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

TL;DR

The paper establishes a correspondence between groups with root subgroups indexed by the root system __ and odd unitary Steinberg groups derived from odd form rings, extending known cases.

Contribution

It proves that groups with _-root subgroups satisfying natural conditions are isomorphic to odd unitary Steinberg groups from odd form rings, generalizing previous results.

Findings

Groups with _ root subgroups correspond to odd unitary Steinberg groups.

The construction applies to groups with _ root systems and satisfies natural conditions.

Extends known results from _ root systems to _ cases.

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.