Lie Groups is intended as 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 analysed 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, roots, 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.
Wulf Rossmann is in the Department of Mathematics and Statistics, University of Ottawa.
'Rossmann's book is a pioneering treatment of elementary Lie theory.' A. W. Knapp, The American Mathematical Monthly
'The book is very well written, covering essential facts needed for understanding the theory, and is easy to use when preparing courses on Lie groups ... This is an excellent addition to the existing literature and should be useful for teachers and students, and for mathematicians and mathematical physicists.' EMS
'Anyone who has been privileged to work with Rossmann's research ideas knows that he is beholden to no one's traditions but his own. In capable hands, that kind of free thinking invariably leads to considerable and unanticipated advances. This time Rossmann has turned his attention to the exposition of elementary Lie theory and has, indeed, advanced the subject considerably.' Bulletin of the American Mathematical Society
'Review from previous edition Rossmann adopts a new approach and works with arbitrary sets of invertible matrices closed under inversion and multiplication but with absolutely no topological assumptions.' Bulletin of the American Mathematical Society
'Rossmann's book is a gem ... His development has some elements that are new to me, and I regard those elements as a breakthrough in making elementary Lie theory more widely accessible.' A. W. Knapp, The American Mathematical Monthly
'This is a significant accomplishment.' Bulletin of the American Mathematical Society
'The book benefits from a considerable number of exercises in each chapter. It is well worth considering as a course text.' The Mathematical Gazette