# An Analogue of a Reductive Algebraic Monoid Whose Unit Group Is a Kac-Moody Group

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*Claus Mokler*

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

# Table of Contents

## An Analogue of a Reductive Algebraic Monoid Whose Unit Group Is a Kac-Moody Group

- Contents v6 free
- Introduction 18
- Contents v6 free
- Chapter 1. Preliminaries 714 free
- Chapter 2. The monoid G and its structure 1926
- 2.1. The face lattice of the Tits cone 1926
- 2.2. The definition of the monoid G 2229
- 2.3. Formulas for computations in G 2330
- 2.4. The unit regularity of G 3138
- 2.5. The Weyl monoid W and the monoids T, N 3239
- 2.6. Some double coset partitions of G 3441
- 2.7. Constructing G from the twin root datum 3643
- 2.8. The action of G on the admissible modules of ο 3845
- 2.9. The submonoids G[sub(J)] (J ⊆ I) 3946
- 2.10. The monoid G for a decomposable matrix A 4249

- Chapter 3. An algebraic geometric setting 4552
- Chapter 4. A generalized Tannaka-Krein reconstruction 5360
- Chapter 5. The proof of G = G and some other theorems 6168
- 5.1. The coordinate rings and closures of T, T[sub(J)] (J ⊆ I), and T[sub(rest)] 6168
- 5.2. The orbits G[sub(J)](L(Λ)[sub(Λ)]) (J ⊆ I) and G (L(Λ)[sub(Λ)]) 6471
- 5.3. The coordinate rings and closures of U[sup(+)][sub(J)], (U[sup(J)])[sup(+)] (J ⊆ I) 6673
- 5.4. The closures of G[sub(J)] (J ⊆ I) and G 6976
- 5.5. The closures of N[sub(J)] (J ⊆ I) and N 7784
- 5.6. The openness of the unit group 7885
- 5.7. The Kac-Peterson-Slodowy part of SpecmF [G] 7885

- Chapter 6. The proof of Lie(G) ≅ g 8188
- Bibliography 8996