Lie groups, Lie algebras, and representation theory are the main focus
of this text. In order to keep the prerequisites to a minimum, the
author restricts attention to matrix Lie groups and Lie algebras. This
approach keeps the discussion concrete, allows the reader to get to the
heart of the subject quickly, and covers all of the most interesting
examples. The book also introduces the often-intimidating machinery of
roots and the Weyl group in a gradual way, using examples and
representation theory as motivation. The text is divided into two parts.
The first covers Lie groups and Lie algebras and the relationship
between them, along with basic representation theory. The second part
covers the theory of semisimple Lie groups and Lie algebras, beginning
with a detailed analysis of the representations of SU(3). The author
illustrates the general theory with numerous images pertaining to Lie
algebras of rank two and rank three, including images of root systems,
lattices of dominant integral weights, and weight diagrams. This book is
sure to become a standard textbook for graduate students in mathematics
and physics with little or no prior exposure to Lie theory. Brian Hall
is an Associate Professor of Mathematics at the University of Notre
Dame.

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