QUIVER BIALGEBRAS and MONOIDAL CATEGORIES 3 K N ≥ 0
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Introduction to Linear Bialgebra
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of New Mexico University of New Mexico UNM Digital Repository Mathematics and Statistics Faculty and Staff Publications Academic Department Resources 2005 INTRODUCTION TO LINEAR BIALGEBRA Florentin Smarandache University of New Mexico, [email protected] W.B. Vasantha Kandasamy K. Ilanthenral Follow this and additional works at: https://digitalrepository.unm.edu/math_fsp Part of the Algebra Commons, Analysis Commons, Discrete Mathematics and Combinatorics Commons, and the Other Mathematics Commons Recommended Citation Smarandache, Florentin; W.B. Vasantha Kandasamy; and K. Ilanthenral. "INTRODUCTION TO LINEAR BIALGEBRA." (2005). https://digitalrepository.unm.edu/math_fsp/232 This Book is brought to you for free and open access by the Academic Department Resources at UNM Digital Repository. It has been accepted for inclusion in Mathematics and Statistics Faculty and Staff Publications by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected], [email protected], [email protected]. INTRODUCTION TO LINEAR BIALGEBRA W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 600036, India e-mail: [email protected] web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico Gallup, NM 87301, USA e-mail: [email protected] K. Ilanthenral Editor, Maths Tiger, Quarterly Journal Flat No.11, Mayura Park, 16, Kazhikundram Main Road, Tharamani, Chennai – 600 113, India e-mail: [email protected] HEXIS Phoenix, Arizona 2005 1 This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N. -
Change of Rings Theorems
CHANG E OF RINGS THEOREMS CHANGE OF RINGS THEOREMS By PHILIP MURRAY ROBINSON, B.SC. A Thesis Submitted to the School of Graduate Studies in Partial Fulfilment of the Requirements for the Degree Master of Science McMaster University (July) 1971 MASTER OF SCIENCE (1971) l'v1cHASTER UNIVERSITY {Mathematics) Hamilton, Ontario TITLE: Change of Rings Theorems AUTHOR: Philip Murray Robinson, B.Sc. (Carleton University) SUPERVISOR: Professor B.J.Mueller NUMBER OF PAGES: v, 38 SCOPE AND CONTENTS: The intention of this thesis is to gather together the results of various papers concerning the three change of rings theorems, generalizing them where possible, and to determine if the various results, although under different hypotheses, are in fact, distinct. {ii) PREFACE Classically, there exist three theorems which relate the two homological dimensions of a module over two rings. We deal with the first and last of these theorems. J. R. Strecker and L. W. Small have significantly generalized the "Third Change of Rings Theorem" and we have simply re organized their results as Chapter 2. J. M. Cohen and C. u. Jensen have generalized the "First Change of Rings Theorem", each with hypotheses seemingly distinct from the other. However, as Chapter 3 we show that by developing new proofs for their theorems we can, indeed, generalize their results and by so doing show that their hypotheses coincide. Some examples due to Small and Cohen make up Chapter 4 as a completion to the work. (iii) ACKNOW-EDGMENTS The author expresses his sincere appreciation to his supervisor, Dr. B. J. Mueller, whose guidance and helpful criticisms were of greatest value in the preparation of this thesis. -
Characterization of Hopf Quasigroups
Characterization of Hopf Quasigroups Wei WANG and Shuanhong WANG ∗ School of Mathematics, Southeast University, Jiangsu Nanjing 210096, China E-mail: [email protected], [email protected] Abstract. In this paper, we first discuss some properties of the Galois linear maps. We provide some equivalent conditions for Hopf algebras and Hopf (co)quasigroups as its applications. Then let H be a Hopf quasigroup with bijective antipode and G be the set of all Hopf quasigroup automorphisms of H. We introduce a new category CH(α, β) with α, β ∈ G over H and construct a new braided π-category C (H) with all the categories CH (α, β) as components. Key words: Galois linear map; Antipode; Hopf (co)quasigroup; Braided π-category. Mathematics Subject Classification 2010: 16T05. 1. Introduction The most well-known examples of Hopf algebras are the linear spans of (arbitrary) groups. Dually, also the vector space of linear functionals on a finite group carries the structure of a Hopf algebra. In the case of quasigroups (nonassociative groups)(see [A]) however, it is no longer a Hopf algebra but, more generally, a Hopf quasigroup. In 2010, Klim and Majid in [KM] introduced the notion of a Hopf quasigroup which was arXiv:1902.10141v1 [math.QA] 26 Feb 2019 not only devoted to the development of this missing link: Hopf quasigroup but also to under- stand the structure and relevant properties of the algebraic 7-sphere, here Hopf quasigroups are not associative but the lack of this property is compensated by some axioms involving the antipode S. This mathematical object was studied in [FW] for twisted action as a gen- eralization of Hopf algebras actions. -
Right Ideals of a Ring and Sublanguages of Science
RIGHT IDEALS OF A RING AND SUBLANGUAGES OF SCIENCE Javier Arias Navarro Ph.D. In General Linguistics and Spanish Language http://www.javierarias.info/ Abstract Among Zellig Harris’s numerous contributions to linguistics his theory of the sublanguages of science probably ranks among the most underrated. However, not only has this theory led to some exhaustive and meaningful applications in the study of the grammar of immunology language and its changes over time, but it also illustrates the nature of mathematical relations between chunks or subsets of a grammar and the language as a whole. This becomes most clear when dealing with the connection between metalanguage and language, as well as when reflecting on operators. This paper tries to justify the claim that the sublanguages of science stand in a particular algebraic relation to the rest of the language they are embedded in, namely, that of right ideals in a ring. Keywords: Zellig Sabbetai Harris, Information Structure of Language, Sublanguages of Science, Ideal Numbers, Ernst Kummer, Ideals, Richard Dedekind, Ring Theory, Right Ideals, Emmy Noether, Order Theory, Marshall Harvey Stone. §1. Preliminary Word In recent work (Arias 2015)1 a line of research has been outlined in which the basic tenets underpinning the algebraic treatment of language are explored. The claim was there made that the concept of ideal in a ring could account for the structure of so- called sublanguages of science in a very precise way. The present text is based on that work, by exploring in some detail the consequences of such statement. §2. Introduction Zellig Harris (1909-1992) contributions to the field of linguistics were manifold and in many respects of utmost significance. -
TENSOR TOPOLOGY 1. Introduction Categorical Approaches Have Been Very Successful in Bringing Topological Ideas Into Other Areas
TENSOR TOPOLOGY PAU ENRIQUE MOLINER, CHRIS HEUNEN, AND SEAN TULL Abstract. A subunit in a monoidal category is a subobject of the monoidal unit for which a canonical morphism is invertible. They correspond to open subsets of a base topological space in categories such as those of sheaves or Hilbert modules. We show that under mild conditions subunits endow any monoidal category with a kind of topological intuition: there are well-behaved notions of restriction, localisation, and support, even though the subunits in general only form a semilattice. We develop universal constructions complet- ing any monoidal category to one whose subunits universally form a lattice, preframe, or frame. 1. Introduction Categorical approaches have been very successful in bringing topological ideas into other areas of mathematics. A major example is the category of sheaves over a topological space, from which the open sets of the space can be reconstructed as subobjects of the terminal object. More generally, in any topos such subobjects form a frame. Frames are lattices with properties capturing the behaviour of the open sets of a space, and form the basis of the constructive theory of pointfree topology [35]. In this article we study this inherent notion of space in categories more general than those with cartesian products. Specifically, we argue that a semblance of this topological intuition remains in categories with mere tensor products. For the special case of toposes this exposes topological aspects in a different direction than considered previously. The aim of this article is to lay foundations for this (ambitiously titled) `tensor topology'. Boyarchenko and Drinfeld [10, 11] have already shown how to equate the open sets of a space with certain morphisms in its monoidal category of sheaves of vector spaces. -
Left Hopf Algebras
JOURNAL OF ALGEBRA 65, 399-411 (1980) Left Hopf Algebras J. A. GREEN Mathematics Institute, University of Warwick, Coventry CV4 7AL, England WARREN D. NICHOLS Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802; and Department of Mathematics, Florida State University, Tallahassee, Florida 32306 AND EARL J. TAFT Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903; and School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540 Communicated by N. Jacobson Received March 5, 1979 I. INTRODUCTION Let k be a field. If (C, A, ~) is a k-coalgebra with comultiplication A and co- unit ~, and (A, m,/L) is an algebra with multiplication m and unit/z: k-+ A, then Homk(C, A) is an algebra under the convolution productf • g = m(f (~)g)A. The unit element of this algebra is/zE. A bialgebra B is simultaneously a coalgebra and an algebra such that A and E are algebra homomorphisms. Thus Homk(B, B) is an algebra under convolution. B is called a Hopf algebra if the identify map Id of B is invertible in Homk(B, B), i.e., there is an S in Homk(B, B) such that S * Id =/zE = Id • S. Such an S is called the antipode of the Hopf algebra B. Using the notation Ax = ~ x 1 @ x~ for x ~ B (see [8]), the antipode condition is that Y'. S(xa) x 2 = E(x)l = • XlS(X~) for all x E B. A bialgebra B is called a left Hopf algebra if it has a left antipode S, i.e., S ~ Homk(B, B) and S • Id =/,E. -
Quasitriangular Structure of Myhill–Nerode Bialgebras
Axioms 2012, 1, 155-172; doi:10.3390/axioms1020155 OPEN ACCESS axioms ISSN 2075-1680 www.mdpi.com/journal/axioms Article Quasitriangular Structure of Myhill–Nerode Bialgebras Robert G. Underwood Department of Mathematics/Informatics Institute, Auburn University Montgomery, P.O. Box 244023, Montgomery, AL 36124, USA; E-Mail: [email protected]; Tel.: +1-334-244-3325; Fax: +1-334-244-3826 Received: 20 June 2012; in revised form: 15 July 2012 / Accepted: 17 July 2012 / Published: 24 July 2012 Abstract: In computer science the Myhill–Nerode Theorem states that a set L of words in a finite alphabet is accepted by a finite automaton if and only if the equivalence relation ∼L, defined as x ∼L y if and only if xz 2 L exactly when yz 2 L; 8z, has finite index. The Myhill–Nerode Theorem can be generalized to an algebraic setting giving rise to a collection of bialgebras which we call Myhill–Nerode bialgebras. In this paper we investigate the quasitriangular structure of Myhill–Nerode bialgebras. Keywords: algebra; coalgebra; bialgebra; Myhill–Nerode theorem; Myhill–Nerode bialgebra; quasitriangular structure 1. Introduction ^ Let Σ0 be a finite alphabet and let Σ0 denote the set of words formed from the letters in Σ0. Let ^ L ⊆ Σ0 be a language, and let ∼L be the equivalence relation defined as x ∼L y if and only if xz 2 L ^ exactly when yz 2 L; 8z 2 Σ0. The Myhill–Nerode Theorem of computer science states that L is accepted by a finite automaton if and only if ∼L has finite index (cf. -
From Combinatorial Monoids to Bialgebras and Hopf Algebras, Functorially
From combinatorial monoids to bialgebras and Hopf algebras, functorially Laurent Poinsot Abstract. This contribution provides a study of some combinatorial monoids, namely finite decomposition and locally finite monoids, using some tools from category theory. One corrects the lack of functoriality of the construction of the large algebras of finite decomposition monoids by considering the later as monoid objects of the category of sets with finite-fiber maps. Moreover it is proved that an algebraic monoid (i.e., a commutative bialgebra) may be as- sociated to any finite decomposition monoid, and that locally finite monoids furthermore provide algebraic groups (i.e., commutative Hopf algebras), by attaching in the first case a monoid scheme to the large algebra, and in the second case a group scheme to a subgroup of invertible elements in the large algebra. 1. Introduction The field of algebraic combinatorics deals with the interface between algebra and combinatorics, i.e., it solves problems from algebra with help of combinatorial methods, and vice versa. The algebras considered are often algebras of \combinato- rial" monoids or their deformations. A combinatorial monoid is roughly speaking a usual monoid together with an informal notion of a natural integer-valued size. The size may be the length of elements of a monoid (when this makes sense), but also the number of decompositions of an element into a product of \smaller" pieces. Hence in some sense in a combinatorial monoid the construction or the decomposition of the elements is combinatorially controlled. Two main classes of such combinatorial monoids have been recognized, namely finite decomposition and locally finite monoids. -
Deformations of Bimodule Problems
FUNDAMENTA MATHEMATICAE 150 (1996) Deformations of bimodule problems by Christof G e i ß (M´exico,D.F.) Abstract. We prove that deformations of tame Krull–Schmidt bimodule problems with trivial differential are again tame. Here we understand deformations via the structure constants of the projective realizations which may be considered as elements of a suitable variety. We also present some applications to the representation theory of vector space categories which are a special case of the above bimodule problems. 1. Introduction. Let k be an algebraically closed field. Consider the variety algV (k) of associative unitary k-algebra structures on a vector space V together with the operation of GlV (k) by transport of structure. In this context we say that an algebra Λ1 is a deformation of the algebra Λ0 if the corresponding structures λ1, λ0 are elements of algV (k) and λ0 lies in the closure of the GlV (k)-orbit of λ1. In [11] it was shown, us- ing Drozd’s tame-wild theorem, that deformations of tame algebras are tame. Similar results may be expected for other classes of problems where Drozd’s theorem is valid. In this paper we address the case of bimodule problems with trivial differential (in the sense of [4]); here we interpret defor- mations via the structure constants of the respective projective realizations. Note that the bimodule problems include as special cases the representa- tion theory of finite-dimensional algebras ([4, 2.2]), subspace problems (4.1) and prinjective modules ([16, 1]). We also discuss some examples concerning subspace problems. -
Cocommutative Vertex Bialgebras
COCOMMUTATIVE VERTEX BIALGEBRAS JIANZHI HAN, HAISHENG LI, YUKUN XIAO Abstract. In this paper, the structure of cocommutative vertex bialgebras is investigated. For a general vertex bialgebra V , it is proved that the set G(V ) of group-like elements is naturally an abelian semigroup, whereas the set P (V ) of primitive elements is a vertex Lie algebra. For g ∈ G(V ), denote by Vg the connected component containing g. Among the main results, it is proved that if V is a cocommutative vertex bialgebra, then V = ⊕g∈G(V )Vg, where V1 is a vertex subbialgebra which is isomorphic to the vertex bialgebra VP (V ) associated to the vertex Lie algebra P (V ), and Vg is a V1-module for g ∈ G(V ). In particular, this shows that every cocommutative connected vertex bialgebra V is isomorphic to VP (V ) and hence establishes the equivalence between the category of cocommutative connected vertex bialgebras and the category of vertex Lie algebras. Furthermore, under the condition that G(V ) is a group and lies in the center of V , it is proved that V = VP (V ) ⊗ C[G(V )] as a coalgebra where the vertex algebra structure is explicitly determined. 1. Introduction Vertex algebras are analogous and closely related to classical algebraic systems such as Lie algebras and associative algebras on one hand, and they are also highly nonclassical in nature on the other hand. Especially, vertex algebras can be associated to (infinite-dimensional) Lie algebras of a certain type, including the Virasoro algebra and the affine Lie algebras. Vertex algebra analogues of Lie algebras, called vertex Lie algebras, were studied by Primc (see arXiv:2107.07290v1 [math.QA] 15 Jul 2021 [17]), and the same structures, called Lie conformal algebras, were studied independently by Kac (see [11]). -
Coalgebras from Formulas
Coalgebras from Formulas Serban Raianu California State University Dominguez Hills Department of Mathematics 1000 E Victoria St Carson, CA 90747 e-mail:[email protected] Abstract Nichols and Sweedler showed in [5] that generic formulas for sums may be used for producing examples of coalgebras. We adopt a slightly different point of view, and show that the reason why all these constructions work is the presence of certain representative functions on some (semi)group. In particular, the indeterminate in a polynomial ring is a primitive element because the identity function is representative. Introduction The title of this note is borrowed from the title of the second section of [5]. There it is explained how each generic addition formula naturally gives a formula for the action of the comultiplication in a coalgebra. Among the examples chosen in [5], this situation is probably best illus- trated by the following two: Let C be a k-space with basis {s, c}. We define ∆ : C −→ C ⊗ C and ε : C −→ k by ∆(s) = s ⊗ c + c ⊗ s ∆(c) = c ⊗ c − s ⊗ s ε(s) = 0 ε(c) = 1. 1 Then (C, ∆, ε) is a coalgebra called the trigonometric coalgebra. Now let H be a k-vector space with basis {cm | m ∈ N}. Then H is a coalgebra with comultiplication ∆ and counit ε defined by X ∆(cm) = ci ⊗ cm−i, ε(cm) = δ0,m. i=0,m This coalgebra is called the divided power coalgebra. Identifying the “formulas” in the above examples is not hard: the for- mulas for sin and cos applied to a sum in the first example, and the binomial formula in the second one. -
Bimodule Categories and Monoidal 2-Structure Justin Greenough University of New Hampshire, Durham
University of New Hampshire University of New Hampshire Scholars' Repository Doctoral Dissertations Student Scholarship Fall 2010 Bimodule categories and monoidal 2-structure Justin Greenough University of New Hampshire, Durham Follow this and additional works at: https://scholars.unh.edu/dissertation Recommended Citation Greenough, Justin, "Bimodule categories and monoidal 2-structure" (2010). Doctoral Dissertations. 532. https://scholars.unh.edu/dissertation/532 This Dissertation is brought to you for free and open access by the Student Scholarship at University of New Hampshire Scholars' Repository. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of University of New Hampshire Scholars' Repository. For more information, please contact [email protected]. BIMODULE CATEGORIES AND MONOIDAL 2-STRUCTURE BY JUSTIN GREENOUGH B.S., University of Alaska, Anchorage, 2003 M.S., University of New Hampshire, 2008 DISSERTATION Submitted to the University of New Hampshire in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematics September, 2010 UMI Number: 3430786 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMT Dissertation Publishing UMI 3430786 Copyright 2010 by ProQuest LLC. All rights reserved. This edition of the work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 This thesis has been examined and approved.