Introduction to Lie Algebras, ISSN 1615-2085, 2006, 251 pages, K. Erdmann, Mark J.Wildon, 1846280400, 9781846280405, Springer, 2006

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Lie groups and Lie algebras have become essential to many parts of mathematics and theoreticalphysics, with Lie algebras a central object of interest in their own right. Based on a lecture coursegiven to fourth-year undergraduates, this book provides an elementary introduction to Liealgebras. It starts with basic concepts. A section on low-dimensional Lie algebras provides readerswith experience of some useful examples. This is followed by a discussion of solvable Lie algebrasand a strategy towards a classification of finite-dimensional complex Lie algebras. The nextchapters cover Engel's theorem, Lie's theorem and Cartan's criteria and introduce somerepresentation theory. The root-space decomposition of a semisimple Lie algebra is discussed, andthe classical Lie algebras studied in detail. The authors also classify root systems, and give anoutline of Serre's construction of complex semisimple Lie algebras. An overview of furtherdirections then concludes the book and shows the high degree to which Lie algebras influencepresent-day mathematics. The only prerequisite is some linear algebra and an appendixsummarizes the main facts that are needed. The treatment is kept as simple as possible with noattempt at full generality. Numerous worked examples and exercises are provided to testunderstanding, along with more demanding problems, several of which have solutions.Introduction to Lie Algebras covers the core material required for almost all other work in Lietheory and provides a self-study guide suitable for undergraduate students in their final year andgraduate students and researchers in mathematics and theoretical physics.

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