Book Review: Lie groups: An introduction through linear groups

Abstract

This book is an introduction to the theory of Lie groups and theirrepresentations at the advanced undergraduate or beginning graduate level. Itcovers the essentials of the subject starting from basic undergraduatemathematics. The correspondence between linear Lie groups and Lie algebras isdeveloped in its local and global aspects. The classical groups are analyzedin detail, first with elementary matrix methods, then with the help of thestructural tools typical of the theory of semisimple groups, such as Cartansubgroups, root, weights and reflections. The fundamental groups of theclassical groups are worked out as an application of these methods. Manifoldsare introduced when needed, in connection with homogeneous spaces, and theelements of differential and integral calculus on manifolds are presented,with special emphasis on integration on groups and homogeneous spaces.Representation theory starts from first principles, such as Schur's lemma andits consequences, and proceeds from there to the Peter-Weyl theorem, Weyl'scharacter formula, and the Borel-Weil theorem, all in the context of lineargroups.