Steinberg groups for Jordan pairs

Abstract

We introduce categories of groups with commutator relations with respect to root groups and Weyl elements, permuting the root groups. This allows us to view the classical Steinberg groups, for example the Steinberg group of a ring, as an initial object in an appropriate category. The general framework is then specialized to groups associated to Jordan pairs, first for arbitrary Jordan pairs and then later for Jordan pairs with Peirce gradings or more general gradings by root systems, for example Jordan pairs covered by grids.