Lie groups, Lie algebras, and representations: An Elementary Introduction
To present a circle of ideas around Lie groups, Lie algebras and their representations, it is necessary to make a few principal choices. The first question is how to describe a relation between Lie groups and Lie algebras. To make the book accessible to a broader audience, the author does not suppose knowledge of theory of manifolds. He restricts the attention to matrix groups. Lie algebra of G is then defined using simple properties of the exponential map. As for the correspondence between Lie group homomorphisms and Lie algebra homomorphisms, the author is using the Baker-Campbell-Hausdorff theorem for its description. In the main part of the book, finite dimensional representations of classical semisimple Lie groups are classified by their highest weights. Their construction is given in three different ways (as quotients of Verma modules, or by the Peter-Weyl theorem, or by the Borel-Weil realization). The proof of the complete reducibility is based on properties of representations of compact groups. The Weyl character formula and the classification of complex semisimple Lie algebras end the main part of the book. To keep prerequisites minimal, the author also offers a few appendices. The book is written in a systematic and clear way, each chapter ends with a set of exercises. The book could be valuable for students of mathematics and physics as well as for teachers, for preparation of courses. It is a nice addition to the existing literature.