Ebook: Representations of Compact Lie Groups

This book is based on several courses given by the authors since 1966. It introduces the reader to the representation theory of compact Lie groups. We have chosen a geometrical and analytical approach since we feel that this is the easiest way to motivate and establish the theory and to indicate relations to other branches of mathematics. Lie algebras, though mentioned occasionally, are not used in an essential way. The material as well as its presentation are classical; one might say that the foundations were known to Hermann Weyl at least 50 years ago. Prerequisites to the book are standard linear algebra and analysis, including Stokes' theorem for manifolds. The book can be read by German students in their third year, or by first-year graduate students in the United States. Generally speaking the book should be useful for mathematicians with geometric interests and, we hope, for physicists. At the end of each section the reader will find a set of exercises. These vary in character: Some ask the reader to verify statements used in the text, some contain additional information, and some present examples and counter­ examples. We advise the reader at least to read through the exercises.

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This book is an introduction to the representation theory of compact Lie groups, following Hermann Weyl's original approach. Although the authors discuss all aspects of finite-dimensional Lie theory, the emphasis throughout the book is on the groups themselves. The presentation is consequently more geometric and analytic than algebraic in nature. The central results, culminating the Weyl character formula, are reached directly and quickly, and they appear in forms suitable for applications to physics and geometry. This book is a good reference and a source of explicit computations, for physicists and mathematicians. Each section is supplemented by a wide range of exercices, and geometric ideas are illustrated with the help of 24 figures.