Lie Groups And Compact Groups

These short notes, already well-known in their original French edition, give the basic theory of semisimple Lie algebras over the complex numbers, including classification theorem. The author begins with a summary of the general properties of nilpotent, solvable, and semisimple Lie algebras. Subsequent chapters introduce Cartan subalgebras, root systems, and linear representations. The last chapter discusses the connection between Lie algebras, complex groups and compact groups; it is intended to guide the reader towards further study.

These short notes, already well-known in their original French edition, give the basic theory of semisimple Lie algebras over the complex numbers, including classification theorem. The author begins with a summary of the general properties of nilpotent, solvable, and semisimple Lie algebras. Subsequent chapters introduce Cartan subalgebras, root systems, and linear representations. The last chapter discusses the connection between Lie algebras, complex groups and compact groups; it is intended to guide the reader towards further study.

These short notes, already well-known in their original French edition, give the basic theory of semisimple Lie algebras over the complex numbers, including classification theorem. The author begins with a summary of the general properties of nilpotent, solvable, and semisimple Lie algebras. Subsequent chapters introduce Cartan subalgebras, root systems, and linear representations. The last chapter discusses the connection between Lie algebras, complex groups and compact groups; it is intended to guide the reader towards further study.

These short notes, already well-known in their original French edition, give the basic theory of semisimple Lie algebras over the complex numbers, including classification theorem. The author begins with a summary of the general properties of nilpotent, solvable, and semisimple Lie algebras. Subsequent chapters introduce Cartan subalgebras, root systems, and linear representations. The last chapter discusses the connection between Lie algebras, complex groups and compact groups; it is intended to guide the reader towards further study.

This book aims to be a course in Lie groups that can be covered in one year with a group of good graduate students. I have attempted to address a problem that anyone teaching this subject must have, which is that the amount of essential material is too much to cover. One approach to this problem is to emphasize the beautiful representation theory of compact groups, and indeed this book can be used for a course of this type if after Chapter 25 one skips ahead to Part III. But I did not want to omit important topics such as the Bruhat decomposition and the theory of symmetric spaces. For these subjects, compact groups are not sufficient. Part I covers standard general properties of representations of compact groups (including Lie groups and other compact groups, such as finite or p­ adic ones). These include Schur orthogonality, properties of matrix coefficients and the Peter-Weyl Theorem.

Blending algebra, analysis, and topology, the study of compact Lie groups is one of the most beautiful areas of mathematics and a key stepping stone to the theory of general Lie groups. Assuming no prior knowledge of Lie groups, this book covers the structure and representation theory of compact Lie groups. Coverage includes the construction of the Spin groups, Schur Orthogonality, the Peter-Weyl Theorem, the Plancherel Theorem, the Maximal Torus Theorem, the Commutator Theorem, the Weyl Integration and Character Formulas, the Highest Weight Classification, and the Borel-Weil Theorem. The book develops the necessary Lie algebra theory with a streamlined approach focusing on linear Lie groups.