Presentation and uniqueness of Kac-Moody groups over local rings

2025/10/13 by Marquis, Timothée, Mühlherr, Bernhard
#19C20 #20E42 #20F05 #20G44 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2510.11272

Abstract

To any generalised Cartan matrix (GCM) A and any ring R, Tits associated a Kac-Moody group \mathfrakG_A(R) defined by a presentation à la Steinberg. For a domain R with field of fractions \mathbbK, we explore the question of whether the canonical map φ_R\colon\thinspace \mathfrakG_A(R)→ \mathfrakG_A(\mathbbK) 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 φ_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.

Citations

• Twin masures associated with Kac-Moody groups over Laurent polynomials
• Spherical Hecke algebras for Kac-Moody groups over local fields

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