Introduction to Lie algebras : finite and infinite dimension / J. I. Hall.

Includes bibliographical references and index.

Algebras -- Examples of Lie algebras -- Lie groups -- Lie algebra basics -- Cartan decomposition -- Semisimple Lie algebras -- Root systems -- PBW and free Lie algebras -- Chevalley bases -- Kac-Moody Lie algebras -- Integrable representations -- Character theory -- Infinite-dimensional Lie algebras -- Virasoro algebra.

This graduate-level textbook presents a modern introduction to the theory of Lie algebras, covering both finite-dimensional and infinite-dimensional structures. Beginning with the algebraic foundations and classical examples of Lie algebras and Lie groups, the book develops the structural theory of semisimple Lie algebras, including Cartan decompositions, root systems, and the Killing-Cartan classification. The author further explores representation theory, Chevalley bases, universal enveloping algebras, and integrable representations. A substantial portion of the text is devoted to infinite-dimensional Lie algebras, including Kac-Moody algebras, affine Lie algebras, loop algebras, and the Virasoro algebra, together with applications to geometry, mathematical physics, and differential equations. Combining rigorous mathematical development with historical context, examples, and exercises, the book serves as a comprehensive resource for graduate students and researchers in algebra, geometry, and theoretical physics.