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El. knyga: Complex Semisimple Quantum Groups and Representation Theory

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  • Formatas: EPUB+DRM
  • Serija: Lecture Notes in Mathematics 2264
  • Išleidimo metai: 24-Sep-2020
  • Leidėjas: Springer Nature Switzerland AG
  • Kalba: eng
  • ISBN-13: 9783030524630
Kitos knygos pagal šią temą:
Complex Semisimple Quantum Groups and Representation TheoryDidesnis paveikslėlis
  • Formatas: EPUB+DRM
  • Serija: Lecture Notes in Mathematics 2264
  • Išleidimo metai: 24-Sep-2020
  • Leidėjas: Springer Nature Switzerland AG
  • Kalba: eng
  • ISBN-13: 9783030524630
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This book provides a thorough introduction to the theory of complex semisimple quantum groups, that is, Drinfeld doubles of q-deformations of compact semisimple Lie groups. The presentation is comprehensive, beginning with background information on Hopf algebras, and ending with the classification of admissible representations of the q-deformation of a complex semisimple Lie group.





 The main components are:





-   a thorough introduction to quantized universal enveloping algebras over general base fields and generic deformation parameters, including finite dimensional representation theory, the Poincaré-Birkhoff-Witt Theorem, the locally finite part, and the Harish-Chandra homomorphism,





-   the analytic theory of quantized complex semisimple Lie groups in terms of quantized algebras of functions and their duals,





-   algebraic representation theory in terms of category O, and





-   analytic representationtheory of quantized complex semisimple groups.





 Given its scope, the book will be a valuable resource for both graduate students and researchers in the area of quantum groups.

Recenzijos

The book is largely self-contained. It is highly recommended for mathematicians of all levels wishing to learn about these topics, in the algebraic setting and/or in the analytic setting. (Huafeng Zhang, zbMATH 1514.20006, 2023)

1 Introduction
1(4)
2 Multiplier Hopf Algebras
5(20)
2.1 Hopf Algebras
5(3)
2.2 Multiplier Hopf Algebras
8(4)
2.2.1 Essential Algebras
8(1)
2.2.2 Algebraic Multiplier Algebras
9(1)
2.2.3 Multiplier Hopf Algebras
10(2)
2.3 Integrals
12(6)
2.3.1 The Definition of Integrals
13(1)
2.3.2 The Dual Multiplier Hopf Algebra
14(4)
2.4 The Drinfeld Double: Algebraic Level
18(7)
3 Quantized Universal Enveloping Algebras
25(170)
3.1 q-Calculus
25(2)
3.2 The Definition of Uq(g)
27(28)
3.2.1 Semisimple Lie Algebras
27(2)
3.2.2 The Quantized Universal Enveloping Algebra Without Serre Relations
29(6)
3.2.3 The Serre Elements
35(4)
3.2.4 The Quantized Universal Enveloping Algebra
39(4)
3.2.5 The Restricted Integral Form
43(12)
3.3 Verma Modules
55(5)
3.3.1 The Parameter Space h*q
56(2)
3.3.2 Weight Modules and Highest Weight Modules
58(1)
3.3.3 The Definition of Verma Modules
59(1)
3.4 Characters of Uq(g)
60(1)
3.5 Finite Dimensional Representations of Uq(sl(2, K))
61(4)
3.6 Finite Dimensional Representations of Uq(g)
65(4)
3.6.1 Rank-One Quantum Subgroups
65(1)
3.6.2 Finite Dimensional Modules
66(3)
3.7 Braid Group Action and PBW Basis
69(41)
3.7.1 The Case of sl(2, K)
70(5)
3.7.2 The Case of General g
75(11)
3.7.3 The Braid Group Action on Uq(g)
86(8)
3.7.4 The Poincare-Birkhoff-Witt Theorem
94(14)
3.7.5 The Integral PBW-Basis
108(2)
3.8 The Drinfeld Pairing and the Quantum Killing Form
110(27)
3.8.1 Drinfeld Pairing
110(5)
3.8.2 The Quantum Killing Form
115(4)
3.8.3 Computation of the Drinfeld Pairing
119(18)
3.8.4 Nondegeneracy of the Drinfeld Pairing
137(1)
3.9 The Quantum Casimir Element and Simple Modules
137(5)
3.9.1 Complete Reducibility
141(1)
3.10 Quantized Algebras of Functions
142(3)
3.11 The Universal R-Matrix
145(9)
3.11.1 Universal R-Matrices
145(2)
3.11.2 The R-Matrix for Uq(g)
147(7)
3.12 The Locally Finite Part of Uq(g)
154(6)
3.13 The Centre of Uq(g) and the Harish-Chandra Homomorphism
160(8)
3.14 Noetherianity
168(10)
3.14.1 Noetherian Algebras
168(4)
3.14.2 Noetherianity of Uq(q)
172(1)
3.14.3 Noetherianity of (Gq)
173(2)
3.14.4 Noetherianity of FUq(s)
175(3)
3.15 Canonical Bases
178(7)
3.15.1 Crystal Bases
179(2)
3.15.2 Global Bases
181(4)
3.16 Separation of Variables
185(10)
3.16.1 Based Modules
186(2)
3.16.2 Further Preliminaries
188(3)
3.16.3 Separation of Variables
191(4)
4 Complex Semisimple Quantum Groups
195(40)
4.1 Locally Compact Quantum Groups
195(4)
4.1.1 Hopf C*-Algebras
195(1)
4.1.2 The Definition of Locally Compact Quantum Groups
196(3)
4.2 Algebraic Quantum Groups
199(13)
4.2.1 The Definition of Algebraic Quantum Groups
199(1)
4.2.2 Algebraic Quantum Groups on the Hilbert Space Level
200(4)
4.2.3 Compact Quantum Groups
204(2)
4.2.4 The Drinfeld Double of Algebraic Quantum Groups
206(6)
4.3 Compact Semisimple Quantum Groups
212(5)
4.4 Complex Semisimple Quantum Groups
217(6)
4.4.1 The Definition of Complex Quantum Groups
217(4)
4.4.2 The Connected Component of the Identity
221(2)
4.5 Polynomial Functions
223(4)
4.6 The Quantized Universal Enveloping Algebra of a Complex Group
227(5)
4.7 Parabolic Quantum Subgroups
232(3)
5 Category
235(52)
5.1 The Definition of Category
235(9)
5.1.1 Category Is Artinian
236(3)
5.1.2 Duality
239(1)
5.1.3 Dominant and Antidominant Weights
240(4)
5.2 Submodules of Verma Modules
244(5)
5.3 The Shapovalov Determinant
249(6)
5.4 Jantzen Filtration and the BGG Theorem
255(5)
5.5 The PRV Determinant
260(23)
5.5.1 The Spaces F Hom(M, N)
260(3)
5.5.2 Conditions for F Hom(M, N) = 0
263(3)
5.5.3 Multiplicities in F Hom(M, N)
266(3)
5.5.4 Hilbert-Poincare Series for the Locally Finite Part of Uq(g)
269(7)
5.5.5 The Quantum PRV Determinant
276(7)
5.6 Annihilators of Verma Modules
283(4)
6 Representation Theory of Complex Semisimple Quantum Groups
287(70)
6.1 Verma Modules for URq(g)
288(6)
6.1.1 Characters of URq(b)
288(1)
6.1.2 Verma Modules for URq(g)
289(5)
6.2 Representations of Gq
294(3)
6.2.1 Harish-Chandra Modules
294(1)
6.2.2 Unitary Gq-Representations
295(2)
6.3 Action of URq(g) on Kq-Types
297(4)
6.4 Principal Series Representations
301(13)
6.4.1 The Definition of Principal Series Representations
301(2)
6.4.2 Compact Versus Noncompact Pictures
303(3)
6.4.3 The Action of the Centre on the Principal Series
306(1)
6.4.4 Duality for Principal Series Modules
307(2)
6.4.5 Relation Between Principal Series Modules and Verma Modules
309(5)
6.5 An Equivalence of Categories
314(10)
6.6 Irreducible Harish-Chandra Modules
324(7)
6.7 The Principal Series for SLq (2, C)
331(4)
6.8 Intertwining Operators in Higher Rank
335(15)
6.8.1 Intertwiners in the Compact Picture
335(2)
6.8.2 Intertwiners Associated to Simple Roots
337(5)
6.8.3 Explicit Formulas for Intertwiners
342(8)
6.9 Submodules and Quotient Modules of the Principal Series
350(3)
6.10 Unitary Representations
353(4)
List of Symbols and Conventions 357(14)
References 371
Christian Voigt is a Senior Lecturer at the School of Mathematics, University of Glasgow. His main research area is noncommutative geometry, with a focus on quantum groups, operator K-theory, and cyclic homology.





Robert Yuncken is Maītre de Conférences at the Laboratoire de Mathématiques Blaise Pascal, Univerité Clermont Auvergne in France.  His main research interests are in operator algebras, geometry, and representation theory.
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