Presentation and uniqueness of Kac-Moody groups over local rings
Timothée Marquis, Bernhard Mühlherr
TL;DR
This work proves that for a 2-spherical generalized Cartan matrix $A$ and a valuation ring $R$ satisfying condition $(co)$, the canonical map $\mathfrak{G}_A(R)\to\mathfrak{G}_A^{\min}(R)$ is injective, yielding $\mathfrak{G}_A(R)\cong \mathfrak{G}_A^{\min}(R)\subseteq \mathfrak{G}_A(\mathbb{K})$ where $\mathbb{K}$ is the field of fractions of $R$. The authors establish this by presenting $\mathfrak{G}_A^{\min}(R)$ as a Curtis–Tits amalgam $\mathrm{CT}_A(R)$ built from rank-1 and rank-2 Chevalley groups, and by proving that the natural map $\mathrm{CT}_A(R)\to \mathfrak{G}_A(R)\to \mathfrak{G}_A^{\min}(R)$ is an isomorphism. A core geometric input is the construction of simply connected twin chamber systems associated to $G^{\min}_R$, together with a general result: if a twin chamber system is simply connected, then its opposition system $\mathrm{Opp}(\mathcal{C})$ is simply connected; this underpins the Curtis–Tits presentation in the ring setting. As a byproduct, Laurent polynomial rings $R[t,t^{-1}]$ are shown to be universal for root systems not of type $A_1$ in this context, generalizing Morita-type universality results to the Kac–Moody setting over valuation 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 $\varphi_R\colon\thinspace \mathfrak{G}_A(R)\to \mathfrak{G}_A(\mathbb{K})$ 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 $\varphi_R$ 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.