Linear Algebraic Monoids

The theory of linear algebraic monoids culminates in a coherent blend of algebraic groups, convex geometry, and semigroup theory. The book discusses all the key topics in detail, including classification, orbit structure, representations, universal constructions, and abstract analogues. An explicit cell decomposition is constructed for the wonderful compactification, as is a universal deformation for any semisimple group. A final chapter summarizes important connections with other areas of algebra and geometry. The book will serve as a solid basis for further research. Open problems are discussed as they arise and many useful exercises are included.

“From the reviews:"This account of algebraic monoids … starts by recalling the basics of algebraic geometry, algebraic groups and semigroups … . The main question, how a monoid is composed of units and idempotents, is discussed at length … . A most useful account, giving current developments of the subject." (Mathematika, Vol. 52, 2005)”

From the reviews:

"This account of algebraic monoids … starts by recalling the basics of algebraic geometry, algebraic groups and semigroups … . The main question, how a monoid is composed of units and idempotents, is discussed at length … . A most useful account, giving current developments of the subject." (Mathematika, Vol. 52, 2005)

The object of this monograph is to document what is most interesting about linear monoids. We show how these results ?t together into a coherent blend of semigroup theory, groups with BN-pair, representation theory, convex - ometry and algebraicgrouptheory.The intended reader is one who is familiar with some of these topics, and is willing to learn about the others. The intention of the author is to convince the reader that reductive monoids are among the darlings of algebra. We do this by systematically assembling many of the major known results with many proofs,examples and explanations. To further entice the reader, we have included many exercises. The theory of linear algebraic monoids is quite recent, originating around 1980. Both Mohan Putcha and the author began the systematic study in- pendently. But this development would not have been possible without the pioneering work of Chevalley, Borel and Tits on algebraic groups. Also, there is the related, but more general theory of spherical embeddings, developed largely by Brion, Luna and Vust. These theories were developed somewhat independently, but it is always a good idea to interpret monoid results in the combinatorial apparatus of spherical embeddings. Each chapter of this monograph is focussed on one or more of the major themes of the subject. These are: classi'cation, orbits, geometry, represen- tions, universal constructions and combinatorics. There is an inherent div- sity and richness in the subject that usually rewards a stalwart investigation.

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