Lie Groups, Lie Algebras, and Representations - An Elementary Introduction (Relié)

This textbook treats Lie groups, Lie algebras, and their representations in an elementary but rigorous fashion, requiring minimal prerequisites. It develops the theory of matrix Lie groups and their Lie algebras using only linear algebra, and provides more motivation and intuition for proofs than in most texts on the subject. In addition to providing an accessible treatment of basic Lie theory, the book also : derives the Baker-Campbell-Hausdorff formula and uses it in place of the Frobenius theorem to establish deeper results about the relationship between Lie groups and Lie algebras ; motivates the machinery of roots, weights, and the Weyl group through a detailed exposition of the representation theory of sl(3 ; C) ; rapidly develops the structure of semisimple Lie algebras by using a definition of semisimplicity that is unconventional but equivalent to the standard one gives independent constructions of the representations of semisimple Lie algebras and compact Lie groups.
The second edition of Lie Groups, Lie Algebras, and Representations contains many substantial improvements and additions, among them : an entirely new part devoted to the structure and representation theory of compact Lie groups ; a full derivation of the main properties of root systems ; a complete construction of the representations of semisimple Lie algebras using Verma modules ; and detailed proofs of the Weyl character formula, the Weyl dimension formula, and the Kostant multiplicity formula.
The new edition also includes many additional figures.