Lie Groups: An Introduction through Linear Groups (Oxford Graduate Texts in Mathematics)

Lie Groups: An Introduction through Linear Groups (Oxford Graduate Texts in Mathematics)

This book is an introduction to the theory of Lie groups and their representations at the advanced undergraduate or beginning graduate level. It covers the essentials of the subject starting from basic undergraduate mathematics. The correspondence between linear Lie groups and Lie algebras is developed in its local and global aspects. The classical groups are analyzed in detail, first with elementary matrix methods, then with the help of the structural tools typical of the theory of semisimple groups, such as Cartan subgroups, root, weights and reflections. The fundamental groups of the classical groups are worked out as an application of these methods. Manifolds are introduced when needed, in connection with homogeneous spaces, and the elements 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 and its consequences, and proceeds from there to the Peter-Weyl theorem, Weyl's character formula, and the Borel-Weil theorem, all in the context of linear groups. "[A]n excellent and original introductory course on Lie groups."-- Zentralblatt MATH "This excellent book is a welcome edition to the leterature on Lie groups and their representations and will greatly help to make the subject more accessible to less experienced readers."-- Matematical Reviews Wulf Rossmann is at University of Ottawa, Department of Mathematics and Statistics.