Groups with ${\mathsf{BC}}_{\ell }$-commutator relations

Abstract

Isotropic odd unitary groups generalize Chevalley groups of classical types over commutative rings and their twisted forms. Such groups have root subgroups parameterized by a root system ${\mathsf{BC}}_{\ell }$ and may be constructed by so-called odd form rings with Peirce decompositions. We show the converse: if a group $G$ has root subgroups indexed by roots of ${\mathsf{BC}}_{\ell }$ and satisfying natural conditions, then there is a homomorphism $\mathrm{StU}(R,\Delta )\to G$ inducing isomorphisms on the root subgroups, where $\mathrm{StU}(R,\Delta )$ is the odd unitary Steinberg group constructed by an odd form ring $(R,\Delta )$ with a Peirce decomposition. For groups with root subgroups indexed by ${\mathsf{A}}_{\ell }$ (the already known case) the resulting odd form ring is essentially a generalized matrix ring.

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• Groups with ${\mathsf{A}}_{\ell }$-commutator relations