A Tour of Representation Theory
About this Title
Martin Lorenz, Temple University, Philadelphia, PA
Publication: Graduate Studies in Mathematics
Publication Year: 2018; Volume 193
ISBNs: 978-1-4704-3680-3 (print); 978-1-4704-4905-6 (online)
MathSciNet review: MR3837537
MSC: Primary 00-02; Secondary 16-02, 16G, 16T, 17-02, 20-02, 20C, 20G
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.
Graduate students and researchers interested in various aspects of representation theory.
Table of Contents
- Lie algebras and enveloping algebras
- Semisimple Lie algebras
- Root systems
- Representations of semisimple Lie algebras
- Coalgebras, bialgebras, and Hopf algebras
- Representations and actions
- Affine algebraic groups
- Finite-dimensional Hopf algebras
- The language of categories and functors
- Background from linear algebra
- Some commutative algebra
- The Diamond Lemma
- The symmetric ring of quotients