Commutators of relative and unrelative elementary unitary groups

TL;DR

This paper determines generators for mixed commutator subgroups of relative elementary unitary groups, simplifying their structure and unifying previous results in the context of Bak's hyperbolic unitary groups over form rings.

Contribution

It provides explicit generator descriptions for commutator subgroups in Bak's unitary groups, showing certain generators are redundant and establishing their centrality properties.

Findings

Generators for commutator subgroups are expressible as elementary conjugates and commutators.

The commutator of elementary subgroups equals the commutator of their relative elementary subgroups.

Elementary commutators behave as symbols and are central modulo certain subgroups.

Abstract

In the present paper we find generators of the mixed commutator subgroups of relative elementary groups and obtain unrelativised versions of commutator formulas in the setting of Bak's unitary groups. It is a direct sequel of our similar results were obtained for $GL(n,R)$ and for Chevalley groups over a commutative ring with 1, respectively. Namely, let $(A,\Lambda)$ be any form ring and $n\ge 3$. We consider Bak's hyperbolic unitary group $GU(2n,A,\Lambda)$. Further, let $(I,\Gamma)$ be a form ideal of $(A,\Lambda)$. One can associate with $(I,\Gamma)$ the corresponding elementary subgroup $FU(2n,I,\Gamma)$ and the relative elementary subgroup $EU(2n,I,\Gamma)$ of $GU(2n,A,\Lambda)$. Let $(J,\Delta)$ be another form ideal of $(A,\Lambda)$. In the present paper we prove an unexpected result that the non-obvious type of generators for $\big[EU(2n,I,\Gamma),EU(2n,J,\Delta)\big]$, as constructed in our previous papers with Hazrat, are redundant and can be expressed as products of the obvious generators, the elementary conjugates $Z_{ij}(ab,c)=T_{ji}(c)T_{ij}(ab)T_{ji}(-c)$ and $Z_{ij}(ba,c)$, and the elementary commutators $Y_{ij}(a,b)=[T_{ji}(a),T_{ij}(b)]$, where $a\in(I,\Gamma)$, $b\in(J,\Delta)$, $c\in(A,\Lambda)$. It follows that $\big[FU(2n,I,\Gamma),FU(2n,J,\Delta)\big]= \big[EU(2n,I,\Gamma),EU(2n,J,\Delta)\big]$. In fact, we establish much more precise generation results. In particular, even the elementary commutators $Y_{ij}(a,b)$ should be taken for one long root position and one short root position. Moreover, $Y_{ij}(a,b)$ are central modulo $EU(2n,(I,\Gamma)\circ(J,\Delta))$ and behave as symbols. This allows us to generalise and unify many previous results,including the multiple elementary commutator formula, and dramatically simplify their proofs.