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.
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