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Publisher:

Imperial College Press

Publication Date:

2010

Number of Pages:

263

Format:

Hardcover

Price:

65.00

ISBN:

9781848165175

Category:

Textbook

[Reviewed by , on ]

Fernando Q. Gouvêa

04/14/2010

Sometimes the best way to really understand a theory is to take a special case and understand it through-and-through. Ideally, the special case should be simple enough to work through but complex enough to illustrate the general situation well. This is also a good way to prepare to *create* a general theory: work out a (more) concrete example to the end, in the hope that it will provide guidance for the general case.

This book applies that method to a sophisticated subject, the representation theory of Lie algebras. The chosen Lie algebra is sl_{2}(**C**), which can be viewed as the Lie algebra of the complex Lie group SL_{2}(**C**) or as the complexification of the real Lie algebra of SU(2). The goal is to describe (up to isomorphism) all simple sl_{2}(**C**)-modules. The catch is that the author really does mean *all*, which includes the infinite-dimensional ones.

The theory of finite-dimensional sl_{2}(**C**)-modules is a standard topic, and this book disposes of it quite quickly in the first chapter. The second chapter introduces the universal enveloping algebra and shows that representations of sl_{2}(**C**) are the same as representations of its enveloping algebra. And then the fun starts. The ensuing chapters develop the full theory of simple sl_{2}(**C**)-modules, leading to a description of all the simple modules in chapter 6.

This brings us to the frontiers of research. The description of all the simple sl_{2}(**C**)-modules was obtained only in the last few years, and sl_{2}(**C**) is the *only* simple complex Lie algebra for which such a description is known.

Reading through this book would be an excellent way of learning this material. The exposition is brisk but not unfriendly, and there are many exercises. Very brief hints for the exercises are provided in an appendix, and contextualizing notes appear at the end of each chapter. It is very nicely done.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.

- Finite-Dimensional Modules
- The Universal Enveloping Algebra
- Weight Modules
- The Primitive Spectrum
- Category O
- Description of All Simple Modules
- Categorification of Simple Finite-Dimensional Modules

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