#### Vol. 10, No. 3, 2016

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Presentation of affine Kac–Moody groups over rings

### Daniel Allcock

Vol. 10 (2016), No. 3, 533–556
##### Abstract

Tits has defined Steinberg groups and Kac–Moody groups for any root system and any commutative ring $R$. We establish a Curtis–Tits-style presentation for the Steinberg group $\mathfrak{S}\mathfrak{t}$ of any irreducible affine root system with rank $\ge 3$, for any $R$. Namely, $\mathfrak{S}\mathfrak{t}$ is the direct limit of the Steinberg groups coming from the $1$- and $2$-node subdiagrams of the Dynkin diagram. In fact, we give a completely explicit presentation. Using this we show that $\mathfrak{S}\mathfrak{t}$ is finitely presented if the rank is $\ge 4$ and $R$ is finitely generated as a ring, or if the rank is $3$ and $R$ is finitely generated as a module over a subring generated by finitely many units. Similar results hold for the corresponding Kac–Moody groups when $R$ is a Dedekind domain of arithmetic type.

##### Keywords
affine Kac–Moody group, Steinberg group, Curtis–Tits presentation
##### Mathematical Subject Classification 2010
Primary: 20G44
Secondary: 14L15, 22E67, 19C99