This textbook treats Lie groups, Lie algebras and their representations
in an elementary but fully rigorous fashion requiring minimal
prerequisites. In particular, the theory of matrix Lie groups and their
Lie algebras is developed using only linear algebra, and more motivation
and intuition for proofs is provided than in most classic texts on the
subject.
In addition to its accessible treatment of the basic theory of Lie
groups and Lie algebras, the book is also noteworthy for including:
- a treatment of the Baker-Campbell-Hausdorff formula and its use in
place of the Frobenius theorem to establish deeper results about the
relationship between Lie groups and Lie algebras
- motivation for the machinery of roots, weights and the Weyl group via
a concrete and detailed exposition of the representation theory of
sl(3;C)
- an unconventional definition of semisimplicity that allows for a rapid
development of the structure theory of semisimple Lie algebras
- a self-contained construction of the representations of compact
groups, independent of Lie-algebraic arguments
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 complete derivation of the main properties of root
systems; the construction of finite-dimensional representations of
semisimple Lie algebras has been elaborated; a treatment of universal
enveloping algebras, including a proof of the Poincaré-Birkhoff-Witt
theorem and the existence of Verma modules; complete proofs of the Weyl
character formula, the Weyl dimension formula and the Kostant
multiplicity formula.
Review of the first edition:
This is an excellent book. It deserves to, and undoubtedly will, become
the standard text for early graduate courses in Lie group theory ... an
important addition to the textbook literature ... it is highly
recommended.
-- The Mathematical Gazette
