Steinberg groups as amalgams

Abstract

For any root system and any commutative ring, we give a relatively simple presentation of a group related to its Steinberg group $\mathfrak{S}\mathfrak{t}$. This includes the case of infinite root systems used in Kac–Moody theory, for which the Steinberg group was defined by Tits and Morita–Rehmann. In most cases, our group equals $\mathfrak{S}\mathfrak{t}$, giving a presentation with many advantages over the usual presentation of $\mathfrak{S}\mathfrak{t}$. This equality holds for all spherical root systems, all irreducible affine root systems of rank $>2$, and all $3$-spherical root systems. When the coefficient ring satisfies a minor condition, the last condition can be relaxed to $2$-sphericity.

Our presentation is defined in terms of the Dynkin diagram rather than the full root system. It is concrete, with no implicit coefficients or signs. It makes manifest the exceptional diagram automorphisms in characteristics $2$ and $3$, and their generalizations to Kac–Moody groups. And it is a Curtis–Tits style presentation: it is the direct limit of the groups coming from $1$- and $2$-node subdiagrams of the Dynkin diagram. Over nonfields this description as a direct limit is new and surprising. Our main application is that many Steinberg and Kac–Moody groups over finitely generated rings are finitely presented.