Presentation of affine Kac–Moody groups over rings

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.