Synopses & Reviews
This is the first monograph to exclusively treat Kac-Moody (K-M) groups, a standard tool in mathematics and mathematical physics. K-M Lie algebras were introduced in the mid-sixties independently by V. Kac and R. Moody, generalizing finite-dimensional semisimple Lie algebras. K-M theory has since undergone tremendous developments in various directions and has profound connections with a number of diverse areas, including number theory, combinatorics, topology, singularities, quantum groups, completely integrable systems, and mathematical physics. This comprehensive, well-written text moves from K-M Lie algebras to the broader K-M Lie group setting, and focuses on the study of K-M groups and their flag varieties. In developing K-M theory from scratch, the author systematically leads readers to the forefront of the subject, treating the algebro-geometric, topological, and representation-theoretic aspects of the theory. Most of the material presented here is not available anywhere in the book literature. {\it Kac--Moody Groups, their Flag Varieties and Representation Theory} is suitable for an advanced graduate course in representation theory, and contains a number of examples, exercises, challenging open problems, comprehensive bibliography, and index. Research mathematicians at the crossroads of representation theory, geometry, and topology will learn a great deal from this text; although the book is devoted to the general K-M case, those primarily interested in the finite-dimensional case will also benefit. No prior knowledge of K-M Lie algebras or of (finite-dimensional) algebraic groups is required, but some basic knowledge would certainly be helpful. For the reader's convenience some of the basic results needed from other areas, including ind-varieties, pro-algebraic groups and pro-Lie algebras, Tits systems, local cohomology, equivariant cohomology, and homological algebra are included.
Review
"Most of these topics appear here for the first time in book form. Many of them are interesting even in the classical case of semi-simple algebraic groups. Some appendices recall useful results from other areas, so the work may be considered self-contained, although some familiarity with semi-simple Lie algebras or algebraic groups is helpful. It is clear that this book is a valuable reference for all those interested in flag varieties and representation theory in the semi-simple or Kac-Moody case." --MATHEMATICAL REVIEWS "A lot of different topics are treated in this monumental work. . . . many of the topics of the book will be useful for those only interested in the finite-dimensional case. The book is self contained, but is on the level of advanced graduate students. . . . For the motivated reader who is willing to spend considerable time on the material, the book can be a gold mine. " --ZENTRALBLATT MATH
Table of Contents
Introduction * Kac--Moody Algebras -- Basic Theory * Representation Theory of Kac--Moody Algebras * Lie Algebra Homology and Cohomology * An Introduction to ind-Varieties and pro-Groups * Tits Systems -- Basic Theory * Kac--Moody Groups -- Basic Theory * Generalized Flag Varieties of Kac--Moody Groups * Demazure and Weyl--Kac Character Formulas * BGG and Kempf Resolutions * Defining Equations of G/P and Conjugacy Theorems * Topology of Kac-Moody Groups and Their Flag Varieties * Smoothness and Rational Smoothness of Schubert Varieties * An Introduction to Affine Kac-Moody Lie Algebras and Groups * Appendix A. Results from Algebraic Geometry * Appendix B. Local Cohomology * Appendix C. Results from Topology * Appendix D. Relative Homological Algebra * Appendix E. An Introduction to Spectral Sequences * Bibliography * Index of Notation * Index