # III = II :In In .
! In this case, In (In, respectively) denotes an n × (n + 1) matrix obtained from the identity In by adding a column of zeroes. In fact, the last column (the first column, respectively) " # in these matrices consists only of zeroes. In the same way, In (In, respectively) denotes an (n + 1) × n matrix obtained from the identity In by adding a row of zeroes.
If n = 0, then U0 = k and V0 = 0. The cases II and III constitute the non-regular cases of this classification, whereas cases 0 and I constitute the regular one.
Several generalizations of the Kronecker problem have been posed and solved through- out the years. Perhaps the semilinear Kronecker problem is one of the most remarkable generalizations of such a matrix problem. It is defined as follows.
To classify indecomposable pairs (A, B) of rectangular matrices of equal size over a division ring k with an automorphism σ with respect to transformations of the form:
(A, B) 7! (X−1AY, X−1BY σ). (4.16) with non-singular matrices X and Y .
Problem (4.16) was solved by Djokovi´cin 1978 and generalized by Sergeichuk to pseudo linear bundle in 1989 [24, 54]. In such a work, Sergeichuk investigated systems of linear differential equations of the form A(t)y0 + B(t)y = f(t) and determined the canonical form of the corresponding pairs (A(t),B(t)) once the equation has been multiplied by a meromorphic coefficient.
The Djokovi´c's solution to the semilinear Kronecker problem was simplified by Zavadskij in 2007, see [70]. In such a work, he defined an interesting sequence of integer matrices CHAPTER 4. APPLICATIONS OF EQUIPPED POSETS TO NUMBER THEORY AND DIFFERENTIAL EQUATIONS107 denoted Θn and a corresponding polynomial transformation named Θ-transformation. Such results allowed to give a description of solutions of some variants of the biquadratic homogeneous problem (whose solution is still an open problem).
Now, we recall the solutions of the matrix problem (4.16) proposed by Djokovi´cand Zavadskij in [24] and [70], respectively. In order to give the Zavadskij's solution, we define first some useful definitions and notations and in particular the Θ-transformation.
2 n−1 n If f(t) = a0 + a1t + a2t + ··· + an−1t + t is a monic non-constant polynomial over the field F , denote by C(f) its standard companion matrix which is a square matrix of the form:
0 0 0 ··· 0 −a0 1 0 0 ··· 0 −a1 0 1 0 ··· 0 −a2 C(f) = 0 0 1 ··· 0 −a3 ...... 0 0 0 ··· 1 −an−1
We let F = ∪Fn denotes the set of all indecomposable polynomials (with right coefficients) n of the skew polynomial ring R = F [t, σ], where Fn is the subset of polynomials of degree n. Notice that, if f ∈ F, then det C(f) = 0 if and only if f(t) = 0 if and only if f(t) = tn.
· For a subset X ⊂ F, we let X denotes any fixed maximal subset of non-similar in pairs polynomials in X. Besides, given a ring σ, we let σhai denotes the subset of all polynomials in σ not vanishing in a.
The following is the Zavadskij's version of the Djokovi´c's solution of the semilinear Kro- necker problem [70]:
Theorem 31. The indecomposable pair of matrices (A, B) for the semilinear Kronecker problem (4.16) over an arbitrary skew field k with an automorphism σ are exhausted up to equivalence by the pairs of the following types:
· ∗ n 0 = 0 :(In,C(f)), f ∈ F\{t }, n ≥ 1, ∗ 1 = 1 :(In,Jn(0)), (Jn(0),In), n ≥ 1, (4.17) ∗ " # 2 = 3 :(In,In), n ≥ 0, ! ∗ 3 = 2 :(In, In), n ≥ 0.
These types are non-equivalent two-by-two.
A short proof of Theorem 31 was given by Zavadskij in [70]. In such a work, Zavadskij gives the solution to the homogeneous biquadratic problem over a pair of fields (F,G), where F ⊂ G and G is a quadratic extension of F . In this case, any element z ∈ G can be written in the form z = a + ξb with a, b ∈ F and ξ ∈ G is a root of a minimal quadratic polynomial p(t) = t2 + q, q ∈ F , a is said to be the real part of z, whereas b is the corresponding imaginary part of z. This point of view allows to obtain a more CHAPTER 4. APPLICATIONS OF EQUIPPED POSETS TO NUMBER THEORY AND DIFFERENTIAL EQUATIONS108 general classification of equipped posets (to the case (F,G) instead of the pair of fields (R, C) used in [67] and [68]) which have been well studied by Zavadskij, Ca˜nadaset al [12, 13, 67, 68, 71], and the author with the present research.
We remember here that the homogeneous biquadratic problem (which already was defined in (1.1)) consists of finding a canonical form for real matrices (in the sense described above) of even size 2m × 2n with respect to formally complex transformations of the form:
AB X −qY AB Z −qT 7−! , (4.18) CD YX CD TZ where all the matrices are over F , the blocks A, B, C and D are all of the same size m × n, the matrix blocks X,Y,Z and T have a suitable order such that the corresponding matrices defined by them in (4.18) are non-singular.
Regarding the homogeneous biquadratic problem (4.18), Zavadskij obtained the following result:
Theorem 32. Let F ⊂ F (ξ) be a quadratic field extension of characteristic λ 6= 2 with the AB minimal polynomial t2 +q of the element ξ. Then, the indecomposable matrices CD for the homogeneous biquadratic problem (4.18) are exhausted, up to equivalence, by the pairwise non-equivalent ones of the following types:
I · 1. n , f ∈ Θ(F h−1i) ∪ (t + 1)n, Im C(f) Re C(f) n
" In 2. # , n ≥ 0, In
! In 3. , n ≥ 0. In
2 n In this case, to each polynomial f = a0 + a1t + a2t + ··· + ant one can attach the t almost Θ-transformation, [Θn+1(f)] = Θn+1[f] with [f] = (a0, a1, a2, . . . , an) and Θn is a sequence of integer matrices that are defined in the next section.
4.2.2 The Θ-Transformation
In this part, we recall the definition of the sequences of integer matrices {Θn} and some of their properties described by Zavadskij in [70].
i,j m The entries Θn of {Θn} are given by the following identity ( l = 0 if l and m do not satisfy the condition 0 ≤ l ≤ m):
X j − 1 n − j Θi,j = (−1)k . (4.19) n k i − k − 1 k CHAPTER 4. APPLICATIONS OF EQUIPPED POSETS TO NUMBER THEORY AND DIFFERENTIAL EQUATIONS109
The Θ-transformation of f is given by the equality: a Θf = (Θ f) n (4.20) n+1 f(−1)
Thus, the Θ-transformation appears to be an involution on Rh−1i preserving degrees and leading coefficients of polynomials.
i,j For all the following properties, it is assumed that the entries Θn of the corresponding matrices {Θn} are well defined in the sense that all binomial coefficients can be computed. In particular, the following equalities have place:
1,j Θn = 1, n,j j−1 Θn = (−1) , n − 1 Θi,1 = , n i − 1 i,n i−1 i,1 Θn = (−1) Θn , (4.21) 1 2m − 2 Θm,2 = C = , 2m m−1 m m − 1 i,j i+1,j i+1,j Θn + Θn = Θn+1 , i,j i−1,j i,j+1 Θn − Θn = Θn+1 .
Cm−1 is the (m − 1)th Catalan number. Among other properties regarding polynomials and entries of the matrices, Zavadskij also found the following results.
# " # Θn+1(In + In) = 2InΘn, # " " Θn+1(In − In) = 2InΘn, 2 n−1 Θn = 2 In, (4.22) 1 Θ−1 = Θ , n 2n−1 n n(n+1)/2 det Θn = (−2) .
The following are the matrices Θn, with n ≤ 5.
1 1 1 1 1 1 1 1 1 1 1 1 4 2 0 −2 −4 1 1 3 1 −1 −3 1 , , 2 0 −2 , , 6 0 −2 0 6 1 −1 3 −1 −1 3 1 −1 1 4 −2 0 2 −4 1 −1 1 −1 1 −1 1 −1 1
Remark 4.4. In [55], Severo proved the following result regarding the eigenvalues and eigenvectors of generalized Krawtchouk matrices:
The jth eigenvalue and associated eigenvector, j = 0,...,N for the Krawtchouk matrix N−j j are, respectively, w2 w1 and A(w2 − 1, 1 − w1; N, j) × A(−1, 1; j, 0). Where, for integers N ≥ M ≥ 0 and for complex numbers α and β, A(α, β; N,M) denotes the N + 1 by M + 1 matrix with (i, j)−element given by the following formula: CHAPTER 4. APPLICATIONS OF EQUIPPED POSETS TO NUMBER THEORY AND DIFFERENTIAL EQUATIONS110
( N−M N−M−i+j i−j i−j α β ,N − M ≤ i − j ≤ 0, aij(α, β; N,M) = 0, elsewhere for i = 0,...,N and j = 0,...,M. And
1 p4p2 + 1 1 p4p2 + 1 w = 1 − + , w = 1 − − (4.23) 1 2p 2p 2 2p 2p
1 The spectrum analysis of K Z-matrices is given by replacing p = 2 . In the next section, we describe additional properties of the sequence of integer matrices {Θn} and of the Θ-transformation which altogether constitute the Zavadskij calculus.
4.2.3 Zavadskij calculus to solve some differential equations
We recall that in general to give explicit solutions of non-linear differential equations is a cumbersome problem, however some systems of non-linear differential equations can be solved by using K Z-matrices. In particular, systems of the form:
00 0 2 X (t) ± αX(t) = 0, α ∈ R,X (t) − X (t)Θk = Θk, (4.24) where X(t) = (xi,j(t)) is a k × k matrix.
We call Zavadskij calculus all the tools allowing such solutions.
The following proposition regards the spectrum analysis of K Z-matrices.
Proposition 6. For any k ≥ 1, the matrix Θk is diagonalizable. Moreover the corre- k−1 d k e k−1 b k c sponding characteristic polynomial Pk(t) has the form Pk(t) = (t − 2 2 ) 2 (t + 2 2 ) 2 .
Proof. The result follows from the identities (4.22), Remark 4.4 and the fact that the k−1 2 k k geometric multiplicity of the characteristic values ±2 are d 2 e and b 2 c, respectively, for any k ≥ 1.
Henceforth, we let sh (x) and ch (x) denote the usual hyperbolic functions.
Theorem 33. For any k ≥ 1 and t ∈ R, it holds that
1 k−1 k−1 Θkt 2 2 1. e = k−1 sh 2 t Θk + ch 2 t Ik, 2 2 (4.25) 1 k−1 k−1 −Θkt 2 2 2. e = − k−1 sh 2 t Θk + ch 2 t Ik. 2 2 CHAPTER 4. APPLICATIONS OF EQUIPPED POSETS TO NUMBER THEORY AND DIFFERENTIAL EQUATIONS111
Proof.
∞ n ∞ 2j ∞ 2j+1 X(Θkt) X(Θkt) X(Θkt) 1. eΘkt = = + n! (2j)! (2j + 1)! n=0 j=0 j=0 ∞ 2j 2j ∞ 2j+1 2j X(t) Θ X(t) Θ Θk = k + k (2j)! (2j + 1)! j=0 j=0 ∞ k−1 2j j ∞ k−1 2j+1 j X((2 2 )t) I X((2 2 )t) I Θk = k + k (2j)! k−1 j=0 j=0 (2j + 1)!(2 2 ) ∞ k−1 2j ∞ k−1 2j+1 X((2 2 )t) Θk X((2 2 )t) = I + k (2j)! k−1 (2j + 1)! j=0 (2 2 ) j=0 k−1 k−1 sh (2 2 t) 2 = ch 2 t Ik + k−1 Θk. 2 2 k−1 k−1 sh (2 2 (−t)) −Θkt 2 2. e = ch 2 (−t) Ik + k−1 Θk 2 2 k−1 k−1 sh (2 2 (t)) 2 = ch 2 (t) Ik − k−1 Θk. 2 2
k−1 k−1 k−1 2 k−1 sh 2 2 (t) Θkt −Θkt 2 sh 2 t 2 Note that, e e = [(ch 2 t)Ik + k−1 Θk][(ch 2 (t))Ik − k−1 Θk] = 2 2 2 2 k−1 k−1 k−1 sh2 2 2 (t) k−1 sh2 2 2 (t) k−1 2 2 2 2 2 2 2 k−1 2 2 2 (ch 2 t)Ik − 2k−1 Θk = (ch 2 t)Ik − 2k−1 2 Ik = (ch 2 t)Ik − 2 k−1 2 k−1 2 k−1 sh 2 2 (t)Ik = [(ch 2 2 t) − sh 2 2 (t)]Ik = Ik.
eΘkt+e−Θkt eΘkt−e−Θkt For any k ≥ 1 and t ∈ R, we let ch (Θkt) = 2 and sh (Θkt) = 2 . Besides, we assume the following notations:
−1 th Θkt := (sh Θkt)(ch Θkt) , −1 cth Θkt := (ch Θkt)(sh Θkt) , for t 6= 0, −1 (4.26) sech Θkt := (ch Θkt) , −1 csch Θkt := (sh Θkt) , for t 6= 0.
Corollary 10. For any k ≥ 1 and t ∈ R, the following identities hold:
2 2 ch Θkt − sh Θkt = Ik, 2 2 ch Θkt + sh Θkt = ch 2Θkt, k−1 th 2 2 t th Θkt = [ k−1 ]Θk, 2 2 k−1 cth 2 2 t (4.27) cth Θkt = [ k−1 ]Θk, 2 2 k−1 sch Θkt = [sch 2 2 t]Ik, k−1 csch 2 2 t csch Θkt = [ k−1 ]Θk. 2 2 CHAPTER 4. APPLICATIONS OF EQUIPPED POSETS TO NUMBER THEORY AND DIFFERENTIAL EQUATIONS112
Proof.
2 k−1 k−1 Θ k−1 2 2 2 2 2 2 k 2 2 2 sh 2 t k−1 1. ch Θkt − sh Θkt = [ch 2 t]Ik − [ k−1 ]Θk = [ch 2 t]Ik − [ 2k−1 ]2 Ik = 2 2 2 k−1 2 k−1 (ch 2 2 t − sh 2 2 t)Ik = Ik,
2 2 2 k−1 2 k−1 k+1 2. ch Θkt + sh Θkt = (ch 2 2 t + sh 2 2 t)Ik = (ch 2 2 t)Ik = ch 2Θkt,
k−1 −1 k−1 −1 (sh 2 2 t)Θk Ik th 2 2 t 3.( sh Θkt)(ch Θkt) = [ k−1 ][ k−1 ] = [ k−1 ]Θk, 2 2 ch 2 2 t 2 2
k−1 −1 k−1 k−1 k−1 2 2 Θ 2 2 −1 2 k 2 ch 2 t 4.( ch Θkt)(sh Θkt) = [(ch 2 t)Ik][ k−1 ] = [ 2k−1 ][ k−1 ]Θk = sh 2 2 t sh 2 2 t k−1 1 2 [ k−1 cth 2 t]Θk, 2 2
I k−1 −1 k 2 5.( ch Θkt) = k−1 = (sch 2 t)Ik, ch 2 2 t
k−1 −1 2 2 −1 Θk 6.( sh Θkt) = [ k−1 ]Θk = [ k−1 ]. sh 2 2 t 2k−1sh 2 2 t √ We also note that, for i = −1, the following identities hold:
k−1 k−1 [sh 2 2 t]Θk iΘkt 2 e = [ch 2 t]Ik + i k−1 , 2 2 k−1 (4.28) k−1 [sh 2 2 t]Θk −iΘkt 2 e = [ch 2 t]Ik − i k−1 . 2 2
eiΘkt−e−iΘkt eiΘkt+e−iΘkt As in the classical case, we let sin (Θkt) := 2 and cos (Θkt) := 2 . Thus, the following identities have place, for any k ≥ 1 and t ∈ R:
Corollary 11. For any k ≥ 1 and t ∈ R, the following identities hold:
k−1 sin 2 2 t sin Θkt = [ k−1 ]Θk, 2 2 k−1 cos Θkt = [cos 2 2 t]Ik, Θ t Θ t (4.29) sin Θ t = 2sin k cos k , k 2 2 2 2 sin Θkt + cos Θkt = Ik, 2 2 cos Θkt − sin Θkt = cos 2Θkt.
Proof. All the identities can be verified by using the same arguments as in Corol- k−3 Θ t Θ t 2 k−3 k k sin 2 t 2 lary 10. Note that, in particular, 2sin 2 cos 2 = 2[ k−1 ][cos 2 t]IkΘk = 2 2 k−3 k−1 sin 2(2 2 t) sin 2 2 t [ k−1 ]IkΘk = [ k−1 ]Θk = sin Θkt. 2 2 2 2 CHAPTER 4. APPLICATIONS OF EQUIPPED POSETS TO NUMBER THEORY AND DIFFERENTIAL EQUATIONS113
Corollaries 10 and 11 allow to establish the following identities (for some n ∈ Z):
k−1 tg 2 2 t k−1 π −1 2 (sin Θkt)(cos Θkt) = tan Θkt = [ k−1 ]Θk, for 2 t 6= (2n + 1) , 2 2 2 k−1 ctg 2 2 t k−1 −1 2 (cos Θkt)(sin Θkt) = ctg Θkt = [ k−1 ]Θk, for 2 t 6= nπ, (4.30) 2 2 −1 k−1 k−1 π sec Θ t = (cos Θ t) = sec 2 2 tI , for 2 2 t 6= (2n + 1) , k k k 2 −1 k−1 k−1 csec Θkt = (sin Θkt) = csec 2 2 tΘk for 2 2 t 6= nπ.
In particular, it holds that:
2 k−1 tg 2 2 t I + tg2 Θ t = I + [ ]Θ2 k k k 2k−1 k 2 k−1 tg 2 2 t k−1 2 k−1 (4.31) = I + [ ]2 I = I + [tg 2 2 t]I k 2k−1 k k k 2 k−1 2 = [1 + tg 2 2 t]Ik = sec Θkt.
By applying derivatives component-wise, the following identities hold:
Lemma 8. d (sh Θ t) = [ch Θ t]Θ . dt k k k d (sin Θ t) = [cos Θ t]Θ . (4.32) dt k k k d (ch Θ t) = [sh Θ t]Θ . dt k k k d (cos Θ t) = −[sin Θ t]Θ . dt k k k d (th Θ t) = [sch2 Θ t]Θ . dt k k k d (tg Θ t) = [sec2 Θ t]Θ . dt k k k d (cth Θ t) = −[csch2 Θ t]Θ . dt k k k d (ctg Θ t) = −[csec2 Θ t]Θ . (4.33) dt k k k d (sch Θ t) = −[sch Θ t][th Θ t]Θ . dt k k k k d (sec Θ t) = [sec Θ t][tg Θ t]Θ . dt k k k k d (csch Θ t) = −[csch Θ t][cth Θ t]Θ . dt k k k k d (csec Θ t) = −[csec Θ t][ctg Θ t]Θ . dt k k k k CHAPTER 4. APPLICATIONS OF EQUIPPED POSETS TO NUMBER THEORY AND DIFFERENTIAL EQUATIONS114
Proof. It is enough to verify the first six identities:
k−1 k−1 d d sh 2 2 t k−1 ch 2 2 t k−1 2 2 (sh Θkt) = [ k−1 Θk] = [2 k−1 Θk] = [ch 2 t]Θk. dt dt 2 2 2 2 k−1 k−1 d d sin 2 2 t k−1 cos 2 2 t k−1 2 2 (sin Θkt) = [ k−1 Θk] = [2 k−1 Θk] = [cos 2 t]Θk. dt dt 2 2 2 2 k−1 d d k−1 k−1 k−1 Θk sh 2 2 t 2 2 2 (ch Θkt) = ([ch 2 t]Ik) = 2 [sh 2 t][ k−1 ]Θk = [ k−1 ]Θk. dt dt 2 2 2 k−1 d d k−1 k−1 k−1 Θk sin 2 2 t 2 2 2 (cos Θkt) = ([cos 2 t]Ik) = −2 [sin 2 t][ k−1 ]Θk = −[ k−1 ]Θk. dt dt 2 2 2 k−1 2 k−1 d d th 2 2 t k−1 sch 2 2 t k−1 2 2 2 (th Θkt) = ([ k−1 ]Θk) = 2 ([ k−1 Ik]Θk) = [(sch 2 t)Ik]Θk. dt dt 2 2 2 2 k−1 2 k−1 d d tg 2 2 t k−1 sec 2 2 t k−1 2 2 2 (tg Θkt) = ([ k−1 ]Θk) = 2 ([ k−1 Ik]Θk) = [(sec 2 t)Ik]Θk. dt dt 2 2 2 2
The remaining identities can be verified by using the same arguments.
The following result follows directly from Lemma 8.
Theorem 34. For any k ≥ 1, the second order system of the linear differential equations 00 k−1 X (t) + 2 X(t) = 0, has a solution the matrix product X(t) = [cos Θkt + sin Θkt]Θk. 0 2 Moreover X(t) = tg Θkt is a solution of the system X (t)−X (t)Θk = Θk. X(t) is a k ×k matrix. CHAPTER 5
Applications of the algorithm of differentiation VII in cybersecurity
This chapter is devoted to some applications of the algorithm of differentiation VII in computer science. In section 5.1.6, we use the algorithm of differentiation VII to gener- ate schemes of fragile digital watermarks. Whereas, in section 5.2.4 such an algorithm is used to model some interactions between informatic virus and worms (infections and detections). For the sake of clarity, we give theoretical frameworks of digital watermark- ing, wavelet transform and virus detection as Cox, Miller et al, Ritter, and Konstantinou describe in [22, 37, 49].
5.1 Digital watermarking
In this section, we present an explanation of how the theory of algorithms of differentiation is a theoretical framework for digital watermarking procedures.
5.1.1 The wavelets
Nowadays, images are widely used in many areas. It is oftentimes desirable or necessary to embed data or marks into the images, and this embedding can be done in a way that is perceptible or imperceptible.
We recall that according to Cox, Miller et al [22], the sudden increase in watermarking interest is most likely due to the increase in concern over copyright protection of content and copy prevention, so, these have been the main driving forces of research in the field of watermarks.
Content owners are eagerly seeking technologies that promise to protect their rights. Cryp- tography is the first technology used by content owners to protect data, it ensures the secrecy of the data, whereas the steganography allows to ensure the privacy of the infor- mation.
115 CHAPTER 5. APPLICATIONS OF THE ALGORITHM OF DIFFERENTIATION VII IN CYBERSECURITY116
Unfortunately, encryption cannot help the seller monitor how a legitimate customer han- dles the content after decryption. A pirate can actually purchase the product, use the decryption key to obtain an unprotected copy of the content, and then proceed to dis- tribute illegal copies. In other words, cryptography can protect content in transit, but once decrypted, the content has no further protection.
Thus, there is a strong need for an alternative or complement to cryptography: a technol- ogy that can protect content even after it is decrypted. Watermarking has the potential to fulfill this need because it places information within the content where it is never removed during normal usage.
In general, if it is useful to associate some additional information with a Work, this meta- data can be embedded as a watermark. Of course, there are other ways to associate information with a Work, such as placing it in the header of a digital file, encoding it in a visible bar code on an image, or speaking it aloud as an introduction to an audio clip.
Watermarking is distinguished from other techniques in three important ways. First, wa- termarks are imperceptible. Unlike bar codes, they do not detract from the aesthetics of an image. Second, watermarks are inseparable from the Works in which they are embedded. Unlike header fields, they do not get removed when the Works are displayed or converted to other file formats. Finally, watermarks undergo the same transformations as the Works. It is these three attributes that make watermarking invaluable for certain applications. Watermarking can be used in a wide variety of applications, for instance, broadcast moni- toring, owner identification, proof of ownership, transaction tracking, authentication, copy control, device control, and legacy enhancements (see [22] to more details).
From the above, we conclude that digital watermarks complement the encryption work to enable security in the transmission of data through the Internet (images, texts). Now, a digital watermark is a piece of information hiding into the content of a message in such a way that it will be imperceptible to the human eye but easy to detect by a computer. The advantage here is that the digital watermark cannot be separated from the content.
Cox, Miller et al in [22] defined Watermarking as the practice of imperceptibly altering a Work to embed a message about that Work. However, note that some authors define this term in other ways. First, imperceptibility is not always considered a defining character- istic of watermarking. Second, the term “watermarking” is sometimes considered to cover only applications in which the same message is embedded in every copy of a Work (i.e., watermarking is not considered to include transaction tracking). In this case, the term “marking” is used to encompass both watermarking and transaction tracking.
The idea is not new, the approach via superposition of functions has existed since the nineteenth century when Joseph Fourier discovered that sine and cosine functions can overlap, to represent other functions. However, in the Wavelet analysis, the scale used to observe the data plays a special role. Wavelet algorithms process data at different scales and resolutions. If we observe a signal through a large window, notable features can be noticed in it but if we observe it through a small window, in turn, small features will be noticed. The objective of the Wavelet analysis is to observe, as it were, trees and forests. CHAPTER 5. APPLICATIONS OF THE ALGORITHM OF DIFFERENTIATION VII IN CYBERSECURITY117
5.1.2 Wavelet transform
A wavelet transform is a linear transformation that generalizes the properties of the Haar's transform. A wavelet in the sense of the discrete wavelet transform (DWT), is an orthog- onal function that can be applied to a finite group of data [49].
There are several types of wavelets, for example, Haar, Coiflet, Daubechies, etc. Just as the base functions in the Fourier transform are sine and cosine, the wavelet base is a set of functions defined by the difference equation:
M−1 X φ(x) = ckφ(2x − k), k=0 where M is the number of coefficients other than zero. The value of coefficients is deter- mined by orthogonality restrictions and standardization.
The wavelet transformation uses wavelets, as a base, and it is a tool that cuts data or functions into different frequency components; and then, these data can be studied by using a resolution associated with its scale.
The discrete representation of the wavelet transform DWT is used in digital image pro- cessing and it is defined as:
X X j −j F (i) = F (j, k)2 2 φ(2 i − k), ∀j ∀k where i, j and k are integer numbers.
The 2D algorithm is based on the prioritization of the x and y addresses, while the scaling function is defined as: φ(x, y) = φ(x)φ(y). The Haar wavelet transform is a linear transformation that is based on the following scaling function [49]: ( 1 if 0 ≤ x < 1, g(x) = 0 otherwise. of the dyadic translations and dilations of Haar's function:
1 1 if 0 ≤ x < 2 , 1 h(x) = −1 if 2 ≤ x < 1, 0 otherwise.
The following is a matrix interpretation of the Haar's transform.
k + If n = 2 , k ∈ Z and a ∈ RZn is a signal with a dimensional real value, we can associate t to a the column vector (a0, a1, . . . , an−1) , where ai = a(i), i ∈ Zn.
n In order to build an orthonormal base {ui} in the space RZ , we consider the identity: CHAPTER 5. APPLICATIONS OF THE ALGORITHM OF DIFFERENTIATION VII IN CYBERSECURITY118
p hpq(x) ≡ h(2 x − q), p with p ∈ N and 0 ≤ q < 2 for fixed p, q.
p The hpq(x) function is a translation, by q, of the h(2 x) function, which is a dilation of the Haar's wavelet function h(x). In fact, hpq(x) is supported (not canceled) in a range −p p of length 2 . The infinite family of functions {hpq(x) | p ∈ N, 0 ≤ q < 2 }, together with the scaling function g(x), constitute an orthogonal base (known as the Haar's base in the functional space L2[0, 1] of integrable functions over the unit interval) which can be 2 extend to a base in the space L [R] [49].
If we want to obtain a discrete version of the Haar base, we must proceed as follows:
Let us keep in mind that every positive integer i can be written in the form i = 2p + q, p where p ∈ N and 0 ≤ q < 2 . Using this fact, we define the following base: 1 u (j) := √ , for all j ∈ , 0 n Zn (5.1) r2p j u (j) := h = 2(p−k)/2h (2−kj), for all j ∈ . i n pq n pq Zn
In the case that i = 2p + q, where i = 1, 2, . . . , n − 1, the factor 2(p−k)/2 normalizes the Euclidean norm of the vector ui. Therefore kuik2 = 1, for all i ∈ Zn.
The vector u0 is a discrete version of the normalization of the function g(x). As well as the vectors ui are a normalized discrete version of the hpq(x) functions, which are also n orthonormal (ui.uj). Thus, B = {ui | i ∈ Zn} is an orthonormal base of the space RZ .
k A matrix Hn, with n = 2 and k > 0, whose i-th row is the vector ui ∈ B is called a ' 0 0 Haar s Matrix. Note that, for each n, it follows HnHn = In = HnHn, where In is the identity n × n. Thus H0 = H−1. Note that for η = √1 , the normalization factor acquires n n 2 the form: 2(p−k)/2 = 2−(k−p)/2 = ηk−p.
For example, the following matrices are the Haar's matrices Hn, for n = 2, 4, 8.
3 3 3 3 3 3 3 3 η η η η η η η η η η H = 3 3 3 3 3 3 3 3 2 η −η η η η η −η −η −η −η 2 2 2 2 η η −η −η 0 0 0 0 0 0 0 0 η2 η2 −η2 −η2 2 2 2 2 H8 = η η η η η −η 0 0 0 0 0 0 2 2 2 2 η η −η −η 0 0 η −η 0 0 0 0 H4 = η −η 0 0 0 0 0 0 η −η 0 0 0 0 η −η 0 0 0 0 0 0 η −η
The signal a can be written as a linear combination of the basic vectors ui:
n X a = ciui, (5.2) i=0 CHAPTER 5. APPLICATIONS OF THE ALGORITHM OF DIFFERENTIATION VII IN CYBERSECURITY119 where ci are unknown coefficients that must be determined. In this case, the equation 5.2 takes the form: 0 a = [u0, u1, . . . , un−1]c = Hnc, (5.3) where the vector c of coefficients can be obtained by performing the following calculation:
c = Hna. (5.4)
The equation (5.4) defines the Haar's transform for unidimensional signals of length n = 2k (which is denoted HaarTransform1D(a, n)).
An image can be reconstructed from the vector c of Haar's coefficients by using equation (5.3). For that reason, it is necessary to define the inverse Haar's transform (which is 0 denoted InverseHaarTransform1D(c, n) ≡ Hnc).
k l The Haar's transform for two-dimensional images, with m = 2 , n = 2 and a ∈ RZm ×RZn , is a matrix m × m of coefficients c = [cij], given by:
0 c = Hm[aij]Hn, (5.5) where aij = a(i, j), for all (i, j) ∈ Zm × Zn.
The equation (5.5) defines the Haar's transform of an m × n image. The image a can be recovered by performing the inverse transform (InverseHaarTransform2D(c, m, n) ≡ 0 0 Hm[cij]Hn). That is, a = Hm[cij]Hn.
Consider the following matrices k2, k4, and k8: η η η η 0 0 0 0 0 0 K = 2 η −η 0 0 η η 0 0 0 0 0 0 0 0 η η 0 0 0 0 0 0 0 0 η η K8 = η η 0 0 η −η 0 0 0 0 0 0 0 0 η η 0 0 η −η 0 0 0 0 K4 = η −η 0 0 0 0 0 0 η −η 0 0 0 0 η −η 0 0 0 0 0 0 η −η
Then, the Haar's matrices H2, H4 and H8 can be factored in the following form:
H2 = K2, K2 02 H4 = K4, 02 I2 (5.6) H4 04 H8 = K8. 04 I4
From the previous factorizations (5.6), it can be deduced that it is only required an even number of sums or differences with the multiplication by η, to obtain a Haar's transform [49]. In fact, the Haar's transform of a signal of length 2k can be computed in k steps. CHAPTER 5. APPLICATIONS OF THE ALGORITHM OF DIFFERENTIATION VII IN CYBERSECURITY120
Figure 5.1. Example of a wavelet decomposition. MATLAB routines describing the process can be found in Appendix B.2
In the wavelet decomposition, the original image decomposes by using high pass filters and low pass filters, followed by first sampling of rows and then columns. The result of the wavelet decomposition is an approximation of the original image, and the three decomposition processes can be expressed in such a way that:
(h) (v) (d) cIj = cIj+1 + cDj+1 + cDj+1 + cDj+1.
The next step is to decompose cIj+1 and so on.
5.1.3 Digital watermarks based on wavelets
Watermarks based on wavelets consist of parts encoded and decoded. The following steps are distinguished in the encoded part:
1. a wavelet frequency domain image is decomposed;
2. a permuted image is inserted into the decomposed image, which acts as a watermark;
3. the permutation is invertible and acts as a key, for the correct extraction of the watermark;
4. for each coefficient, in the frequency domain of the wavelet transform, the key takes a value, of one or zero, according to if it has been modified or not.
In the decoding process, the wavelet reverse transform is used to obtain the already marked image.
Often, in order to obtain more robustness of the watermark the wavelet method is com- bined with a suitable singular value decomposition procedure.
We recall that, any matrix A can be decomposed into a product of 3 matrices, this is: T T T A = UDV , where U U = I, V V = I and D = diag(λ1, λ2,...).
The diagonal entries are called singular values of A, the columns of U are the singular vectors to the left of A and the columns of V are singular vectors to the right of A. The CHAPTER 5. APPLICATIONS OF THE ALGORITHM OF DIFFERENTIATION VII IN CYBERSECURITY121 decomposition A = UDV T , is the Singular Value Decomposition of A, frequently denoted by SVD.
The SVD decomposition can be written in the form:
T T T A = λ1U1V1 + λ2U2V2 + ··· + λrUrVr , where r is the range of the matrix A. It should be noted that each singular value λr, gives the illuminance of a layer of an image, while the singular vectors specify the geometry of the image layer.
An important feature of SVD marking is that there is not considerably changes in the size of the singular values when different types of attacks are made. Below, we report some of these threats to the security of digital watermarks.
1. Transpose: A and its transpose AT have the same singular values.
2. Rotation: A and Ar have the same singular values.
3. Scaling: If B is a scaled version√ of A repeating each row L1-times. For each singular value λ 6= 0 of A, B has L1λ as singular value. If C is a column-scaled version of A√, by repeating each column of A, L2-times, then for each value λ 6= 0 of A, C has L2λ as singular value. If F is obtained of A, by repeating L1 rows and L√2 columns of A, then for each not null singular value of A, F has the singular value L1L2λ.
Sometimes, it is possible to combine the two SVD and DWT methods to make digital watermarks.
The degree of resistance of a water-marking method, to host medium modifications, is usually called robustness. Depending on the level of robustness offered, one can distinguish between the following categories of watermarking techniques [9].
1. Robust: In this class, the watermarks are designed in such a way to resist the host signal manipulations, and they are usually employed in intellectual property rights (IPR) protection applications. Obviously, no watermarking scheme can resist all types of modifications, regardless of their severity. As a consequence, robustness refers to a subset of all the possible manipulations and up to a certain degree of host signal degradation.
2. Fragile: In this case, the watermarks are designed to be vulnerable to all modifica- tions, i.e., they become undetectable by even the slightest host data modification. Fragile watermarks are more easy to devise than robust ones, and they are usually applied in authentication scenarios.
3. Semifragile: This class of watermarks provides selective robustness to a certain set of manipulations, which are considered as legitimate and allowable while being vul- nerable (fragile) to others. Such watermarks can also be used in the authentication cases instead of the fragile ones. In practice, all robust watermarks are essentially semifragile; but in the former case, the selective robustness is not a requirement imposed by the system designer but rather something that cannot be avoided. CHAPTER 5. APPLICATIONS OF THE ALGORITHM OF DIFFERENTIATION VII IN CYBERSECURITY122
5.1.4 DWT-SVD watermark insertion and extraction
Assume, for example, that the size of the watermark is n × n and that the size of the host is 2n × 2n. Then, we will proceed as follows to embed a digital watermark in a given host:
Algorithm 5.1.1: DWT-SVD watermark insertion 1. By using DWT, decompose the host A at 4 LL, LH, HL and HH sub-bands.
k k k kT 2. Apply SVD to each sub-band of the image A = Ua DaVa , for k = 1, 2, 3, 4, where k k denotes the bands LL, HL, LH, HH. Moreover, let λi , for i = 1, 2, . . . , n, be the k singular values of Da. T 3. Apply SVD to the watermark W = UW DW VW , where λwi, for i = 1, 2, . . . , n are the singular values of DW . 4. Modify the singular values of the recipient image in each sub-band by using the ∗k k singular values of W : λi = λi + αkλwi, for i = 1, 2, . . . , n and k = 1, 2, 3, 4. 5. Obtain the coefficients of the step 4. and modified the DWT sets: ∗k k ∗k kT A = Ua Da Va , for k = 1, 2, 3, 4. 6. Apply the inverse transform DWT by using the 4 sets of modified coefficients, to produce the marked image.
Algorithm 5.1.2: DWT-SVD watermark extraction 1. Use the DWT decomposition to decompose the image marked A∗ into four LL, LH, HL and HH sub-bands.
∗k k ∗k kT 2. Apply SVD to each sub-band of the image A = Ua Da Va , for k = 1, 2, 3, 4. k ∗k k 3. Extract the singular values of each sub-band λwi = (λi − λi )/αk, for i = 1, 2, . . . , n and k = 1, 2, 3, 4.
k T 4. Build the four watermarks using the singular vectors W = UW DW VW .
5.1.5 Image algebra
The image algebra is an area of recent development, which is a mathematical theory related to the transformation and the analysis of image. Serra and Sternberg were the first to unify the morphological concepts and methods into a coherent algebraic theory specifically designed for the image processing and the image analysis. Sternberg was also the first to use the term “image algebra”. Here, there are some basic and important definitions and properties regarding image algebras, as Ritter describes in [49].
The image algebra is a heterogeneous or many-valued algebra, in the sense of Birkhoff and Lipson [8], with multiple sets of operands and operators. The manipulation of the images CHAPTER 5. APPLICATIONS OF THE ALGORITHM OF DIFFERENTIATION VII IN CYBERSECURITY123 for purposes of image enhancement, analysis, and understanding involves operations not only on the images but also on different types of values and quantities associated with these images. Roughly speaking, an image consists of two things: a collection of points and a set of values associated with these points. The images are therefore endowed with two types of information, namely, the spatial relationship of the points, and also some type of numeric or other descriptive information associated with these points. Consequently, the field of image algebra bridges two broad mathematical areas: the theory of point sets and the algebra of value sets, and investigates their interrelationship.
The primary operands in the image algebra are images, templates, and neighborhoods. Of these three classes of operands, the images are the most fundamental since templates and neighborhoods can be viewed as special cases of the general concept of an image. In order to provide a mathematically rigorous definition of an image that covers the plethora of objects called an “image” in signal processing and image understanding, we define an image in general terms, with a minimum of specification. More details and definitions of template and neighborhoods can be found in [49].
For what follows, we recall the following definitions. A point set is simply a topological space. The most common point sets occurring in image processing are discrete subsets n of the n-dimensional Euclidean space R , with n = 1, 2 or 3, together with the discrete + + topology. Also, we encounter regularly the following point sets Zm × Zn and Zm × Zn . As mentioned, some of the more pertinent point sets are discrete subsets of the vector space n R . These point sets inherit the usual elementary vector space operations, for instance, x + y, k · x, and k + x = (k ± x1, . . . , k ± xn).
In addition to these standard vector space operations, the image algebra also incorporates three basic types of point multiplication. These are the Hadamard's product, the cross 3 3 product (or vector product) for points in Z (or R ), and the dot product which are defined, respectively, by:
x · y = (x1 · y1, . . . , xn · yn),
x × y = (x2 · y3 − x3 · y2, x3 · y1 − x1 · y3, x1 · y2 − x2 · y1), (5.7)
x • y = x1 · y1 + ··· + xn · yn.
Point arithmetic leads in a natural way to the notion of set arithmetic. Given a vector space Z, for X, Y ∈ 2Z (i.e., X, Y ⊂ Z) and an arbitrary point p ∈ Z, there are defined the following arithmetic operations: X ± Y, X ± p, X ∪ Y, X ∩ Y, X∆Y, X × Y, if X is finite, min X (denoted by ∧X) , max X (denoted by ∨X), and some more operations.
A heterogeneous algebra is a collection of nonempty sets of possibly different types of el- ements together with a set of finitary operations which provide the rules of combining various elements in order to form a new element. Note that, the collection of point sets, points, and scalars, together with the operations described previously, form a heteroge- neous algebra.
A homogeneous algebra is a heterogeneous algebra with only one set of operands. In other words, a homogeneous algebra is simply a set together with a finite number of operations. Homogeneous algebras will be referred to as value sets and will be denoted by capital font CHAPTER 5. APPLICATIONS OF THE ALGORITHM OF DIFFERENTIATION VII IN CYBERSECURITY124 letters, e.g., E, F, and G. There are several value sets that occur more often than others in digital image processing. These are the set of integers, the real numbers (floating point numbers), the complex numbers, the binary numbers of fixed length k, the extended real numbers (which include the symbols +∞ and/or −∞), and the extended non-negative real numbers. We denote these sets by Z, R, C, Z2k , R+∞ = R ∪ {+∞}, R−∞ = R ∪ {−∞}, ≥0 + R±∞ = R ∪ {+∞, −∞}, and R∞ = R ∪ {0, +∞}, respectively.
The following Table 5.1 provides a listing of the algebraic structures of the most often used value sets in which we focus our attention:
(R, ∨, ∧, +, ·) (C, +, ·) (Z2K , ∨, ∧, +, ·) (Z, ∨, ∧, +, ·)
0 ≥0 0 ∗ 0 (R±∞, ∨, ∧, +, + ) (R∞ , ∨, ∧, ×, × ) (Z2, ∨, ∧, +e, +e ) Table 5.1. The summary of pertinent numeric value sets.
As for point sets, for a given value set F, the operations on 2F are again the usual operations of union, intersection, set difference, etc. If, in addition, F is a lattice, the operations of infimum and supremum are also included.
In what follows, we use the notation AB to denote the set of all functions B −! A (i.e., AB = {f | f is a function from B to A}). Definition 27. Let F be a homogeneous algebra and X be a point set. An F-valued image X X on X is any element of F . For a given F-valued image a ∈ F (i.e., a : X −! F), F is called the set of possible range values of a and X the special domain of a.
X It is often convenient to let the graph of an image a ∈ F represent a. The graph of an image is also referred to as the data structure representation of the image. Given the data structure representation a = {(x, a(x)) | x ∈ X}, an element (x, a(x)) of the data structure is called a pixel. The first coordinate x of a pixel is called the pixel location or image point, and the second coordinate a(x) is called the pixel value of a at location x.
The above definition of an image covers all mathematical images on topological spaces with range in algebraic system. Requiring X to be a topological space provides us with the notion of nearness of a pixel. Since X is not directly specified, we may substitute any space required for the analysis of an image of imposed by a particular sensor scene. For 3 3 example, X could be a subset of Z or R , with x ∈ X of form x = (x, y, t), where the first coordinates (x, y) denote the spatial location and t a time variable.
Similarly, the replacement of the unspecific value set F with Z2k or F = (Z2k , Z2m , Z2n ) provides us with digital integer-valued and digital vector-valued images, respectively. An implication of these observations is that our image definition also characterizes any type of discrete or continuous physical image. 2 Definition 28. Let X be the set R . The binary images, grayscale images and color images are defined as follows:
(i) The binary images (or black and white images) are applications of type a : X −! Z2, for which in each position x the values of the pixels a(x) take only values 0 or 1. The black (white) color occurs when a(x) = 0 (a(x) = 1). CHAPTER 5. APPLICATIONS OF THE ALGORITHM OF DIFFERENTIATION VII IN CYBERSECURITY125
(ii) The digital grayscale images are applications a : X −! Z255. Each value of a(x) between 0 and 255 corresponds to a shade of gray, being a(x) = 0 the black color, and a(x) = 255 the white color.
3 (iii) The color images (or RGB-color images) are applications of type a : X −! Z255, for which in each position x the values of the pixels a(x) take integer values from 0 to 255. For instance, the black color is represented by (0, 0, 0), the white color is represented by (255, 255, 255), the red color is represented by (255, 0, 0), the green color is represented by (0, 255, 0), and the blue color is represented by (0, 0, 255).
Operations on and between F-valued images are the natural induced operations of the algebraic system F. For example, if γ is a binary operation on F, then γ induces a binary X X operation, again denoted by γ over F defined as follows, if a, b ∈ F , then:
aγb = {(x, c(x)) | c(x) = a(x)γb(x), x ∈ X}.
X For example, suppose a, b ∈ F and our value set is the algebraic structure of the F- numbers (F, +, ·, ∧, ∨). Replacing γ by the binary operations +, ·, ∧, ∨, we obtain the basic binaries operations:
a + b := {(x, c(x)) | c(x) = a(x) + b(x)}, a · b := {(x, c(x)) | c(x) = a(x) · b(x)}, (5.8) a ∧ b := {(x, c(x)) | c(x) = a(x) ∧ b(x)}, a ∨ b := {(x, c(x)) | c(x) = a(x) ∨ b(x)}.
The binary operation γ on F also induces the following scalar operations on images, for k ∈ F: kγa := {(x, c(x)) | c(x) = kγa(x)}, aγk := {(x, c(x)) | c(x) = a(x)γk}, (5.9) k · a := {(x, c(x)) | c(x) = k · a(x)}, k + a := {(x, c(x)) | c(x) = k + a(x)}.
Although a lot of images processing is accomplished using real-, integer-, binary-, or complex-valued images, many higher-level vision tasks require manipulation of vector- and set-valued images. A set-valued-images is of the form a : X −! 2F. Here the underlying value set is (2F, ∪, ∩, ˜), where the tilde symbol denotes complementation. Hence, the operations on set-valued images are those induced by the Boolean algebra of the value set [49].
5.1.6 Extracting and embedding digital watermarks with the algorithm of differentiation VII
In this section, the algorithm of differentiation VII is used to embed and extract fragile digital watermarks. In order to do that, we attached a suitable representation to the following equipped poset P: CHAPTER 5. APPLICATIONS OF THE ALGORITHM OF DIFFERENTIATION VII IN CYBERSECURITY126 bc ⊗
e
⊗ a Figure 5.2. The diagram of the equipped poset P. where a ≺ b is a weak two-chain and b is a strong point. Actually, for the sake of applicability, a representation of P is interpreted in such a way that a vector of the form a + ib is interpreted as a fusion of the form a ∨ b, where a and b are (grayscale, RGB-color) images. The linear combinations are calculated over R.
A representation of P of the corresponding image algebra is a system of marked images (via the ∨ operation), it is defined as follows;
To each point a ∈ P, it is associated the image-subspace:
a a a a a a a Ua = R{Ia1 ,Ia2 ,...,Iai ,Ib1 ∨ Jb1 ,...,Ibs ∨ Jbs }.
The hull of Ua is the subspace:
+ a a a a a a a Ua = R{Ia1 ,Ia2 ,...,Iai ,Ib1 ,...,Ibs ,Jb1 ,...,Jbs }.
The point c has associated the image subspace:
U = U + {Ic ,Ic ,...,Ic ,Lc ∨ M c ,...,Lc ∨ M c }. c a R c1 c2 cl d1 d1 dm dm The point b has associated an image-subspace of the form:
a a a c c c a a a Ub = {I ,I ,...,I ,I ,I ,...,I ,J ,J ,...,J }, R ah1 ah2 aht ch1 ch2 chs bh1 bh2 bhu where {ah1 , ah2 , . . . , aht } ⊂ {a1, a2, . . . , ai}, {ch1 , ch2 , . . . , chs } ⊂ {c1, c2, . . . , cl},
{bh1 , bh2 , . . . , bhu } ⊂ {b1, b2, . . . , bs}.
Ub is generated by a set of watermarks which can be obtained from the subspace Uc by using the algorithm VII. The procedure goes as follows:
Algorithm 5.1.3: Extraction via differentiation VII a a a + 1. identify images I ,I ,...,I in U + Uc; ah1 ah2 aht a c c c + 2. identify images I ,I ,...,I in U + Uc; ch1 ch2 chs a a a a + 3. identify images J ,J ,...,J in Ua + Uc. bh1 bh2 bhu
The images J a ,J a ,...,J a are obtained via linear combinations of the corresponding bh1 bh2 bhu fusion Ia ∨ J a . That is, J a can be obtained by applying a linear combination of the bhi bhi bhi CHAPTER 5. APPLICATIONS OF THE ALGORITHM OF DIFFERENTIATION VII IN CYBERSECURITY127 form: k · (Ia ∨ J a ) + L · J a or k · (Ia ∨ J a ) + L · Ia . bhi bhi bhi bhi bhi bhi
5.1.6.1 Embedding a digital watermark
In order to embed digital watermarks into images generating the subspace Ua. The fol- lowing procedure must be carried out:
Algorithm 5.1.4: Embedding via integration
1. identify all the watermarks in subspace Uc;
a a a + 2. identify the images J ,J ,...,J in Ua + Uc; bh1 bh2 bhu 3. to fusion each image J a with a suitable image Ia (Ia ∨ J a ); bhi bhi bhi bhi
4. to construct the new subspace Ua.
In the following Figure 5.3, we present the matrix form of the integration process of D-VII, to more details, see section 2.3 and [67]. The J a are the watermarks, all empty blocks bhi (those marked by ∗) are supposed to be zero (to be arbitrary images). The process of a embedding the J images of the Ub space is the process of inserting digital watermarks. Of bhi course, lowercase letters, above and below the matrix, refer to the vector space associated Ia bhi a a to the point, for example, a represents Ua. The block a means the fusion Ib ∨ Jb . J hi hi bhi a c b Ia ai Ia bhi J a J a bhi bhi <
a c a ∗ Ia ∗ Ic J ∗ U i hi bhi b − − a c | {z } a+ c+ Figure 5.3. The matrix presentation of embedding a digital watermark process.
Since this procedure is independent of domain, it can be applied in a spatial and frequency domain. Wavelets are functions that satisfy certain mathematical requirements, which are used to represent other data or functions. Experimental results, regarding the subject, are presented in Appendix B. To do that, MATLAB routines are implemented in an ASUS Notebook PC, with model: UX430U. These routines are implemented as digital watermark systems, both for embedding and extracting watermarks. An example of a singular value CHAPTER 5. APPLICATIONS OF THE ALGORITHM OF DIFFERENTIATION VII IN CYBERSECURITY128 decomposition routine is also presented, which is applied after a wavelet procedure to embed a digital watermark.
5.2 Viruses and worms
In this section, we present how the algorithm of differentiation VII, is a model to the process of infection and detection of viruses in a computer network.
Definition 29. Malicious software or malware are programs intentionally designed to perform some unauthorized often harmful or undesirable act [37].
5.2.1 Viruses
In the present session, we make a theoretical introduction, about some of the most popular types of malicious software, which include, computer viruses, worms, Trojan horses and polymorphic viruses, as is presented in [37].
In 1987, Fred Cohen, the pioneer researcher in computer viruses, defined a computer virus as follows.
Definition 30. A virus is a program that can infect other programs by modifying them to include a possibly evolved copy of itself.
The following is the pseudocode of a simple computer virus. This one is a typical structure of a computer virus which contains three subroutines.
The first subroutine, infect-executable, is responsible for finding available executable files and infecting them by copying its code into them.
The subroutine do-damage, also known as the payload of the virus, is the code responsible for delivering the malicious part of the virus.
The last subroutine, trigger-pulled, checks if the desired conditions are met in order to deliver its payload. Examples of conditions could be the day of the week, the number of infections, or the current date.
Nowadays, there are so many different kinds of computer viruses that it is difficult to provide a precise definition. Some of these kinds of viruses are: boot sector viruses, file infecting viruses and memory resident viruses.
A more exact and contemporary definition of virus was given by Grimes in 2005.
Definition 31. A virus is a malicious program that modifies other host files (few excep- tions) or boot areas to replicate. In most cases, the host object is modified to include a complete copy of the malicious code program. The subsequent running of the infected host file or boot area then infects other objects. CHAPTER 5. APPLICATIONS OF THE ALGORITHM OF DIFFERENTIATION VII IN CYBERSECURITY129
Boot sector viruses: Boot sector viruses infect the master Boot sector of hard drives or floppy drives and infect other machines only when the machine boots up from an infected floppy disk. Boot sector viruses were the firs successful viruses created and can infect a machine regardless of what Operating Systems runs on it. Today, they are rarely found because floppy disks are not so common any more. One of the most successful boot sector viruses of all time is the Stoned virus, including his infamous variant Michelangelo. Stones is a very simple boot sector virus, but at one time, it was so prevalent that National Computer Security Association reported that roughly one out of every four virus infections involved some form of Stoned.
File infecting viruses (FIV): Program viruses infect executable programs, such as EXE or COM, by attaching themselves to them. The virus executes and infects other executables when its host file is executed. To infect an EXE file, a virus has to modify the EXE- Header and the Relocation Pointer Table (RPT) [A variable length of pointers to segment references] and add its own code to the Load Module (LM). This can be done in many ways.
Memory resident viruses: Memory resident viruses remain in memory after the initializa- tion of virus code. They take control of the system and allocate a block of memory for their own code. They remain in memory while other programs run and infect them. MRV’s are highly portable because they can jump across both directories and disk drives by taking advantage of the user’s actions. Worth noting that FIV’s are also MRV’s. These viruses distribute the infection process over time better than direct acting viruses. A technique that a virus can use to go resident is to take advantage of a memory hole, which is an unused part of memory that is not likely to be overwritten.
5.2.1.1 Worms
The first researcher who tried to give a formal and mathematical definition of a computer worm was Fred Cohen in 1992. A more easy to understand definition is: Definition 32. A worm is a sophisticated piece of replicating code that uses its own program coding to spread, with minimal user intervention.
A worm usually exists as a standalone program that executes itself automatically on a remote machine without any user interaction. They typically do not require any host files although some types of worms do. It is derived from the word “tapeworm” a parasitic organism that lives inside a host and saps its resources to maintain itself.
What makes worms faster spreading and more dangerous than viruses is that they require no user interaction to replicate. Some worms use active network connections to send themselves and infect new targets. SQL Slammer, which was the fastest spreading worm ever created, infected its victims by randomly selecting IP addresses, eventually finding and infecting all prone hosts.
5.2.1.2 Network worms life cycle
The life cycle of a worm is divided into seven phases: CHAPTER 5. APPLICATIONS OF THE ALGORITHM OF DIFFERENTIATION VII IN CYBERSECURITY130
1. Initialization phase; in which the worm may install software, determine the config- uration of the local machine, instantiate global variables, and begin the main worm process.
2. Payload activation phase; where the worm activates its Payload. This phase is logically distinct from the other phases and it does not usually affect the network behavior of the worm.
3. Network propagation phase; which is the phase that encompasses the behavior that describes how a worm moves through a network. In this phase, the worm selects a set of targets and tries to infect them.
4. The network acquisition phase; describes the process a worm goes through to select targets for infection.
5. Network Reconnaissance Phase; the worm attempts to learn about the environment, particularly with respect to the targeted hosts. This phase includes validating what a worm knows about the environment and enables the worm to make more informed decisions about its operations.
6. Attack phase; is when the worm preforms actions that enable it to acquire elevated privileges on a remote system, usually exploiting other software vulnerabilities.
7. The infection phase; is the last phase in the worms life cycle and is when the worm leverages the acquired privileges on the target host to begin the Initialization phase of a new instance of the worm.
We recall that the infamous Ransomware of type Wannacry is a kind of Cryptoworm spreading itself by using Eternal Blue developed by the NSA.
5.2.1.3 Trojan horses
A Trojan horse program is any program that intentionally hides its malicious actions while pretending to be something else.
Definition 33. A Trojan or Trojan Horse is a non-replicating program masquerading as one type of program with its real intent hidden from the user.
A Trojan is different from a virus as it does not copy its code to other host files or boot areas; instead it exists as a standalone program that masquerades as something else. Also, it does not copy itself in order to replicate. Trojan are often distributed by worms, the following are various types of Trojans.
1. Remote Administration Trojans (RAT): allow remote malicious users to have com- plete control of a machine. Complete control means that a malicious user can read what the user is reading, record key strokes, capture screen shots, copy and delete files, and many more. RATs consist of a client which is installed on the victims com- puter and a server which enables malicious users to remotely control the client some of the most popular RATs found in the wild was NetBus, Subseven, Deepthroat and the infamous Back Orifice. CHAPTER 5. APPLICATIONS OF THE ALGORITHM OF DIFFERENTIATION VII IN CYBERSECURITY131
2. Back Door Trojans: open up a new entry point for hackers by installing a new TCP/IP service or mapping a new drive share. The Trojan can easily open up windows, drive shares, which can be accessed from the Internet.
3. Network Redirection Trojans: as their name suggests, allow network redirection. They allow malicious users to redirect specific attacks, through a compromised in- termediate host, toward a new target.
4. Distributed Attacks Trojans: spread to as many machines as they can and then wait for commands. Usually, the exploit is directed towards another central target. Up to thousand of these malicious programs can be spread over a period of months and can be then used to attack a common target, causing a Denial of Service (DoS) or gather information that can be used later. Infected machines are called Zombies and the set of infected machines is called Zombie Network.
One increasingly popular activity among cyber-criminals is the use of malware to create zombie computers. A zombie computer or bot, is a computer on which a worm or virus has installed programs that run automatically and allow cyber-criminals access and control. Cyber-criminals use these zombie computers to search for other vulnerable computers, install their programs or store data. A zombie network or bot network, is a set of infected machines often compromised weeks or months earlier by attackers using worms or viruses to plant backdoor components. These zombie networks can contain from a few hundreds to several thousands of compromises machines and can be used to launch simultaneous Denial-of-Service (DoS) attacks, send spam, host illegal websites or illegal material, and other criminal activities.
5.2.1.4 Polymorphic viruses
Polymorphism is the next step viruses writers took to challenge antivirus software. The term polymorphic come from Greek word poli, which means many, and morphi, which means form. The 1260 virus was the first known polymorphic virus, see [61].
Definition 34. A polymorphic virus is a kind of virus that can take many forms. Poly- morphic virus can mutate their decryptors to a high number of different instance that take millions of different forms. They use their mutation engine to create a new decryption routine would have exactly the same functionality, but the sequence of instructions could be completely different.
5.2.2 Virus detection mechanisms
This section presents some techniques used by antivirus software to detect computer viruses. Among the mechanisms to detect viruses are wildcards, mismatches, generic detection, and string scanning which is defined as follows (see, [37] for more details).
5.2.2.1 First generation scanners
First Generation Scanners use simple methods to detect computer viruses. Techniques usually involve scanning for pre-defined sequences of bytes called strings. CHAPTER 5. APPLICATIONS OF THE ALGORITHM OF DIFFERENTIATION VII IN CYBERSECURITY132
String scanning: String scanning is the simplest technique used by an anti-virus software to detect computer viruses. It searches for sequence of bytes (strings) that are typical of a specific virus, but not likely to be found in other programs. This sequence of bytes is often called the signature of the virus, which is extracted for each different virus and organized in a database. The virus scanner will then use this database to search files and system areas for presence of the virus. The following example string is a typical pattern of the W32/Beast virus in EXE files, which was published in 1999.
83EB 0274 1683 EBOE 740A 81EB 0301 0000
The virus scanner checks the code of EXE files and when it encounters the previous string it announces detection of the Beast virus. Usually sixteen unique bytes is long enough to detect a 16-bit malicious code without any false positives. But for 32-bit viruses to be detected safely, longer strings might be necessary, especially when a virus is written in a high level language.
5.2.3 Intentional and accidental interactions
Computer virus researches have observed a set of interesting behaviors that are the result of intentional and accidental interactions between various kinds of malicious code.
5.2.3.1 Cooperations
Some computer viruses accidentally cooperate with other malicious code. For example, computer worms might get infected with a standard file infector virus as they pass through already infected nodes. It is common to encounter multiple infections on top of in-the-wild computer worms. It is not uncommon to find three or even more different viruses on the top of a worm carrier. This can help both the network worm and the standard file infector virus in various ways.
A worm can take advantage of the infection of an unknown file infector virus. If the file infector virus is unknown to antivirus products, the computer worm body might not be detectable. For example, in some cases the worm body will be embedded deep inside the virus code, leaving little chance for the antivirus program to find it.
5.2.4 The algorithm of differentiation VII towards secure computer net- works
In this section, differentiation VII is used to model the way that an antivirus program detect and delete malware.
The following is the structure of the three subroutines of a computer virus, which were defined at the beginning of this section.
1. Infect-executable: the computer is infected with a worm. CHAPTER 5. APPLICATIONS OF THE ALGORITHM OF DIFFERENTIATION VII IN CYBERSECURITY133
2. Do-damage: the worm successfully penetrates a new remote system. That computer, however, is already infected with a virus that infects exactly the same type of files in which the worm spread itself. Thus, the file infector virus attaches itself to the worm.
3. Trigger-pulled: the multiple infection arrives at a new computer when the worm is executed, the infected file runs (in most cases, it will run before the worm's code) and infected other files.
5.2.4.1 Some preliminary assumptions
Let P = {C1,C2,...,Cn, b} be a computer network, considered as a poset whose relations are induced by the different connections between points Ci's. We suppose that the net- work is susceptible to a hacker attack, which eventually will infect the network with a polymorphic virus which uses a worm to travel throughout the network. We also suppose that the virus is stored in a file associated to the point b.
⊗ Cn b e ⊗ Cn−1 . .
P = ⊗C3 ⊗ C4 ⊗ C5
AA A A ⊗ C2 A A ⊗ C1
Figure 5.4. The diagram associated with the computer network P.
The P computer network becomes a PZ zombie computer network, after infection. The new order of the computers in PZ = {T1,T2,...,Tn, b} is linearly ordered according to the infection, i.e., the computer network assumes the following shape Tn = {a = T0 ≺ T1 ≺ T2 ≺ · · · ≺ Tn} (where Ti = Cσ(i), σ is a permutation of the indexes {1, 2, . . . , n}), in such a way that the files inside the network of each computer is constructed using the files of lower-ranking computers, i.e., the files of {T0,T1,T2 ...,Tn−1} contributes with the construction of the files of Tn, where T0 is going to be the lowest ranking computer or minimal point in the chain Tn.
⊗ Tn b e ⊗ Tn−1 . . PZ = ⊗ T2
⊗ T1
Figure 5.5. The diagram associated with the zombie computer network PZ . CHAPTER 5. APPLICATIONS OF THE ALGORITHM OF DIFFERENTIATION VII IN CYBERSECURITY134
The interaction between worms an viruses (e.g., trojans) can infect other susceptible files in a lowest rank computer.
The susceptible files mutate within it becoming an infected file so that a susceptible file is nothing more than a change in the appearance of an infected one (susceptible files cannot be fused). In this case, it is assumed that a file can be determined from the fusion of two files. The interaction between files can cause mutations in the susceptible file and also within the network there are those uninfected files in the computer Ti, which are shared with the computer Tj, i ≤ j, and these files tend to mutate to an infected file in the computer Tj.
5.2.4.2 The infection
In the section 5.2.1, we defined what are the three classic subroutines of the action of a virus, according to our model these three subroutines are configured as follows:
A virus or virus cooperation affects the computer network P whose structure is given by the files in Cn in the following way:
1. by inserting itself into a susceptible file of a computer Ci of the network;
2. by making a fusion with susceptible files in any of the computers Ci; 3. there are two types of infections, worm infections and Trojan infections, sometimes Trojan virus use worms to spread out the infection throughout the computer network;
i 4. initial susceptible files will be called of type V0 .
Fusions between files make that the detection task be a difficult task. In this case, The fusion is carried out by files of type ti or W i (files infected by Trojans and files infected by worms respectively).
The following Figure 5.6 is an example of the way that a cooperation, Trojan-worm affects a computer network. In this case, the infection makes of the computer network a weak chain (the order is given by the chronological time of the infection) or an equipped poset whose representation is given by the type of cooperation.
The infection spreads out to the network via fusions of type (W i,V i) between susceptible i i files generally of type EXE, DOC, PDF, JPEG, PNG and infected files V . We let V0 denote those susceptible files with no infections.
For the sake of simplicity, we assume n = 6, but in general, the model have the same behavior. Also, in this figure, the black lines between the computers Ci and Cj means the initial connection of these computers within the network. The red lines mean the connec- tions (order) between the computers Tl and Tm after the virus infection. For example, the C5 (C2) computer becomes the T2 (T3) computer after the virus infection, T2 ≺ T3 in the Zombie network (after the infection), provide that computer T2 has been infected before than the computer T3. CHAPTER 5. APPLICATIONS OF THE ALGORITHM OF DIFFERENTIATION VII IN CYBERSECURITY135
6 ¨¨*(t6, f ) - 6 6 T6 (W ,V ) .....HHj ..... V 6 @@...... @@ 0 6 Computer Network 5 Virus ¨¨*(t5, f ) - 5 5 T5 (W ,V ) .....HHj C5 C6 ------..... V 5 ...... > @@...... @@ 0 @...... @ @...... @ @.....@ @.....@ 6 - 4 @ (t4, f ) - ¨¨* @ - 4 4
- T4 (W ,V )
- @ .....HHj 4
- - ..... @.....@V0
C4 ------C3 @.....@
- Infection ...... 6
- 3
@.....@ @.....@ (t3, f )
- @ - @ @ @
..... - ..... ¨¨* -
- 3 3
@ - T3 - (W ,V ) - -
@ - .....H - - Hj 3 - ..... V @ @@...... @@ 0 C2 C1 6 ...... (t , f 2) ...... ¨¨* 2 @.....@ @.....@ - 2 2 @.....@ @.....@ T2 (W ,V ) .....HHj ..... V 2 @@...... @@ 0 6 1 ¨¨*(t1, f ) - 1 1 T1 (W ,V ) .....HHj ..... V 1 @@...... @@ 0 Figure 5.6. Infection of a virus in a computer network.
5.2.4.3 Tracking and detection
In this model, computers most prone to the action of an infection are those with the lowest rank which have a large number of files. Some of these files are infected after the initialization stage of the infection.
In order to control the infection an antivirus program activates on files of type T0 the trigger-pulled stage of the infection in a emulator. Then, the antivirus, identifies and separates the infected files the success of such separation depends on the capacity of the i i i antivirus program to identify virus signatures and fusions of the type, (ti, f ) and (W ,V ) where the signature of the virus is more difficult to detect.
Each computer Ti, for 0 ≤ i ≤ n, contains the following three types of files:
0 (i) Type Vi which are not infected susceptible files; (ii) Fusions (W i,V i) which are infected files by a worm W i;
i (iii) Fusions of type (ti, f ) which are files infected by a polymorphic Trojan virus. Such infections are activated automatically according to the trigger-pulled restrictions, without take into account the behavior of the worm. CHAPTER 5. APPLICATIONS OF THE ALGORITHM OF DIFFERENTIATION VII IN CYBERSECURITY136
5.2.4.4 Notation
We denote by ⊗ a susceptible computer, which we consider a weak point of the computer network. ◦ denotes a zombie computer or zombie computer network.
We can represent files as vectors with coordinates in a square extension G of a field F . This is the structure of the computers associated with a representation of PZ on an F -vector space U0. So, the matrix representation associated to a weak or susceptible computer Ti are described in the form:
I I UT = i iI
For instance, 1 0 * + 0 1 UT = , i 0 i 0 0 G In this case, we can see that the first vector corresponds to a signature of a virus easy to identify whereas the second vector corresponds to an infected file by a virus whose signature is hard to detect. 1 0 1 0 0 1 i 1 , −! , . 0 0 0 0 0 0 0 0 From an algebraic point of view an antivirus develops a linear combination to extract virus signatures as shown above.
Therefore, we can use arguments from the theory of representation of equipped posets. And it is possible to interpret the network computer as an equipped poset with a pair (a, b)-suitable for the differentiation VII. The action of the algorithm VII permits us, to i i i detect the infected files, which are of the form, (ti, f ) or (W ,V ), and the user decides whether they must be put in quarantine or delete it.
The action of the antivirus program consists of identifying infected files to be deleted from the computer. In the next example, we illustrate the identification of a worm fusion. CHAPTER 5. APPLICATIONS OF THE ALGORITHM OF DIFFERENTIATION VII IN CYBERSECURITY137 ¨H 6 6 ¨ H ¨¨*(t6, f ) ¨¨*(t6, f ) - 6 6 - 6 6 (W 6,V 6) T6 (W ,V ) T6 (W ,V ) .....HHj .....HHj ..... V 6 ..... V 6 H ¨ @@...... @@ 0 @@...... @@ 0 H¨ 6 6 ¨H6 5 5 ¨ H ¨¨*(t5, f ) ¨¨*(t5, f ) - 5 5 - 5 5 (W 5,V 5) T5 (W ,V ) T5 (W ,V ) .....HHj .....HHj ..... V 5 ..... V 5 H ¨ @@...... @@ 0 @@...... @@ 0 H¨ 6 6 6 ¨H 4 4 ¨ H ¨¨*(t4, f ) ¨¨*(t4, f ) - 4 4 - 4 4 (W 4,V 4) T4 (W ,V ) T4 (W ,V ) ..... ..... .....HHj 4 .....HHj 4 H ¨ @.....@V0 @.....@V0 H¨ @.....@ - @.....@ - 6 6 ¨H6 3 Antivirus 3 Quarantine ¨ H ¨¨*(t3, f ) ¨¨*(t3, f ) - 3 3 - 3 3 (W 3,V 3) T3 (W ,V ) T3 (W ,V ) .....HHj .....HHj ..... V 3 ..... V 3 H ¨ @@...... @@ 0 @@...... @@ 0 H¨ 6 6 ¨H6 2 2 ¨ H ¨¨*(t2, f ) ¨¨*(t2, f ) - 2 2 - 2 2 (W 2,V 2) T2 (W ,V ) T2 (W ,V ) .....HHj .....HHj ..... V 2 ..... V 2 H ¨ @@...... @@ 0 @@...... @@ 0 H¨ 6 6 ¨H6 1 1 ¨ H ¨¨*(t1, f ) ¨¨*(t1, f ) - 1 1 - 1 1 (W 1,V 1) T1 (W ,V ) T1 (W ,V ) .....HHj .....HHj ..... V 1 ..... V 1 H ¨ @@...... @@ 0 @@...... @@ 0 H¨ 6 6 Zombie network ¨¨HH 1 -(t1, f ) V i T0 - 0 V 1 ...... 0 H ¨ @@...... @@ H¨
Figure 5.7. Antivirus action in the infected computer network.
+ + Figure 5.8. Differentiation VII allows to recognize infected files, in this case, c1 b and c2 b. MAT- LAB routines describing the process can be found in Appendix B.1 CHAPTER 5. APPLICATIONS OF THE ALGORITHM OF DIFFERENTIATION VII IN CYBERSECURITY138
Figure 5.9. Integration is a way of an infection process. In this figure, we show how integration + allows that file c1 b infects files a, c1, c2 and c3. MATLAB routines describing the process can be found in Appendix B.1 APPENDIX A
The Auslander-Reiten quiver of some ordinary posets
In this section it is shown tables with some sincere ordinary posets, some sincere equipped posets, and its evolvents, we point out that Zavadskij gave the classification of these sincere posets in [68]. For some ordinary posets of the finite representation type, we give their indecomposable projective and injective representations, which are vertices of Auslander- Reiten quivers. For the ordinary poset of finite growth representation type K1, we give its indecomposable projective and injective representations, which have been used to build its corresponding preprojective and preinjective components.
A.1 Some sincere ordinary and equipped posets
In this section, we present tables that regard some sincere non-trivially equipped posets, its evolvents and some sincere trivially equipped posets as Zavadskij classified in [68].
F1 F2 F3 F4 F5 F6
d d d d @ d d d d d@ d d d d d d d d d d d d d d d d d d d d F7 F8 F9 F10 F11 F12
d ¡ d d d d d ¡ J @ d @ d d@ d d d¡ d JJ d d d d@ d d d d d d d d d d d d d d d d d d Table A.1. The Hasse diagrams of the sincere trivially equipped posets of finite representation type.
139 APPENDIX A. THE AUSLANDER-REITEN QUIVER OF SOME ORDINARY POSETS 140
K1 K2 K3 K4 K5 d d d d d d d @ d d d d d d@ d d d d d d d d d d d d d d d d d d d d
Table A.2. The Hasse diagrams of the trivially equipped posets of finite growth representation type (Kleiner's critical posets).
F13 F14 F15 F16 F17 F18
⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ d ⊗ ⊗ d d d d
Table A.3. The diagrams of the sincere non-trivially equipped posets of finite representation type.
F13 F14 F15 F16 F17 F18 4 ⊗ 3 3 2 ⊗ ⊗ ⊗ 3 ⊗ ⊗ ⊗ ⊗ d ⊗ ⊗ 1 1 2 1 2 1 2 1 1 2 d d d d
Fd13 = F7 Fd14 = F2 Fd15 = F3 Fd16 = F8 Fd17 = (2, 2) Fd18 = F 0 00 4 4
0 00 0 00 3 3 3 2 2 0 A ¡ 00 3 d¡A d3 d d d d d d¡ Ad 0 00 0 00 0 00 0 00 0 00 0 00 1 1 1 1 2 1 2 2 1 1 2 1 1 1 2 2 d d d d d d d d d d d d d d d d
Table A.4. The diagrams of the sincere non-trivially equipped posets of finite representation type and its corresponding evolvent. APPENDIX A. THE AUSLANDER-REITEN QUIVER OF SOME ORDINARY POSETS 141
K6 K7 K8 K9
⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ d d d d d
Table A.5. The diagrams of the sincere critical non-trivially equipped posets of finite growth representation type (one parameter).
K6 K7 K8 K9 4 ⊗ 2 4 ⊗ 3 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ d 1 2 1 2 1 2 3 1 3 d d d d
Kc6 = K1 Kc7 = K3 Kc8 = K1 Kc9 = K2 0 00 4 4
0 00 0 00 2 2 4 3 d d3 d d d d d 0 00 0 00 0 00 0 00 0 00 1 1 2 2 1 2 2 1 1 2 3 1 1 3 d d d d d d d d d d d d d d
Table A.6. The diagrams of the sincere critical non-trivially equipped posets of finite growth representation type (one parameter) and its corresponding evolvent.
A.2 Some ordinary posets of finite representation type
In this section, it is used the classic Auslander-Reiten theory described before in order to show the Auslander-Reiten quiver of some ordinary posets of finite representation type. To do that, we present the indecomposable projective representations, its radicals and the indecomposable injective representations of these posets. Although, this work is not devoted to the description of the Auslander-Reiten quiver of ordinary posets, we take into account such quivers in order to apply the lifting algorithm to obtain Auslander-Reiten quivers of equipped posets via the evolvent associated in the procedure.
Below, we describe the ordinary poset F7, its indecomposable projective and injec- tive representations, the radical of the projective representations and the corresponding Auslander-Reiten quiver Γ(F7). In the section 3.2, we present 1 F7.
F7 = 1e0 1 e00 APPENDIX A. THE AUSLANDER-REITEN QUIVER OF SOME ORDINARY POSETS 142
P (10 ) = 1 | 1 0 I(10 ) = 1 | 0 1 P (100 ) = 1 | 0 1 I(100 ) = 1 | 1 0
P (∅) = 1 | 0 0 I(F7) = 1 | 1 1
0 00 0 00 00 0 rad P (1 ) = rad P (1 ) = P (∅); P (1 ) = I(1 ); P (1 ) = I(1 ).
1 | 1 0 ¡¡ A ¡ AAU Γ(F7) : 1 | 0 0 − − 1 | 1 1 A ¡¡ AAU ¡ 1 | 0 1
Figure A.1. The Auslander-Reiten quiver Γ(F7) of the ordinary poset F7.
Now, we present the ordinary poset F2. The indecomposable projective representations, its radicals, and the injective representations, the corresponding Auslander-Reiten quiver Γ(F2) is described as well.
F2 = 1e0 1 e00 2 e
P (10 ) = 1 | 1 0 0 I(10 ) = 1 | 0 1 1 P (100 ) = 1 | 0 1 0 I(100 ) = 1 | 1 0 1 P (2) = 1 | 0 0 1 I(2) = 1 | 1 1 0
P (∅) = 1 | 0 0 0 I(F2) = 1 | 1 1 1
0 00 rad P (1 ) = rad P (1 ) = rad P (2) = P (∅).
1 | 1 0 0 − − 1 | 0 1 1 ¡¡ A ¡¡ A ¡ AAU ¡ AAU Γ(F2) : 1 | 0 0 0 − −2 | 1 1 1 − − 1 | 1 1 1 CA CA ¡¡££ ¡¡££ C AAU ¡ £ C AAU ¡ £ C C C1 | 0 1 0 −£ − C1 | 1 0 1 £ £ £ C £ C £ CCW £ CCW £ 1 | 0 0 1 − − 1 | 1 1 0
Figure A.2. The Auslander-Reiten quiver Γ(F2) of the ordinary poset F2. APPENDIX A. THE AUSLANDER-REITEN QUIVER OF SOME ORDINARY POSETS 143
In this part, we describe the ordinary poset F3, their indecomposable projective represen- tations and their indecomposable injective representations, the radicals of the projective representations and the corresponding Auslander-Reiten quiver Γ(F3). In the section 3.2, we present 1 F3. 30 300
F3 = e e
1e 2 e0 2 e00
0 0 0 1 0 1 1 0 0 1 P (1) = 1 P (3 ) = 1 I(1) = 1 I(3 ) = 1 1 0 0 0 0 0 0 1 1 1 0 1
0 1 0 00 0 1 0 1 1 00 1 0 P (2 ) = 1 P (3 ) = 1 I(2 ) = 1 I(3 ) = 1 0 1 0 0 0 0 1 0 1 1 1 0
00 0 1 0 0 00 1 1 1 1 P (2 ) = 1 P (∅) = 1 I(2 ) = 1 I(F3) = 1 0 0 1 0 0 0 1 1 0 1 1 1
0 00 0 0 00 00 rad P (1) = rad P (3 ) = rad P (3 ) = P (∅); rad P (2 ) = P (3 ); rad P (2 ) = P (3 ).
1 0 0 1 1 1 1 0 1 0 1 0 − − −1 1 0 0 − − −1 0 0 1 − − − 1 1 1 0 @ @ @ @ @@R @@R @@R @@R 1 0 1 1 1 2 2 1 1 1 1 0 0 0 −−−2 1 1 0 −−−2 1 0 1 −−−2 1 1 1 −−− 1 1 1 0 @ @ @ @ @ @@R @@R @@R @@R @@R 0 0 1 1 2 2 2 2 2 2 1 1 Γ(F3) : 1 0 0 0 −−−2 1 0 0 −−−3 1 1 1 −−−3 2 1 1 −−−2 1 1 1 −−− 1 1 1 1 B@ ¢ B@ ¢ B@ ¢ B@ ¢ B@ ¢ ¢ ¢ ¢ ¢ ¢ B @@R B @@R B @@R B @@R B @@R B 0 0 ¢ B 1 1 ¢ B 1 1 ¢ B 1 1 ¢ B 1 1 ¢ B 1 1 0 0 ¢−−−B 1 0 0 0 ¢ −−−B 2 1 1 1 ¢ −−−B 1 1 0 0 ¢ −−−B 1 0 1 1 ¢ B ¢ B ¢ B ¢ B ¢ B ¢ B ¢ B ¢ B ¢ B ¢ B ¢ BN ¢ BN ¢ BN ¢ BN ¢ BN ¢ 0 1 1 1 2 1 1 2 1 1 1 0 0 0 −−−2 1 0 1 −−−2 1 1 0 −−−2 1 1 1 −−− 1 1 0 1 @ @ @ @ @@R @@R @@R @@R 0 1 1 0 1 1 0 1 1 0 0 1 − − −1 1 0 0 − − −1 0 1 0 − − − 1 1 0 1
Figure A.3. The Auslander-Reiten quiver Γ(F3) of the ordinary poset F3.
The following are the ordinary poset F8, its indecomposable projective and injective rep- resentations, the radicals of the projective representations, as well, and the corresponding Auslander-Reiten quiver Γ(F8). APPENDIX A. THE AUSLANDER-REITEN QUIVER OF SOME ORDINARY POSETS 144
3
F8 = e
1e0 1 e00 2 e
0 0 1 0 1 0 P (1 ) = 1 P (3) = 1 I(1 ) = 1 I(3) = 1 1 0 0 0 0 0 0 1 1 1 1 0
00 0 0 00 1 1 P (1 ) = 1 P (∅) = 1 I(1 ) = 1 I(F8) = 1 0 1 0 0 0 0 1 0 1 1 1 1
1 1 P (2) = 1 I(2) = 1 0 0 1 1 1 0
0 00 0 0 00 00 rad P (1) = rad P (3 ) = rad P (3 ) = P (∅); rad P (2 ) = P (3 ); rad P (2 ) = P (3 ).
0 − − −1 − − − 1 1 1 0 0 1 0 1 0 1 1 0 1 @ @ @ @ @ @ @R @R @R 0 − − −1 − − −2 − − − 1 Γ(F8) : 1 0 0 0 2 1 1 0 2 1 1 1 1 1 1 1 B ¢¢ B@ ¢¢ B@ ¢¢@ B @ B @ B @ B @R ¢ B @R ¢ B @R ¢ 0 ¢ 1 ¢ 1 ¢ B 1 ¢− − −B 1 ¢ − − −B 1 ¢ B 0 1 0 B 1 0 0 B 0 1 1 B ¢ B ¢ B ¢ ¢ ¢ ¢ BBN ¢ BBN ¢ BBN ¢ 1 − − −1 − − − 1 1 0 0 0 2 1 1 1 1 1 1 0 @ @ @ @ @R @R 1 − − − 0 1 0 0 1 1 1 1 0
Figure A.4. The Auslander-Reiten quiver Γ(F8) of the ordinary poset F8.
In the sequel, it is described the poset (2, 2) and the corresponding indecomposable pro- jective and injective representations, also the radical of the projective representations and the corresponding Auslander-Reiten quiver Γ(2, 2). In the section 3.2, we present 1 (2, 2). APPENDIX A. THE AUSLANDER-REITEN QUIVER OF SOME ORDINARY POSETS 145
20 200
(2, 2) = e e
1e0 1 e00
0 1 0 00 0 1 1 1 00 1 0 P (1 ) = 1 P (2 ) = 1 I(10) = 1 I(2 ) = 1 1 0 0 0 0 1 1 0
00 0 1 0 0 00 1 1 1 1 P (1 ) = 1 P (∅) = 1 I(1 ) = 1 I(2, 2) = 1 0 1 0 0 1 0 1 1
0 1 0 0 0 1 P (2 ) = 1 I(2 ) = 1 0 0 0 1
0 00 0 0 00 00 0 00 rad P (2 ) = rad P (2 ) = P (∅); rad P (1 ) = P (2 ); rad P (1 ) = P (2 ); P (1 ) = I(2 ); P (100 ) = I(20 ).
1 0 1 1 0 @ @ @R 1 0 − − − 1 1 1 0 0 1 1 0 @ @ @ @ @R @R 0 0 − − −1 1 − − − 1 1 Γ(2, 2) : 1 0 0 1 0 0 1 1 1 @ @ @ @ @R @R 0 1 − − − 1 1 1 0 0 1 0 1 @ @ @R 0 1 1 0 1 Figure A.5. The Auslander-Reiten quiver Γ(2, 2) of the ordinary poset (2, 2).
In this part, it is presented the ordinary poset F, their indecomposable projective represen- tations and their indecomposable injective representations, the radicals of the projective representations and the corresponding Auslander-Reiten quiver Γ(F), as well as, the cal- culus of the 1 F by using the lifting algorithm. APPENDIX A. THE AUSLANDER-REITEN QUIVER OF SOME ORDINARY POSETS 146
0 00 4 4
A ¡ eA ¡ e 0 A ¡ 00 F = 3 ¡A 3 e¡ A e ¡ A
1e 2 e0 2e00
0 0 0 1 1 1 1 1 P (1) = 1 0 0 P (300 ) = 1 0 1 I(1) = 1 1 1 I(300 ) = 1 1 0 1 0 0 0 0 0 0 1 1 1 1 0
1 1 1 0 1 1 0 1 P (20 ) = 1 1 0 P (40 ) = 1 0 0 I(20 ) = 1 1 1 I(40 ) = 1 0 1 0 1 0 0 0 0 1 0 1 1 0 0
1 1 0 1 1 1 1 0 P (200 ) = 1 0 1 P (400 ) = 1 0 0 I(200 ) = 1 1 1 I(400 ) = 1 1 0 0 0 1 0 0 0 1 1 0 1 0 0
1 0 0 0 1 1 1 1 0 0 P (3 ) = 1 1 0 P (∅) = 1 0 0 I(3 ) = 1 0 1 I(F) = 1 1 1 0 0 0 0 0 0 1 0 1 1 1 1
0 00 0 0 00 00 rad P (1) = rad P (4 ) = rad P (4 ) = P (∅); rad P (3 ) = P (4 ); rad P (3 ) = P (4 ). APPENDIX A. THE AUSLANDER-REITEN QUIVER OF SOME ORDINARY POSETS 147 1 1 1 1 ¢ 1 1 1 ¢ ¢ ¢ @R ¢ 1 ¢ @
1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 0 1 BN B @R @R 1 1 1 B B @ @
B B 1 1 1 0 2 2 2 2 1 1 0 1 ¢ 1 1 0 1 1 1 1 0 1 ¢ ¢ ¢ @R @R ¢ 1 2 1 ¢ @ @
. 2 2 2 1 1 1 1 1 2 2 1 2 F 1 1 1 1 0 0 1 1 1 BN B @R @R 2 1 2 B B @ @
B B 1 1 1 1 3 3 2 2 1 1 1 1 ¢ 0 0 1 2 1 1 0 1 0 ¢ ¢ ¢ @R @R ¢ 1 3 1 ¢ @
@ 2 2 1 2 2 2 1 1 2 2 2 1 1 0 1 1 1 1 1 1 0 BN B @R @R 2 2 2 B B @ @
B B ) of the ordinary poset 1 1 0 1 3 3 2 2 1 1 1 0 F ¢ 1 0 0 1 1 1 1 0 0 ¢ ¢ ¢ @R @R @R ¢ 1 3 1 ¢ @
@
@ 0 1 0 1 2 2 1 1 1 1 1 1 2 2 1 1 1 0 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 0 0 BN B @R @R @R 1 2 1 2 1 B B @ @
@
B B 1 2 1 1 2 2 1 1 2 1 1 1 ¢ 1 1 0 1 0 0 1 0 1 ¢ ¢ ¢ − − − − − − @R @R @R ¢ 2 2 2 ¢
@
@ @ 1 1 1 0 1 2 1 1 1 1 0 0 2 1 1 1 1 1 0 1 0 1 0 1 0 0 1 0 0 1 0 0 0 0 1 BN B @R @R @R 1 2 1 2 1 B B @ @ @
B B 1 1 1 0 2 2 1 1 1 1 0 1 The Auslander-Reiten quiver Γ( ¢ 0 0 0 2 0 0 0 0 0 ¢ ¢ ¢ @R @R ¢ 1 3 1 ¢ @
@ 2 1 1 0 1 1 1 1 1 2 0 1 1 0 0 1 0 0 1 0 0 BN B @R @R 2 2 2 B B @
@
Figure A.6. B B 1 0 0 0 2 2 1 1 0 1 0 0 ¢ 1 0 0 1 0 0 1 0 0 ¢ ¢ ¢ @R @R ¢ 3 1 1 ¢ @
@ 1 1 0 1 1 1 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 BN B −−− −−− −−− −−− −−− −−− −−− −−− −−− −−− −−− −−− @R @R 2 1 2 B B @ @
B B 0 1 0 1 1 1 0 0 1 0 1 0 ¢ 0 0 0 1 0 0 0 0 0 ¢ ¢ ¢ −−− −−− −−− −−− −−− −−− −−− −−−−−− −−− −−− −−− −−− −−− −−− −−− −−− −−− −−− −−− −−− @R @R ¢ 1 1 2 ¢ @
@ 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 BN B −−− −−− −−− −−− −−− −−− −−− −−− @R 1 1 1 B B @
B B 0 0 0 0 0 0 0 ): 1 F Γ( APPENDIX A. THE AUSLANDER-REITEN QUIVER OF SOME ORDINARY POSETS 148 2 2 1 2 ¢ ¢ ¢ 1 ¢ @R ¢ ¢ @
2 2 2 2 2 2 1 1 0 2 1 1 BN 1 1 1 B @R @R B B @ @
B 2 1 4 4 2 1 B 1 1 1 2 1 1 ¢ ¢ ¢ 1 2 1 ¢ @R @R ¢ ¢ @
@ 4 3 2 2 4 3 1 2 1 0 1 1 BN 2 1 2 B @R @R B B @ @
B 2 2 6 4 2 2 B 0 1 2 2 0 1 ¢ ¢ ¢ 1 1 3 ¢ @R @R ¢ ¢ @ @
4 3 4 2 4 3 1 1 1 2 1 1 BN 2 2 2 B @R @R B B @
@
B 2 1 6 4 2 1 B 1 0 1 2 1 0 ¢ ¢ ¢ 1 3 1 ¢ @R @R @R ¢ ¢
@ @
@ 1 1 4 2 1 2 4 2 1 1 1 0 1 1 0 0 1 1 1 0 BN 1 2 1 2 1 B @R @R @R B B @ @ @
B 3 2 4 2 3 2 B 1 1 1 0 1 1 ¢ ¢ ¢ 2 2 2 ¢ − − − − − − @R @R @R ¢ ¢ @ @ @
2 1 3 2 2 0 3 2 2 1 0 1 1 0 1 0 1 0 0 1 BN 1 2 1 2 1 B @R @R @R B B @ @
@
B 2 1 4 2 2 1 B 0 0 2 0 0 0 ¢ ¢ ¢ 1 3 1 ¢ @R @R ¢ ¢ @
@
3 1 2 2 3 1 1 0 1 0 1 0 BN 2 2 2 B @R @R B B @
@
B 1 0 4 2 1 0 B 1 0 1 0 1 0 ¢ ¢ ¢ 3 1 1 ¢ @R @R ¢ ¢ @ @
2 1 2 0 2 1 1 0 0 0 1 0 BN 1 2 2 −−− −−− −−− −−− −−−−−− −−− −−− −−− −−− −−− −−− B @R @R B B @
@
B 1 1 2 0 1 1 B 0 0 1 0 0 0 ¢ ¢ ¢ 1 2 1 ¢ −−− −−−−−− −−−−−− −−− −−− −−− −−− −−− −−− −−− −−− −−− −−− −−− −−− −−− −−− −−− −−− @R @R ¢ ¢ @
@
1 0 0 0 1 0 0 0 1 0 0 0 BN 1 1 1 −−− −−− −−− −−− −−− −−− −−− −−− B @R B B @
B 0 0 B 0 0 1 n −! APPENDIX A. THE AUSLANDER-REITEN QUIVER OF SOME ORDINARY POSETS 149 1 1 0 0 1 1 1 1 1 1 @R @R
@ @ 2 2 1 1 0 1 0 0 2 1 0 1 2 1 0 1 2 1 2 1 @R @R @R @R
@ @ @ @ 2 1 2 2 0 0 0 1 2 1 1 1 2 1 1 1 2 2 2 2 @R @R
@ @ 4 3 1 1 0 1 0 1 2 2 1 0 2 2 1 0 4 1 4 1 @R @R @R @R @ @
@ @ 2 2 3 2 0 1 0 1 0 1 2 1 0 1 2 1 2 3 2 3 @R @R
@ @ 4 3 2 1 0 2 0 0 2 1 1 1 2 1 1 1 4 4 2 2 @R @R @R @R
@ @ @ @ 2 1 3 2 1 1 0 1 2 0 1 1 2 0 1 1 2 3 2 3 @R @R @R @R
@ @ @ @
1 1 4 2 1 1 0 1 1 1 0 1 2 0 2 1 0 0 2 0 2 1 0 0 2 4 1 2 4 1 @R @R @R @R
@ @ @ @ 3 2 2 1 0 1 1 1 2 1 1 0 2 1 1 0 4 2 4 2 − − − − − − @R @R @R @R
@ @ @ @ 2 1 3 2 1 0 0 0 1 2 1 0 0 1 2 0 1 0 0 1 2 0 1 0 2 4 1 2 4 1 @R @R @R @R
@ @ @ @ 2 1 2 1 1 1 1 1 0 0 2 0 0 0 2 0 2 3 2 3 @R @R
@ @
3 1 1 1 2 1 0 1 2 0 1 0 2 0 1 0 4 2 4 2 @R @R @R @R
@ @ @ @ 1 0 2 1 1 0 1 1 2 0 1 0 2 0 1 0 2 3 2 3 @R @R
@ @ 2 1 1 0 1 1 1 0 2 0 0 0 2 0 0 0 4 1 4 1 −−− −−− −−− −−− −−−−−− −−− −−− −−− −−− −−− −−− @R @R @R @R
@ @ @ @ 1 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 2 2 2 2 −−− −−− −−− −−− −−−−−− −−− −−− −−− −−− −−− −−− −−− −−− −−− −−− −−− −−− −−−−−− −−− −−− −−− −−− −−− −−− −−− −−− @R @R 2 2 2 2 2 2 2 2 @ @
1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 1 2 2 −−− −−− −−− −−− −−−2−−− −−− −−− 2 −−− −−− −−− 2 −−− 2 −−− −−− 2 −−− 2 −−− 2 2 @R @R
@ @ 0 0 0 0 0 0 0 0 1 1 1 o −! −! APPENDIX A. THE AUSLANDER-REITEN QUIVER OF SOME ORDINARY POSETS 150
The following setting is an analysis of the poset (n, n), which is the maximal ordinary poset of finite representation type we are working with, the corresponding indecomposable projective and injective representations, the radicals of the projective representations and the Auslander-Reiten quiver Γ(n, n). In section 3.2, we present 1 (n, n).
n0 n00 e e (n − 1)0 (n − 1)00 . . .e .e (n, n) = 30 300 e e 20 200 e e 10 100 e e
1 0 1 0 1 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 ...... 0 . . 0 . . 0 . . ... 0 . . . . P (1 ) = 1 1 0 P (2 ) = 1 1 0 P (3 ) = 1 1 0 P (n ) = 1 0 0 P (∅) = 1 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 ...... 00 . . 00 . . 00 . . ... 00 . . P (1 ) = 1 0 1 P (2 ) = 1 0 1 P (3 ) = 1 0 1 P (n ) = 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0
1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 1 ...... 0 . . 0 . . 0 . . ... 0 . . . . I(1 ) = 1 1 1 I(2 ) = 1 1 1 I(3 ) = 1 0 1 I(n ) = 1 0 1 I(n, n) = 1 1 1 1 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 1 1
1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 ...... 00 . . 00 . . 00 . . ... 00 . . I(1 ) = 1 1 1 I(2 ) = 1 1 1 I(3 ) = 1 1 0 I(n ) = 1 1 0 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0
Note that, the dimension vectors of the representations of (n, n) are very long. Thus, we cannot use such dimensions to calculate the Auslander-Reiten quiver Γ(n, n), instead, we use the suitable notations P (x0), P (x00), and the representation P (x0, y00), because these are the only representations for this kind of poset. We build Γ(n, n) of the classical form by using the following considerations.
We firstly note that the indecomposable projective representations P (10) and P (100) are also indecomposable injective representations; secondly, for 1 ≤ i ≤ n − 1, we have that APPENDIX A. THE AUSLANDER-REITEN QUIVER OF SOME ORDINARY POSETS 151
0 0 00 00 0 00 rad P (i ) = P ((i+1) ), rad P (i ) = P ((i+1) ) and rad P (n ) = rad P (n ) = P (∅). Then, for each n ∈ N, Γ(n, n) has the following shape.
P (10) @ @R P (20) − − −P (10, n00) @ @ @R @R P (30) − − −P (20, n00) −−P (10, (n − 1)00) @ @ @ @R @R @R P (40)− − −P (30, n00) −−P (20, (n − 1)00) −−P (10, (n − 2)00)
0 q q 0 00 P ((n − 2)q q) Pq q(1 , 4 ) @ @R P ((n − 1)0) P (10, 300) @ @ @R @R P (n0)−−P ((n − 1)0, n00) − − −−− −−− −−− −−− −−− −−− −−− − P (20, 300)− − −P (10, 200) @ @ @ @R @R @R 0 00 0 00 0 00 P (∅)− − −P (n , n ) − − −−− −−− −−− −−− −−− −−− −−− −−− −−− −−− − P (2 , 2 ) −−P (1 , 1 ) @ @ @ @R @R @R P (n00)−−P (n0, (n − 1)00) − − −−− −−− −−− −−− −−− −−− −−− − P (30, 200) − − −P (20, 100) @ @ @R @R P ((n − 1)00) P (30, 100) @ @R P ((n − 2)00) P (40, 100)
q P (400)− − −P (n0, 300) −−P ((n − 1)0, 200) −−P ((n − 2)q0,q 100) q q q @ @ @ @R @R @R P (300) − − −P (n0, 200) −−P ((n − 1)0, 100) @ @ @R @R P (200) − − −P (n0, 100) @ @R P (100)
Figure A.7. The Auslander-Reiten quiver Γ(n, n) of the ordinary poset (n, n).
Remark A.1. For the Auslander-Reiten quiver Γ(n, n), we present the following simple observations.
−l 0 00 1. If 1 ≤ l ≤ n then τ (P (∅)) = P ((n − l) , (n − l) ). 2. If 2 ≤ i ≤ n and 1 ≤ i − 1, then the following identities hold:
τ −l(P (i0)) = P ((i − l)0, (n + 1 − l)00) and τ −l(P (i00)) = P ((n + 1 − l)0, (i − l)00). APPENDIX A. THE AUSLANDER-REITEN QUIVER OF SOME ORDINARY POSETS 152 A.3 Some ordinary posets of finite growth representation type
In this last section, we study the critical sincere ordinary poset K1, which is of finite growth representation type, because it is the evolvent of the equipped poset K6 and K8 which are finite growth representation type, as the poset K1 has an infinite number of representations, we show its preprojective P(K1) and preinjective I(K1) components and a closed formula to build these components.
The famous Four Subspace Problem (FSP) associated to the classification of the critical poset k1 has been investigated by many mathematicians, Brener, Gelfand and Ponomarev, Nazarova among others. In particular, Medina and Zavadskij gave an elementary solu- tion of FSP in 2005 [39]. Simson also shows its indecomposable representations and the corresponding Auslander-Reiten quiver in [56].
We devote the following lines to the ordinary poset K1, its indecomposable projective and injective representations, as well as the radical of the projective representations and the corresponding preprojective and preinjective components.
K1 = 1e0 1 e00 2 e0 2 e00
P (10 ) = 1 | 1 0 0 0 I(10 ) = 1 | 0 1 1 1 P (100 ) = 1 | 0 1 0 0 I(100 ) = 1 | 1 0 1 1 P (20 ) = 1 | 0 0 1 0 I(20 ) = 1 | 1 1 0 1 P (200 ) = 1 | 0 0 0 1 I(200 ) = 1 | 1 1 1 0
P (∅) = 1 | 0 0 0 0 I(K1) = 1 | 1 1 1 1
0 00 0 00 rad P (1 ) = rad P (1 ) = rad P (2 ) = rad P (2 ) = P (∅). APPENDIX A. THE AUSLANDER-REITEN QUIVER OF SOME ORDINARY POSETS 153 ) 1 K ( ··· ··· ··· ··· I : CW £ C £ 1 1 1 1 C £ ¡ AU C A ¡ £ | k C £ C A ¡ £ 1 + 1 + 1 + 1 k k + 1 ··· k k k 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 | | | | + 1 + 1 £ CW £ C ¡ AU 1 1 1 1 ¡ A £ C k + 1 + 1 £ C ¡ A £ C £ C k k k k r CW £ C £ 2 2 2 2 + 1 C £ − ¡ AU A ¡ £ C | . C £ τ + 1 + 1 + 1 C A ¡ £ . 1 3 1 k k k k k K 1 | | | | K τ 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 | | | | + 1 + 1 + 1 + 1 £ CW £ CW £ C £ C ¡ AU ¡ AU r r 2 2 2 2 r r ¡ A ¡ A £ C £ C £ C £ C ¡ A ¡ A £ C £ C r r r r £ C £ C | CW £ C £ 3 3 3 3 £ CW C £ ¡ £ C AU A ¡ C £ | £ C ¡ C £ AU − − − − − − − − − − − − ¡ £ C A A ¡ C £ + 1 £ C 5 k ¡ £ C A r 2 k k 2 + 1 τ k k k + 1 1 2 2 2 2 1 2 2 2 2 1 2 2 2 2 1 − − − − − − − − − | | | | + 1 £ CW l £ C ¡ AU 3 3 3 3 ¡ A £ C + 1 − £ C ¡ A £ C £ C τ k k k k k k k k k k | | | | CW £ C £ 4 4 4 4 C £ ¡ − − − − − AU C A ¡ £ − − −− − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − | C £ C A ¡ £ + 1 + 1 + 1 + 1 7 £ CW £ C ¡ AU l l l l ¡ A £ C 3 £ C ¡ A £ C ··· £ C τ l l l l ) of the critical ordinary poset | 3 2 2 2 2 3 2 2 2 2 3 2 2 2 2 3 ) of the critical ordinary poset 1 + 1 | | | | 1 l K 4 4 4 4 ( K + 1 ( P l ··· ··· ··· ··· I 2 + 1 l ··· ··· ··· ··· CW £ + 1 C £ C £ ¡ l AU C A ¡ £ ··· k C £ 3 C £ A ¡ − + 1 + 1 + 1 + 1 τ 2 1 2 2 1 2 2 2 2 2 1 2 2 2 2 1 k k + 1 l | | | | | £ CW k k k £ C ¡ AU 4 4 4 4 ¡ A £ C £ C ¡ A + 1 + 1 £ C £ C + 1 k + 1 + 1 CW £ l C £ 3 3 3 3 C £ k k k k l ¡ 2 AU C £ A ¡ | C £ τ C £ A ¡ + 1 7 + 1 + 1 + 1 2 −− − k k k k k τ | | | | 1 2 1 1 2 1 1 1 1 1 2 1 1 1 1 2 + 1 | | | | r £ CW £ C ¡ AU 3 3 3 3 The preinjective component ¡ A £ C £ C + 1 + 1 + 1 + 1 The preprojective component ¡ A £ C £ CW £ C l l l l £ C ¡ AU ¡ A £ C + 1 £ C ¡ A £ C CW £ C £ £ C 2 2 2 2 r C £ ¡ AU C A £ ¡ | C £ CW £ C A £ ¡ C £ 5 C £ ¡ − − − − − − − − − − − − AU C £ A ¡ + 1 C £ 1 C A ¡ £ r − + 2 + 1 + 1 + 1 τ 1 0 1 1 0 1 1 1 1 1 0 1 1 1 1 0 k k k k | | | | + 1 £ CW £ C ¡ AU 2 2 2 2 ¡ A Figure A.9. £ C r £ C Figure A.8. ¡ A £ C + 1 + 2 + 1 + 1 | £ C k k k k CW £ C £ 1 1 1 1 C £ ¡ AU C A £ ¡ − − −− − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − | + 1 C £ C A ¡ £ + 1 + 1 + 2 + 1 r r 3 τ 2 k k k k 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1 + 1 + 1 + 1 + 2 − − − − − − − − − | | | | £ CW k k k k £ C ¡ AU ··· 1 1 1 1 ¡ A £ C £ C | | | | ¡ A £ C £ C ): 1 0 0 0 0 + 1 + 1 + 1 + 1 K | ( r r r r P 1 ··· ··· ··· ··· APPENDIX A. THE AUSLANDER-REITEN QUIVER OF SOME ORDINARY POSETS 154
Remark A.2. We note that for P(K1) and I(K1), the following conditions hold, k, l, r ∈ Z with k ≥ 0, l = 2k and r = 2k + 1. The following (closed) formulas allow us to make a description of its orbits.
1. In Figure A.8, it is shown the orbits of the indecomposable projective representations 0 00 0 00 P (∅),P (1 ),P (1 ),P (2 ) and P (2 ): ( −l 2l + 1 | l l l l if l = 2k, τ P (∅) = 2r + 1 | r r r r if r = 2k + 1. ( l + 1 | k + 1 k k k if l = 2k, τ −lP (10) = r + 1 | k k + 1 k + 1 k + 1 if r = 2k + 1. ( l + 1 | k k + 1 k k if l = 2k, τ −lP (100) = r + 1 | k + 1 k k + 1 k + 1 if r = 2k + 1. ( l + 1 | k k k + 1 k if l = 2k, τ −lP (20) = r + 1 | k + 1 k + 1 k k + 1 if r = 2k + 1. ( l + 1 | k k k k + 1 if l = 2k, τ −lP (200) = r + 1 | k + 1 k + 1 k + 1 k if r = 2k + 1.
2. Figure A.9 shows the orbits of the indecomposable injective representations 0 00 0 00 I(K1),I(1 ),I(1 ),I(2 ) and I(2 ): ( l 2l + 1 | l + 1 l + 1 l + 1 l + 1 if l = 2k, τ I(k1) = 2r + 1 | r + 1 r + 1 r + 1 r + 1 if r = 2k + 1. ( l + 1 | k k + 1 k + 1 k + 1 if l = 2k, τ lI(10) = r + 1 | k + 2 k + 1 k + 1 k + 1 if r = 2k + 1. ( l + 1 | k + 1 k k + 1 k + 1 if l = 2k, τ lI(100) = r + 1 | k + 1 k + 2 k + 1 k + 1 if r = 2k + 1. ( l + 1 | k + 1 k + 1 k k + 1 if l = 2k, τ lI(20) = r + 1 | k + 1 k + 1 k + 2 k + 1 if r = 2k + 1. ( l + 1 | k + 1 k + 1 k + 1 k if l = 2k, τ lI(200) = r + 1 | k + 1 k + 1 k + 1 k + 2 if r = 2k + 1. APPENDIX B
MATLAB routines
In this Appendix, we present MATLAB routines illustrating how the algorithm of dif- ferentiation VII can be used to embed digital watermarks, and a MATLAB routine that interprets the algorithm VII as a system of digital watermarks in frequency domain. These routines correspond to the digital watermarking procedures defined in section 5.1.
B.1 Differentiation VII in MATLAB
In this section, a simply MATLAB program illustrates the algorithm of differentiation VII as a way to embed and extract digital watermarks in a system of nested images.
First Routine; Infection
%Clear Memory & Command Window clc; clear all; close all; beta = 0.01; gamma = 1; delta = 0.01; epsilon = 0.01; tau = 0.01; iota = 1; psi = 0.01; kappa = 0.01; lambda = 0.01; mu = 1; eta = 0.01; nu = 0.01; xi = 1; ji = 1; 155 APPENDIX B. MATLAB ROUTINES 156 sigma = 1; zeta = 1; xi b = 1; gamma b = 1; nu b = 1; alpha = 1; alpha a = 0.01; A = imread(0Imagen37scan.jpg0); %T = imread(0DV II.png0); R 1 = imread(0Imagen5.jpg0); R 2 = imread(0Imagen6.jpg0); R 3 = imread(0Imagen7.jpg0); R 4 = imread(0Imagen7scan.jpg0); B = imread(0doodle6.jpg0); C = imread(0ty.jpg0); D = imread(0Imagen1.jpg0); W = imread(0Imagen4scanbl.jpg0); V = imread(0Imagen22.jpg0); F 1 = imread(0zvr.jpg0); F 2 = imread(0zk.jpg0); E = imlincomb(zeta, A); F = imlincomb(0.01,B); G = imlincomb(1,C); H = imlincomb(1,D); X = imlincomb(sigma, W ); A = imlincomb(1,V ); F 3 = imlincomb(1,F 1); F 4 = imlincomb(1,F 2); %T1 = imresize(T, [740 740]); R 5 = imresize(R 1, [256 256]); R 6 = imresize(R 2, [256 256]); R 7 = imresize(R 3, [256 256]); R 8 = imresize(R 4, [256 256]); I = imresize(E, [256 256]); J = imresize(F, [256 256]); K = imresize(G, [256 256]); L = imresize(H, [256 256]); Y = imresize(X, [256 256]); B = imresize(A, [256 256]); F 5 = imresize(F 3, [256 256]); F 6 = imresize(F 4, [256 256]); a = imlincomb(xi, R 8, ji, J); a 1 = imlincomb(alpha, a); a 2 = imresize(a 1, [256 256]); c 1 = alpha a ∗ a2 + beta ∗ K + gamma ∗ (I) + delta ∗ R 5; c 2 = epsilon ∗ a2 + tau ∗ (I) + iota ∗ L + psi ∗ K + nu ∗ Y + kappa ∗ R 6; F 9 = imlincomb(1, c 2); G = imresize(F 9, [256 256]); O 1 = eta ∗ F 9 + lambda ∗ R 7 + mu ∗ F 5; APPENDIX B. MATLAB ROUTINES 157
O 2 = imlincomb(1,O 1); c 3 = imresize(O 2, [256 256]); b = xi b ∗ J + gamma b ∗ I + nu b ∗ Y ; figure, imshow(Y, [ ]), title(0cˆ+ 2b0) figure, imshow(I, [ ]), title(0cˆ+ 1b0) 0 0 figure, imshow(a1, [ ]), title( a ) 0 0 figure, imshow(c1, [ ]), title( c 1 ) 0 0 figure, imshow(c2, [ ]), title( c 2 ) 0 figure, imshow(c3, [ ]), title( c 3 (system of susceptibles files)’)
Second Routine; Extraction %Program to extract an infected file from a weak chain of susceptible files %of type jpeg. %Program Description %This program is the main entry of the application. %In this case, images a, c 1, c 2 and c 3 correspond to weak %points of the chain. Note that, elements of the form a + ib in the case %(R,C) are interpreted as system of fragile watermarks given by linear %combinations of images, F , F 6, I, J, L, Y , R 5, R 6, R 7, and R 8. %Infected files correspond to image J which is obtained if the linear %combinations of images J and R 8 are considered independently (i.e, %as elements of the image algebra over R). %Clear Memory & Command Window clc; clear all; close all; alpha = 0.01; beta = 0.01; gamma = 0.01; delta = 0.01; epsilon = 0.01; tau = 0.01; iota = 0.01; psi = 0.01; kappa = 0.01; lambda = 0.01; mu = 0.01; eta = 1; nu = 0.01; xi = 1.2; ji = 0.001; sigma = 1; ro = 1.0; xi b = 1; gamma b = 1; APPENDIX B. MATLAB ROUTINES 158 nu b = 1; zeta = 1;
A = imread(0Imagen37scan.jpg0); %T = imread(0DV II.png0); R 1 = imread(0Imagen5.jpg0); R 2 = imread(0Imagen6.jpg0); R 3 = imread(0Imagen7.jpg0); R 4 = imread(0Imagen7scan.jpg0); B = imread(0doodle6.jpg0); C = imread(0ty.jpg0); D = imread(0Imagen1.jpg0); W = imread(0Imagen4scanbl.jpg0); V = imread(0Imagen22.jpg0); U = imread(0DigitalW atermarking.png0); Z 3 = imread(0poriginalhalftone.jpg0); F 1 = imread(0zvr.jpg0); F 2 = imread(0zk.jpg0); %P 2 = imread(0poriginalHT 1.jpg0); E = imlincomb(zeta, A); F = imlincomb(ro, B); G = imlincomb(1,C); H = imlincomb(1,D); X = imlincomb(sigma, W ); A = imlincomb(1,V ); F 3 = imlincomb(1,F 1); F 4 = imlincomb(1,F 2); U 1 = imresize(U, [740 740]); I = imresize(E, [256 256]); J = imresize(F, [256 256]); K = imresize(G, [256 256]); L = imresize(H, [256 256]); Y = imresize(X, [256 256]); R 5 = imresize(R 1, [256 256]); R 6 = imresize(R 2, [256 256]); R 7 = imresize(R 3, [256 256]); R 8 = imresize(R 4, [256 256]); B = imresize(A, [256 256]); F 5 = imresize(F 3, [256 256]); F 6 = imresize(F 4, [256 256]); a = xi ∗ J + ji ∗ R 8; c 1 = xi ∗ (J) + ji ∗ R 8 + gamma ∗ (I) + delta ∗ R 5; c 2 = xi ∗ (J) + ji ∗ R 8 + gamma ∗ (I) + delta ∗ L + nu ∗ Y + kappa ∗ R 6; F = imlincomb(1, c 2); G = imresize(F, [256 256]); a 1 = imresize(a, [256 256]); O 1 = eta ∗ F + lambda ∗ R 7 + mu ∗ F 6; O 2 = imlincomb(1,O 1); c 3 = imresize(O 2, [256 256]); APPENDIX B. MATLAB ROUTINES 159 b = xi b ∗ J + gamma b ∗ I + nu b ∗ Y ; figure, imshow(Y, [ ]), title(0cˆ+ 2b0) figure, imshow(I, [ ]), title(0cˆ+ 1b0) figure, imshow(J, [ ]), title(0aˆ+ bDigitalW atermark0) figure, imshow(a, [ ]), title(0aˆ+ InfectedfileExtractedasW atermark0) figure, imshow(c 1, [ ]), title(0cˆ+ 10) figure, imshow(c 2, [ ]), title(0cˆ+ 20) figure, imshow(c 3, [ ]), title(0cˆ+ 30)
B.2 Wavelet transformation and singular value decomposi- tion
In this section, we show a MATLAB routine which combine differentiation VII (see c3 in routine B.1) DWT and SVD to embed infected digital watermarks. alpha = 0.55; beta = 0.75; R1 = imread(0cameraman.tif 0); R2 = imread(0rice1.jpg0); R3 = imread(0alberti.jpg0); A = imread(0kirichenko.jpg0); T = imread(032dee.jpg0); B = imread(0zavadskij.jpg0); C = imread(0ty.jpg0); D = imread(0iue.jpg0); W = imread(0zavadskij and bautista.jpg0); V = imread(0Imath.jpg0); F 1 = imread(0zvr.jpg0); F 2 = imread(0zk.jpg0); E = imlincomb(1,A); F = imlincomb(0.01,B); G = imlincomb(1,C); H = imlincomb(1,D); X = imlincomb(1,W ); A = imlincomb(1,V ); F 3 = imlincomb(1,F 1); F 4 = imlincomb(1,F 2); S1 = imresize(R1, [256 256]); S2 = imresize(R2, [256 256]); S3 = imresize(R3, [256 256]);
I = imresize(E, [256 256]); L = imresize(H, [256 256]); Y = imresize(X, [256 256]); B = imresize(A, [256 256]); F 5 = imresize(F 3, [256 256]); APPENDIX B. MATLAB ROUTINES 160
F 6 = imresize(F 4, [256 256]); T 6 = imresize(T, [256 256]); c 3 = imresize(O2, [256 256]);
[LL1,HL1, LH1,HH1] = dwt2(R1,0 haar0); [LL2,HL2, LH2,HH2] = dwt2(LL1,0 haar0); [LL8,HL8, LH8,HH8] = dwt2(LL2,0 haar0); [LL3,HL3, LH3,HH3] = dwt2(D,0 haar0); [LL4,HL4, LH4,HH4] = dwt2(LL3,0 haar0); [LL6,HL6, LH6,HH6] = dwt2(LL3,0 haar0); p = size(LL2); t = size(LL4); r = size(LL6);
[Uy, Sy, V y] = svd(LL2); q = size(Sy); [ny, ay, vy] = svd(LL4); z = size(ay);
[oy, by, py] = svd(LL6); h = size(by);
0 0 P w = imread( c3.jpg ); P w = Pw(:, :, 1); P 1 w = imresize(P w, p); Q w = imread(0zvr.jpg0); Q w = Q w(:, :, 2); Q1 w = imresize(Q w, z); figure(15); imshow(P 1 w); title(0original watermark7a0); figure(13); imshow(Q1 w); title(0original watermark70);
[Uw, Sw, V w] = svd(double(P 1 w)); [nw, sw, vw] = svd(double(Q1 w));
Smark = Sy + alpha ∗ Sw; nmark = ay + alpha ∗ sw;
LL2 1 = Uy ∗ Smark ∗ V y0; LL2 2 = ny ∗ nmark ∗ vy0;
LL1 1 = idwt2(LL2 1,HL2, LH2,HH2,0 haar0); P 1 = idwt2(LL1 1,HL1, LH1,HH1,0 haar0); LL2 3 = idwt2(LL2 2,HL4, LH4,HH4,0 haar0); Q 1 = idwt2(LL2 3,HL3, LH3,HH3,0 haar0); figure(3); imshow(uint8(P 1)); title(0W atermarked Image0); APPENDIX B. MATLAB ROUTINES 161 figure(10); imshow(uint8(Q 1)); title(0W atermarked Image100);
[LL1 wmv, HL1 wmv, LH1 wmv, HH1 wmv] = dwt2(P 1,0 haar0); [LL2 wmv, HL2 wmv, LH2 wmv, HH2 wmv] = dwt2(LL1 wmv,0 haar0); [Uy wmv, Sy wmv, V y wmv] = svd(LL2 wmv); Swrec = (Sy wmv − Sy)/alpha; W My = Uw ∗ Swrec ∗ V w0;
[LL3 wmv, HL3 wmv, LH3 wmv, HH3 wmv] = dwt2(J 1,0 haar0); [LL4 wmv, HL4 wmv, LH4 wmv, HH4 wmv] = dwt2(LL3 wmv,0 haar0); [nywmv, sy wmv, vy wmv] = svd(LL4 wmv); swrec = 2 ∗ alpha ∗ (sy wmv − ay); W My1 = nw ∗ swrec ∗ vw0; figure(4); imshow(uint8(W My)); title(0Extracted W atermark0); figure(12); imshow(uint8(W My1)); title(0Extracted W atermark120); figure(5); imshow(uint8(LL1 1)); title(0W avelet20); figure(6); imshow(H); title(0Original Image0); figure(6); imshow(R1); title(0Original Image(a)0); Bibliography
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1. To describe the categorical properties of the algorithms of differentiation XI-XVII for equipped posets with involution.
2. To complete the lifting algorithm to the tubular component of the equipped posets of finite growth representation type. Extend this algorithm to quivers, that is, build the Auslander-Reiten quiver of valued quivers through the Auslander-Reiten quiver of suitable quivers without valuations.
3. To explore other applications of theory representation of posets with additional struc- tures.
4. To develop the theory of one-peak posets and k-peak posets, initially of equipped posets and subsequently of equipped posets with involution, in the sense of Simson, Kasjan, et al[34].
5. To investigate the theory of representation of posets with additional structures in unit spaces in the sense of Futorny, Samoilenko and Yusenko [30].
6. To give more applications of the Zavadskij calculus and Krawtchouk-Zavadskij ma- trices in the theory of representation of Lie algebras and Jordan superalgebras.
165 Index
(A, B)-cleaving, 28 compositions of type D, 93 Yt type poset, 83 contravariant functor, 6 Θ-transformation, 109 convex envelope, 4 coproduct, 7 additive category, 8 covariant functor, 6 admissible transformations, 17 critical Kleiner's posets, 24 algorithm of differentiation completion, 30 decomposable lattice representation, 101 algorithm of differentiation I, 29 dense functor, 7 algorithm of differentiation IX, 44 derivative, 99 algorithm of differentiation VII, 33 derived poset, 29, 33, 37, 45, 51 algorithm of differentiation VIII, 37 dimension, 16, 18 algorithm of differentiation X, 51 discrete wavelet transform, 117 almost split sequence, 11 dyad, 18 antisimmetric relation, 1 Auslander-Reiten quiver, 11 epimorphism, 6, 59 Auslander-Reiten sequence, 11 equipped poset, 1, 2 Auslander-Reiten translation, 11 equipped poset diagram, 3 equipped poset with involution, 4 big point, 4 equivalence functor, 6 binary images, 124 equivalence of categories, 7 biweak point, 4 equivalent categories, 7 bot, 131 equivalent points, 5 bot network, 131 evolvent, 25 exact category, 10 cardinal sum, 23 exact sequence, 59 category, 5 chain, 4 faithful functor, 7 cokernel, 8, 10, 59 full functor, 7 color images, 125 full subcategory, 6 completely weak chain, 4 complex series, 23 Gabriel's quiver, 11 complexification, 12, 13 genuine complex series, 23 complexification functor, 13, 14 genuine real series, 23 component, 11 genus, 4 composition, 92 grayscale images, 125 166 INDEX 167
Haar wavelet transform, 117 pair of points IX suitable, 44 heterogeneous algebra, 123 pair of points VII suitable, 33 homogeneous algebra, 123 pair of points VIII suitable, 37 homogeneous biquadratic problem, 15 pair of points X suitable, 51 hull, 14 pair splits, 10 hypercritical Nazarova's posets, 25 partition, 92 pixel, 124 I-suitable, 29 polymorphic virus, 131 ideal, 7 poset, 1 idempotent, 8 poset of finite growth representation imaginary part, 14 type, 23 immersion, 7 poset of finite representation type, 23 indecomposable object, 7, 8 poset of tame representation type, 23 indecomposable representation, 15 poset of wild representation type, 23 injective representation, 59 posets m-parametric, 23 interior of a segment, 4 posets one-parametric, 23 irreducible morphism, 9 primitive involution, 5 isomorphic representations, 16 projection, 7 isomorphism, 6 projective representation, 59 proper morphism, 59 k-category, 7 pseudo-cokernel, 12 k-functor, 7 pseudo-kernel, 12 k-quotient category, 7 pure morphism, 59 K Z-matrices, 104 pure subrepresentation, 59 kernel, 8, 10 Krawtchouk-Zavadskij matrices, 104 quasi-inverse, 6 Krull-Schmidt category, 9 radical, 9 lattice representation, 99 radical subspace, 16 lifting, 61 real part, 14 lifting algorithm, 60 real series, 22 reduced involution, 18 malware, 128 reduced representation, 29, 34, 38, 45, 53 matrix problem, 17, 18 reellification, 13 matrix representation, 17 reellification functor, 13, 14 matrix representation equivalents, 17 reflexive relation, 1 minimal left almost split, 11 representation, 15, 16 minimal right almost split, 11 retraction, 9 monomorphism, 6, 59 morphism of lattice representations, 100 section, 9 segment, 4, 97 null object, 8 short pseudo-exact, 12 sincere, 16 object class, 9 singular value decomposition, 121 orbit, 11 sink morphism, 10 order relation, 1 small point, 4 ordinal sum, 23 source morphism, 10 ordinary chain, 4 special domain, 124 ordinary lower cone, 3 special points, 30 ordinary upper cone, 3 split epimorphism, 9 INDEX 168 split monomorphism, 9 value sets, 123 srtong upper cone, 3 valued image, 124 strong lattice path, 98 virus, 128 strong lower cone, 3 strong point, 2 watermarking, 116 strong product, 99 watermarking fragile, 121 strong relation, 2 watermarking robust, 121 watermarking semifragile, 121 sub-representation, 16 wavelet transform, 117 subcategory, 6 weak chain, 4 subsets sum, 23 weak gluing, 61 sum, 7 weak lattice path, 97 weak point, 2, 4 tetrad, 18 weak product, 99 transitive relation, 1 weak relation, 2 triad, 18 wild generator, 23 trivial equipment, 3 worm, 129 Trojan, 130 Trojan Horse, 130 Zavadskij Symbol, 28 truncated lower cone, 3 zombie computer, 131 truncated upper cone, 3 zombie network, 131