$K_2$ of Kac-Moody Groups

TL;DR

This paper explicitly computes the K_2 groups of hyperbolic Kac-Moody groups, showing they can be expressed as products of quotients of K_2 of the base field and K_2(2) of the field.

Contribution

It extends previous presentations of K_2 for Kac-Moody groups to explicitly compute these groups for hyperbolic cases, revealing their structure as products of known K_2 groups.

Findings

K_2(A,F) expressed as products of quotients of K_2(F) and K_2(2,F)

Explicit computation for hyperbolic Cartan matrices

Similar results for matrices with odd entries in each column

Abstract

Ulf Rehmann and Jun Morita, in their 1989 paper "A Matsumoto-type theorem for Kac-Moody groups", gave a presentation of $K_2(A,F)$ for any generalised Cartan matrix $A$ and field $F$. The purpose of this paper is to use this presentation to compute $K_2(A,F)$ more explicitly in the case when $A$ is hyperbolic. In particular, we shall show that these $K_2(A,F)$ can always be expressed as a product of quotients of $K_2(F)$ and $K_2(2,F)$. Along the way, we shall also prove a similar result in the case when $A$ has an odd entry in each column.