For any root system and any commutative ring, we give a relatively
simple presentation of a group related to its Steinberg group
. 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 ,
giving a presentation with many advantages over the usual presentation
of . This
equality holds for all spherical root systems, all irreducible affine root systems of rank
, and all
-spherical root
systems. When the coefficient ring satisfies a minor condition, the last condition can be relaxed
to
-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
and ,
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
- and
-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.