Memoirs of the American Mathematical Society 2005; 90 pp; softcover Volume: 174 ISBN10: 082183648X ISBN13: 9780821836484 List Price: US$66 Individual Members: US$39.60 Institutional Members: US$52.80 Order Code: MEMO/174/823
 By an easy generalization of the TannakaKrein reconstruction we associate to the category of admissible representations of the category \({\mathcal O}\) of a KacMoody algebra, and its category of admissible duals, a monoid with a coordinate ring. The KacMoody group is the Zariski open dense unit group of this monoid. The restriction of the coordinate ring to the KacMoody group is the algebra of strongly regular functions introduced by V. Kac and D. Peterson. This monoid has similar structural properties as a reductive algebraic monoid. In particular it is unit regular, its idempotents related to the faces of the Tits cone. It has Bruhat and Birkhoff decompositions. The KacMoody algebra is isomorphic to the Lie algebra of this monoid. Table of Contents  Introduction
 Preliminaries
 The monoid \(\widehat{G}\) and its structure
 An algebraic geometric setting
 A generalized TannakaKrein reconstruction
 The proof of \(\overline{G}=\widehat{G}\) and some other theorems
 The proof of Lie\((\overline{G})\cong \mathbf g\)
 Bibliography
