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A Tour of Representation Theory 1st Edition by Martin Lorenz 1470436809 9781470436803

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A Tour of Representation Theory draft 1, Draft, January 11, 2017 Edition Martin Lorenz Digital Instant Download

Author(s): Martin Lorenz
ISBN(s): 9781470436803, 1470436809
Edition: 1, Draft, January 11, 2017
File Details: PDF, 3.20 MB
Year: 2017
Language: english
SKU: EB-5724196 Category: Tag:

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ISBN 10: 1470436809 

ISBN 13: 9781470436803

Author: Martin Lorenz

Representation theory investigates the different ways in which a given algebraic object such as a group or a Lie algebra can act on a vector space. Besides being a subject of great intrinsic beauty, the theory enjoys the additional benefit of having applications in myriad contexts outside pure mathematics, including quantum field theory and the study of molecules in chemistry. Adopting a panoramic viewpoint, this book offers an introduction to four different flavors of representation theory: representations of algebras, groups, Lie algebras, and Hopf algebras. A separate part of the book is devoted to each of these areas and they are all treated in sufficient depth to enable and hopefully entice the reader to pursue research in representation theory. The book is intended as a textbook for a course on representation theory, which could immediately follow the standard graduate abstract algebra course, and for subsequent more advanced reading courses. Therefore, more than 350 exercises at various levels of difficulty are included. The broad range of topics covered will also make the text a valuable reference for researchers in algebra and related areas and a source for graduate and postgraduate students wishing to learn more about representation theory by self-study.

Table of contents:

Part I. Algebras
Chapter 1 Representations of Algebras
1.1 Algebras
1.2 Representations
1.3 Primitive Ideals
1.4 Semisimplicity
1.5 Characters
Chapter 2 Further Topics on Algebras
2.1 Projectives
2.2 Frobenius and Symmetric Algebras

Part II. Groups
Chapter 3 Groups and Group Algebras
3.1 Generalities
3.2 First Examples
3.3 More Structure
3.4 Semisimple Group Algebras
3.5 Further Examples
3.6 Some Classical Theorems
3.7 Characters, Symmetric Polynomials, and Invariant Theory
3.8 Decomposing Tensor Powers
Chapter 4 Symmetric Groups
4.1 Gelfand-Zetlin Algebras
4.2 The Branching Graph
4.3 The Young Graph
4.4 Proof of the Graph Isomorphism Theorem
4.5 The Irreducible Representations
4.6 The Murnaghan-Nakayama Rule
4.7 Schur-Weyl Duality

Part III. Lie Algebras
Chapter 5 Lie Algebras and Enveloping Algebras
5.1 Lie Algebra Basics
5.2 Types of Lie Algebras
5.3 Three Theorems about Linear Lie Algebras
5.4 Enveloping Algebras
5.5 Generalities on Representations of Lie Algebras
5.6 The Nullstellensatz for Enveloping Algebras
5.7 Representations of sl_2
Chapter 6 Semisimple Lie Algebras
6.1 Characterizations of Semisimplicity
6.2 Complete Reducibility
6.3 Cartan Subalgebras and the Root Space Decomposition
6.4 The Classical Lie Algebras
Chapter 7 Root Systems
7.1 Abstract Root Systems
7.2 Bases of a Root System
7.3 Classification
7.4 Lattices Associated to a Root System
Chapter 8 Representations of Semisimple Lie Algebras
8.1 Reminders
8.2 Finite-Dimensional Representations
8.3 Highest Weight Representations
8.4 Finite-Dimensional Irreducible Representations
8.5 The Representation Ring
8.6 The Center of the Enveloping Algebra
8.7 Weyl’s Character Formula
8.8 Schur Functors and Representations of sl(V)

Part IV. Hopf Algebras
Chapter 9 Coalgebras, Bialgebras, and Hopf Algebras
9.1 Coalgebras
9.2 Comodules
9.3 Bialgebras and Hopf Algebras
Chapter 10 Representations and Actions
10.1 Representations of Hopf Algebras
10.2 First Applications
10.3 The Representation Ring of a Hopf Algebra
10.4 Actions and Coactions of Hopf Algebras on Algebras
Chapter 11 Affine Algebraic Groups
11.1 Affine Group Schemes
11.2 Affine Algebraic Groups
11.3 Representations and Actions
11.4 Linearity
11.5 Irreducibility and Connectedness
11.6 The Lie Algebra of an Affine Algebraic Group
11.7 Algebraic Group Actions on Prime Spectra
Chapter 12 Finite-Dimensional Hopf Algebras
12.1 Frobenius Structure
12.2 The Antipode
12.3 Semisimplicity
12.4 Divisibility Theorems
12.5 Frobenius-Schur Indicators

Appendices
Appendix A. The Language of Categories and Functors
A.1 Categories
A.2 Functors
A.3 Naturality
A.4 Adjointness
Appendix B. Background from Linear Algebra
B.1 Tensor Products
B.2 Hom-⊗ Relations
B.3 Vector Spaces
Appendix C. Some Commutative Algebra
C.1 The Nullstellensatz
C.2 The Generic Flatness Lemma
C.3 The Zariski Topology on a Vector Space
Appendix D. The Diamond Lemma
D.1 The Goal
D.2 The Method
D.3 First Applications
D.4 A Simplification
D.5 The Poincaré-Birkhoff-Witt Theorem
Appendix E. The Symmetric Ring of Quotients
E.1 Definition and Basic Properties
E.2 The Extended Center
E.3 Comparison with Other Rings of Quotients

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