Infinite-Dimensional Monoids Constructed from Kac-Moody Groups

We construct an infinite dimensional monoid from a Kac-Moody group and an irreducible highest weight representation of the group over the complex number field $\C$. The unit group $G$ of the monoid is the product of the multiplicative group $\C^\times$ and the homomorphic image of the Kac-Moody group under the representation. The idempotents of the monoid, which are intimately connected to the group structure, are determined by the face lattice of the convex hull $H$ of the weight set of the representation. We further establish the Bruhat decomposition for this monoid. The index set of the decomposition is a certain infinite inverse monoid whose unit group is the Weyl group of the Kac-Moody group. Furthermore, we obtain an infinite dimensional monoid version of the Tits System, and we find that the Borel subgroups of $G$ are closely related to the monoid structure in terms of a cross-section lattice defined from fundamental faces of $H$. This leads to the decomposition of the monoid into orbits of the two-sided action of $G$ on the monoid. Moreover, the monoid is unit regular. We describe the parabolic subgroups of $G$ using left and right centralizers of idempotents, and introduce the type map of the monoid. We also provide a method to construct the cross-section lattice using those subsets of the Dynkin diagram that are connected through the highest weight of the representation.