Infinite dimensional Chevalley groups and Kac-Moody groups over ℤ

2018/03/29 by Carbone, Lisa, Liu, Dongwen, Murray, Scott H.
#20G44 #81R10 #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1803.11204

Abstract

Let A be a symmetrizable generalized Cartan matrix, which is not of finite or affine type. Let \mathfrakg be the corresponding Kac-Moody algebra over a commutative ring R with 1. We construct an infinite-dimensional group G_V(R) analogous to a finite-dimensional Chevalley group over R. We use a ℤ-form of the universal enveloping algebra of \mathfrakg and a ℤ-form of an integrable highest-weight module V. We construct groups G_V(ℤ) analogous to arithmetic subgroups in the finite-dimensional case. We also consider a universal representation-theoretic Kac-Moody group G and its completion \widetildeG. For the completion we prove a Bruhat decomposition \widetildeG(ℚ)=\widetildeG(ℤ)\widetildeB(ℚ) over ℚ, and that the arithmetic subgroup \widetildeΓ(ℤ) coincides with the subgroup of integral points \widetildeG(ℤ)

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