Steinberg groups for Jordan pairs
We announce results on classical groups and Steinberg groups over
arbitrary rings from the point of view of Jordan theory. This provides a
unifying framework, avoiding case-by-case arguments, for the linear elementary
groups, the unitary elementary groups. Starting from a Jordan pair V graded by
a 3-graded root system we show that the projective elementary group PE(V) has
commutator relations in the sense of J. R. Faulkner. Since our root systems
are allowed to be infinite but locally finite, we are able to deal with the
infinite elementary groups and Steinberg groups directly, that is, without
having to pass to the limit. To PE(V) we associate a Steinberg group St(V),
following the method of J. Tits for Kac-Moody groups. Our main result concerns
the case where the root system is irreducible of infinite rank and asserts
that St(V) is the universal central extension of PE(V).
O. Loos < ottmar.loos@fernuni-hagen.de >
E. Neher < neher@uottawa.ca >