Lie Groups, Lie Algebras, and Cohomology. (MN-34), Volume 34 (Mathematical Notes Book 108) by Anthony W. Knapp (PDF)

Description

This book starts with the elementary theory of Lie groups of matrices and arrives at the definition, elementary properties, and first applications of cohomological induction, which is a recently discovered algebraic construction of group representations. Along the way it develops the computational techniques that are so important in handling Lie groups. The book is based on a one-semester course given at the State University of New York, Stony Brook in fall, 1986 to an audience having little or no background in Lie groups but interested in seeing connections among algebra, geometry, and Lie theory. These notes develop what is needed beyond a first graduate course in algebra in order to appreciate cohomological induction and to see its first consequences. Along the way one is able to study homological algebra with a significant application in mind; consequently one sees just what results in that subject are fundamental and what results are minor.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Prerequisites for the book are metric spaces, a second course in linear algebra and a bit of knowledge about topological groups. It is one of the three best books I’ve read on the cohomology theory of Lie algebras (the other two are D. Fuch’s book, the Cohomology Theory of Infinite Dimensional Lie Algebras and Borel and Wallach’s book on Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups). The author assumes essentially no background in Lie theory, but in order to understand motivation and context it is perhaps best to have read and understood a first course in Lie theory. I read this book over the summer when I was a graduate student right after I took a course in Lie theory based on Humphrey’s text. For any student who is interested in pursuing research in Lie theory and in particular who wants to learn about the cohomology of Lie algebras, this book is an excellent place to start.One of two great results covered in the book (in the case of U(n)) is the Borel-Weil-Bott Theorem whereby one obtains a realization of all finite dimensional irreducible representations of a compact Lie group as the space of holomorphic sections of certain line bundles. In 1966 Langlands speculated that after some modification the Borel-Weil-Bott Theorem could provide a realization of discrete series representations of noncompact semisimple Lie groups. These representations are a particular fundamental family of infinite dimensional irreducible representations of these noncompact groups. W. Schmid proved in the 1970’s Langland’s conjecture for the discrete series representations and in 1978 G. Zuckerman rephrased Langland’s conjecture in Lie algebraic terms using homological algebra. The main tool used is homological induction. The last chapter of the book deals with how one needs to make changes in order to adjust the arguments to the non-compact case paving the way to a realization of the discrete series representations.

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