**Graduate Studies in Mathematics**

Volume: 193;
2018;
654 pp;
Hardcover

MSC: Primary 16; 17; 20;

**Print ISBN: 978-1-4704-3680-3
Product Code: GSM/193**

List Price: $94.00

AMS Member Price: $75.20

MAA Member Price: $84.60

**Electronic ISBN: 978-1-4704-4905-6
Product Code: GSM/193.E**

List Price: $94.00

AMS Member Price: $75.20

MAA Member Price: $84.60

#### You may also like

#### Supplemental Materials

# A Tour of Representation Theory

Share this page
*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.

#### Readership

Graduate students and researchers interested in various aspects of representation theory.

#### Reviews & Endorsements

Complemented by more than 350 exercises at various levels of difficulty, this text is a valuable reference for researchers and students in algebra and related fields, and is ideally suitable for learning representation theory by self-study.

-- Dongwen Liu, Mathematical Reviews

This excellent book is one that I wish had been written when I was a student...Had I the benefit of a book like this one in my early graduate years, I could have saved myself a lot of time...This is a very nicely written book, with student motivation always in mind. The level of difficulty increases as the book proceeds (as is only reasonable) but at no point does the book become too difficult for a well-prepared graduate student reader...I like this book a lot, and consider it to be a very valuable addition to the existing textbook literature on representation theory. It would not surprise me if it becomes the market leader in books on graduate-level representation theory.

-- Mark Hunacek, MAA Reviews

#### Table of Contents

# Table of Contents

## A Tour of Representation Theory

- Cover Cover11
- Title page iii4
- Preface xi12
- Conventions xvii18
- Part I . Algebras 120
- Part II . Groups 111130
- Part III . Lie Algebras 243262
- Chapter 5. Lie Algebras and Enveloping Algebras 245264
- 5.1. Lie Algebra Basics 246265
- 5.2. Types of Lie Algebras 253272
- 5.3. Three Theorems about Linear Lie Algebras 257276
- 5.4. Enveloping Algebras 266285
- 5.5. Generalities on Representations of Lie Algebras 278297
- 5.6. The Nullstellensatz for Enveloping Algebras 287306
- 5.7. Representations of \fsl₂ 300319

- Chapter 6. Semisimple Lie Algebras 315334
- Chapter 7. Root Systems 341360
- Chapter 8. Representations of Semisimple Lie Algebras 373392
- 8.1. Reminders 374393
- 8.2. Finite-Dimensional Representations 377396
- 8.3. Highest Weight Representations 379398
- 8.4. Finite-Dimensional Irreducible Representations 385404
- 8.5. The Representation Ring 390409
- 8.6. The Center of the Enveloping Algebra 393412
- 8.7. Weyl’s Character Formula 408427
- 8.8. Schur Functors and Representations of \fsl(𝑉) 418437

- Part IV . Hopf Algebras 425444
- Appendices 573592
- Appendix A. The Language of Categories and Functors 575594
- Appendix B. Background from Linear Algebra 587606
- Appendix C. Some Commutative Algebra 599618
- Appendix D. The Diamond Lemma 605624
- Appendix E. The Symmetric Ring of Quotients 615634
- Bibliography 623642
- Subject Index 633652
- Index of Names 645664
- Notation 649668

- Back Cover Back Cover1674