Commutators of relative and unrelative elementary unitary groups

N. Vavilov ; Z. Zhang

Algebra i analiz, Tome 34 (2022) no. 1, pp. 61-104

Résumé

In the present paper, which is an outgrowth of our joint work with Anthony Bak and Roozbeh Hazrat on unitary commutator calculus [9, 27, 30, 31], we find generators of the mixed commutator subgroups of relative elementary groups and obtain unrelativized versions of commutator formulas in the setting of Bak's unitary groups. It is a direct sequel of our papers [71, 76, 78, 79] and [77, 80], where similar results were obtained for $\mathrm{GL}(n,R)$ and for Chevalley groups over a commutative ring with 1, respectively. Namely, let $(A,\Lambda)$ be any form ring and let $n\ge 3$. We consider Bak's hyperbolic unitary group $\mathrm{GU}(2n,A,\Lambda)$. Further, let $(I,\Gamma)$ be a form ideal of $(A,\Lambda)$. One can associate with the ideal $(I,\Gamma)$ the corresponding true elementary subgroup $\mathrm{FU}(2n,I,\Gamma)$ and the relative elementary subgroup $\mathrm{EU}(2n,I,\Gamma)$ of $\mathrm{GU}(2n,A,\Lambda)$. Let $(J,\Delta)$ be another form ideal of $(A,\Lambda)$. In the present paper we prove an unexpected result that the nonobvious type of generators for $\big[\mathrm{EU}(2n,I,\Gamma),\mathrm{EU}(2n,J,\Delta)\big]$, as constructed in our previous papers with Hazrat, are redundant and can be expressed as products of the obvious generators, the elementary conjugates $Z_{ij}(\xi,c)=T_{ji}(c)T_{ij}(\xi)T_{ji}(-c)$, and the elementary commutators $Y_{ij}(a,b)=[T_{ij}(a),T_{ji}(b)]$, where $a\in(I,\Gamma)$, $b\in(J,\Gamma)$, $c\in(A,\Delta)$, and $\xi \in (I,\Gamma)\circ(J,\Delta )$. It follows that $\big[\mathrm{FU}(2n,I,\Gamma),\mathrm{FU}(2n,J,\Delta)\big]= \big[\mathrm{EU}(2n,I,\Gamma),\mathrm{EU}(2n,J,\Delta)\big]$. 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, the $Y_{ij}(a,b)$ are central modulo $\mathrm{EU}(2n,(I,\Gamma)\circ(J,\Gamma))$ and behave as symbols. This allows us to generalize and unify many previous results, including the multiple elementary commutator formula, and dramatically simplify their proofs.

Détail

Keywords: Bak's unitary groups, elementary subgroups, congruence subgroups, standard commutator formula, unrelativized commutator formula, elementary generators, multiple commutator formula.

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