On Constructions of Matrix Bialgebras, © March 2018 DECLARATION
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ONCONSTRUCTIONSOFMATRIXBIALGEBRAS submitted by szabolcs mészáros In partial fulfillment of the requirements for the Degree of Doctor of Philosophy in Mathematics Supervisor: Mátyás Domokos CEU eTD Collection Budapest, Hungary March 2018 CEU eTD Collection Szabolcs Mészáros: On Constructions of Matrix Bialgebras, © March 2018 DECLARATION I, the undersigned Szabolcs Mészáros, candidate for the degree of Doc- tor of Philosophy at the Central European University Department of Math- ematics and its Applications, declare herewith that the present thesis is based on my research and only such external information as properly credited in notes and bibliography. I declare that no unidentified and illegitimate use was made of the work of others, and no part of this thesis infringes on any person’s or institution’s copyright. I also declare that no part of the thesis has been submitted in this form to any other institution of higher education for an academic degree. Budapest, Hungary, March 2018 Szabolcs Mészáros CEU eTD Collection CEU eTD Collection ABSTRACT The thesis consists of two parts. In the first part consisting of Chapter 2 and 3, matrix bialgebras, generalizations of the quantized coordinate ring of n × n matrices are considered. The defining parameter of the construc- tion is an endomorphism of the tensor-square of a vector space. In the investigations this endomorphism is assumed to be either an idempotent or nilpotent of order two. In Theorem 2.2.2, 2.3.2 and 2.5.3 it is proved that the Yang-Baxter equation gives not only a sufficient condition – as it was known before – for certain regularity properties of matrix bialgebras, such as the Poincaré-Birkhoff-Witt basis property or the Koszul property, but it is also necessary, under some technical assumptions. The proofs are based on the methods of the representation theory of finite-dimensional algebras. In the second part consisting of Chapter 4 and 5, the quantized coor- dinate rings of matrices, the general linear group and the special linear group are considered, together with the corresponding Poisson algebras called semiclassical limit Poisson algebras. In Theorem 4.1.1 and 5.1.1 it is proved that the subalgebra of cocommutative elements in the above mentioned algebras and Poisson algebras are maximal commutative, and maximal Poisson-commutative subalgebras respectively. The proofs are based on graded-filtered arguments. CEU eTD Collection v CEU eTD Collection Pötyinek. És mindenkinek, aki szeret. CEU eTD Collection CEU eTD Collection “(...) you’ll begin challenging your own assumptions. Your assumptions are your windows on the world. Scrub them off every once in a while, or the light won’t come in.” — Alan Alda (62nd Commencement Address, Connecticut College, New London, 1980) ACKNOWLEDGMENTS I would like to express my sincere gratitude to a number of people without whom it would not have been possible to complete this thesis. I would like to thank my supervisor, Prof. Mátyás Domokos, who guided me through my doctoral studies and research with his extensive knowledge and professionalism, and devoted time to introduce me to the field of algebra and to the techniques of mathematical research. I would like to say thank you to Prof. Péter Pál Pálfy whose encourage- ment was an immense help from the beginning of my studies. This essay builds on numerous things that I had the opportunity to learn from a variety of exceptional scholars, including but not limited to: Prof. István Ágoston, Prof. Károly Böröczky and Prof. Tamás Szamuely. I am also grateful for the CEU community and the members of the Mathe- matics department. It was a pleasure to be surrounded by truly enthusias- tic people as Martin Allen, László Tóth, Ferenc Bencs (and Eszter Bokányi) who are always glad to help in any form. I would also like to thank János Fokföldy for his helpful advices. I owe a deep debt of gratitude to Attila Guld and Ákos Kyriakos Matszangosz. Their friendship, the inspiring discourses and professional critiques created the motivating environment that aided me during my work. Very special thanks go to my friends and former classmates Barbara A. Balázs, Richárd Seb˝ok,Nóra Sz˝oke,Balázs Takács and Dávid Tóth for their sustained encouragement, particularly for their objective comments. I wish to thank my parents and my family for their invaluable assis- tance through the last two decades. CEU eTD Collection Last, but definitely not the least, I am especially thankful to my wife, Marica. Her unconditional support was my closest colleague along this journey. Thank you. ix CEU eTD Collection CONTENTS preface1 1 preliminaries5 1.1 Conventions . 5 1.2 Graded algebras . 6 1.2.1 PBW-basis . 7 1.3 Finite-dimensional algebras . 10 1.3.1 Representations theory of the four subspace quiver . 10 1.3.2 Modules over biserial algebras . 12 1.3.3 Norm-square on monoids . 14 1.3.4 Endomorphisms . 15 1.4 Monoidal categories . 17 1.4.1 Tensor bialgebra . 17 1.4.2 Ring categories . 19 2 matrix bialgebras 21 2.1 The matrix bialgebra M(p) ................... 21 2.1.1 Universal property . 23 2.1.2 Dual, Schur bialgebra . 26 2.2 Idempotent case . 28 2.2.1 Excursion: relation to the four subspace quiver . 29 2.2.2 Modules of P3 ....................... 31 2.2.3 Yang-Baxter equation . 36 2.2.4 Proof of Theorem 2.2.2 .................. 38 2.2.5 PBW-basis . 47 2.3 Nilpotent case . 49 0 2.3.1 Representations of P3 ................... 50 2.3.2 Proof of Theorem 2.3.2 .................. 54 2.4 Upper bound . 57 2.4.1 Ordered Multiplicities . 57 2.4.2 The case of symmetric groups . 59 2.4.3 Subsymmetric case . 61 2.4.4 Subsymmetric in degree three . 63 CEU eTD Collection 2.5 Orthogonal projection case . 64 2.5.1 Subsymmetric in degree four . 66 2.5.2 Koszul property . 70 2.5.3 Some P4-modules . 73 2.5.4 Proof of Theorem 2.5.3 .................. 76 xi contents 3 examples 80 3.1 Dimension two . 80 3.2 Quantum orthogonal matrices . 83 3.2.1 Dimension three . 84 3.2.2 Dimension four . 85 3.2.3 Higher dimension . 87 3.3 Twisting . 87 4 quantized coordinate rings of Mn, GLn and SLn 91 4.1 Main results of the chapter . 91 4.2 Definitions . 92 4.2.1 Quantized coordinate rings . 92 4.2.2 Quantum minors . 93 4.2.3 PBW-basis . 94 4.2.4 Associated graded ring . 94 4.3 Equivalence of the statements . 95 4.4 Case of Oq(SL2) .......................... 97 4.5 Proof of the main result . 102 5 semiclassical limit poisson algebras 107 5.1 Main results of the chapter . 107 5.2 Definitions . 108 5.2.1 Poisson algebras . 108 5.2.2 Filtered Poisson algebras . 109 5.2.3 The Kirillov-Kostant-Souriau bracket . 110 5.2.4 Semiclassical limits . 111 5.2.5 Semiclassical limits of quantized coordinate rings . 111 5.2.6 Coefficients of the characteristic polynomial . 112 5.3 Equivalence of the statements . 112 5.4 Case of O(SL2) .......................... 114 5.5 Proof of the main result . 115 bibliography 121 index 126 index of symbols 127 CEU eTD Collection xii PREFACE The field of quantum groups and quantum algebra emerged at the in- tersection of ring theory, Lie theory and C∗-algebras in the 1980s from the works of V. G. Drinfeld [Dri], M. Jimbo [Ji], L. D. Faddeev et al. [FRT], Yu. I. Manin [Man], and S. L. Woronowicz [Wo]. Although the frameworks they applied were different in nature and di- verged even further in the last four decades, one of the common guiding principles was to observe phenomena that are in parallel with the clas- sical counterparts, such as the (Brauer-)Schur-Weyl duality (see [Hay] or Sec. 8.6 in [KS]), analogous representation theory (see Sec. 10.1 in [ChP]), existence of a Poincaré-Birkhoff-Witt basis (see I.6.8 in [BG]) or existence of a Haar state (see I.2. in [NT]). In the thesis, we follow the track laid by Yu. I. Manin and M. Takeuchi, and investigate matrix bialgebras (see Def. 2.1.1) from the point of view of properties of quadratic graded algebras and their symptoms on the Hilbert series of the algebra. The terminology on these bialgebras is very diverse, they are also called quantum semigroups in [Man] (Ch. 7), ma- tric bialgebras or conormalizer algebras in [Ta], or matrix-element bial- gebras in [Su]. Moreover, matrix bialgebras are special cases of the FRT- construction in the sense of [Lu], and of the universal coacting bialgebra or coend-construction (see [EGNO] and Subsec. 2.1.1). Main examples of matrix bialgebras include every FRT-bialgebra M(Rˆ ), where Rˆ satisfies the Yang-Baxter equation (see [KS],[Hay]), in particular the quantized coordinate ring Oq(Mn) of n × n matrices for a non-zero scalar q. Further examples are the covering bialgebras of quantum SL2 + Hopf-algebras (see [DVL]) or the quantum orthogonal bialgebra Me q (n) (see [Ta]). A well-investigated case is that of the Hecke-type FRT-bialgebras M(Rˆ ) where Rˆ satisfies both the Yang-Baxter and the Hecke equations. These algebras are known to have several favorable properties under mild con- ditions (see [AA],[Hai1],[Su]). After introducing the conventions and def- initions of the studied topics in Chapter 1, we give results in the reverse direction in Chapter 2. Namely a matrix bialgebra M(p) – associated to CEU eTD Collection an element p 2 End(V ⊗ V) with minimal polynomial of degree two – cannot have the appropriate Hilbert-series implied by the above proper- ties, without being a Hecke-type FRT-bialgebra.