Presentation and uniqueness of Kac-Moody groups over local rings
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909841565417472 |
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| author | Marquis, Timothée Mühlherr, Bernhard |
| author_facet | Marquis, Timothée Mühlherr, Bernhard |
| contents | 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 $φ_R\colon\thinspace \mathfrak{G}_A(R)\to \mathfrak{G}_A(\mathbb{K})$ 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 $φ_R$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_11272 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Presentation and uniqueness of Kac-Moody groups over local rings Marquis, Timothée Mühlherr, Bernhard Group Theory 20G44, 20E42, 20F05, 19C20 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 $φ_R\colon\thinspace \mathfrak{G}_A(R)\to \mathfrak{G}_A(\mathbb{K})$ 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 $φ_R$ 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. |
| title | Presentation and uniqueness of Kac-Moody groups over local rings |
| topic | Group Theory 20G44, 20E42, 20F05, 19C20 |
| url | https://arxiv.org/abs/2510.11272 |