Being both a beautiful theory and a valuable tool, Lie algebras form a very important area of mathematics. This modern introduction targets entry-level graduate students. It might also be of interest to those wanting to refresh their knowledge of the area and be introduced to newer material. Infinite-dimensional algebras are treated extensively along with the finite-dimensional ones. After some motivation, the text gives a detailed and concise treatment of the Killing-Cartan classification of finite-dimensional semisimple algebras over algebraically closed fields of characteristic 0. Important constructions such as Chevalley bases follow. The second half of the book serves as a broad introduction to algebras of arbitrary dimension, including Kac-Moody (KM), loop, and affine KM algebras. Finite-dimensional semisimple algebras are viewed as KM algebras of finite dimension, their representation and character theory developed in terms of integrable representations. The text also covers triangular decomposition (after Moody and Pianzola) and the BGG category $\mathcal{O}$. A lengthy chapter discusses the Virasoro algebra and its representations. Several applications to physics are touched on via differential equations, Lie groups, superalgebras, and vertex operator algebras. Each chapter concludes with a problem section and a section on context and history. There is an extensive bibliography, and appendices present some algebraic results used in the book.
J. I. Hall, Michigan State University, East Lansing, MI
Part I. Preliminaries Algebras Examples of Lie algebras Lie groups Part II. Classification Lie algebra basics The Cartan decomposition Semisimple Lie algebras: Basic structure Classification of root systems Semisimple Lie algebras: Classification Part III. Important constructions Finite degree representations of $\mathfrak{sl}_2(\mathbb{K})$ PBW and free Lie algebras Casimir operators and Weyl's Theorem II Chevalley bases and integration Kac-Moody Lie algebras Part IV. Representation Integrable representations The spherical case and Serre's Theorem Irreducible weight modules for $\mathfrak{sl}_2(\mathbb{K})$ Part V. Infinite dimension Some infinite-dimensional Lie algebras Triangular decomposition and category $\mathcal{O}$ Character theory Representation of the Virasoro algebra Part VI. Appendices Appendix A. Algebra basics Appendix B. Bilinear forms Appendix C. Finite groups generated by reflections Bibliography Index
