An analogue of a reductive algebraic monoid, whose unit group is a   Kac-Moody group
  By a generalized Tannaka-Krein reconstruction we associate to the admissible representations of the category 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 Kac and 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. 
Claus Mokler