Presentation and uniqueness of Kac-Moody groups over local rings

Abstract

To any generalised Cartan matrix (GCM) $A$ and any ring $R$, Tits associated a Kac-Moody group ${\mathfrak{G}}_A(R)$ defined by a presentation à la Steinberg. For a domain $R$ with field of fractions $\mathbb{K}$, we explore the question of whether the canonical map ${\phi }_R:{\mathfrak{G}}_A(R)\to {\mathfrak{G}}_A(\mathbb{K})$ is injective. This question for Cartan matrices has a long history, and for GCMs was already present in Tits' foundational papers on Kac-Moody groups. We prove that for any $2$-spherical GCM $A$, the map ${\phi }_R$ is injective for all valuation rings $R$ (under an additional minor condition (co)). To the best of our knowledge, this is the first such injectivity result beyond the classical setting.

26 pages