Lie Groups, Lie Algebras, and Representations: An Elementary Introduction (Graduate Texts in Mathematics, 222)

Book Details

• Author: Brian Hall

• Publisher: Springer

• Language: English

• Edition: Softcover reprint of the original 2nd ed. 2015

• ISBN: 9783319374338

• Pages: 462

• Cover: Paperback

• Dimensions: 9.1 x 6.0 x 1.2 inches

About The Book

Lie Groups, Lie Algebras, and Representations by Brian Hall is a comprehensive and accessible introduction to one of the most important areas of modern mathematics. Written with clarity and rigor, this textbook provides a foundational treatment of Lie theory, beginning with matrix Lie groups and their corresponding Lie algebras, using only linear algebra as a prerequisite. Its intuitive explanations and detailed motivation behind proofs make it ideal for early graduate-level students.

This second edition has been significantly expanded and enhanced. Key highlights include an innovative approach to semisimplicity, a full derivation of the properties of root systems, and a thorough development of the structure and representation theory of compact Lie groups. The book offers clear treatments of the Baker–Campbell–Hausdorff formula, universal enveloping algebras, and essential results such as the Poincaré–Birkhoff–Witt theorem, Verma modules, Weyl character formula, and Kostant multiplicity formula.

Hall also uniquely motivates advanced concepts like root systems, weights, and the Weyl group through concrete examples, such as the representation theory of sl(3;C). The book’s self-contained style and independent development of compact group representations without relying heavily on Lie algebra theory add to its teaching value.

Highly recommended by The Mathematical Gazette as a future standard for graduate-level study, this book is an essential addition to the library of anyone interested in mathematical physics, advanced algebra, or differential geometry.