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An Analogue of a Reductive Algebraic Monoid Whose Unit Group Is a Kac-Moody Group
Claus Mokler, University of Wuppertal, Germany

Memoirs of the American Mathematical Society
2005; 90 pp; softcover
Volume: 174
ISBN-10: 0-8218-3648-X
ISBN-13: 978-0-8218-3648-4
List Price: US$66
Individual Members: US$39.60
Institutional Members: US$52.80
Order Code: MEMO/174/823
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By an easy generalization of the Tannaka-Krein reconstruction we associate to the category of admissible representations of the category \({\mathcal O}\) of a Kac-Moody algebra, and its category of admissible duals, a monoid with a coordinate ring.

The Kac-Moody group is the Zariski open dense unit group of this monoid. The restriction of the coordinate ring to the Kac-Moody 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 Kac-Moody 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 Tannaka-Krein reconstruction
  • The proof of \(\overline{G}=\widehat{G}\) and some other theorems
  • The proof of Lie\((\overline{G})\cong \mathbf g\)
  • Bibliography
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