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Lectures on sl<sub>2</sub>(C)-modules

Volodymir Mazorchuk
Imperial College Press
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
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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 sl2(C), which can be viewed as the Lie algebra of the complex Lie group SL2(C) or as the complexification of the real Lie algebra of SU(2). The goal is to describe (up to isomorphism) all simple sl2(C)-modules. The catch is that the author really does mean all, which includes the infinite-dimensional ones.

The theory of finite-dimensional sl2(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 sl2(C) are the same as representations of its enveloping algebra. And then the fun starts. The ensuing chapters develop the full theory of simple sl2(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 sl2(C)-modules was obtained only in the last few  years, and sl2(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