Commutators of relative and unrelative elementary unitary groups

Vavilov, N.; Zhang, Z.

doi:10.1090/spmj/1745

Commutators of relative and unrelative elementary unitary groups
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by N. Vavilov and Z. Zhang

St. Petersburg Math. J. 34 (2023), 45-77

DOI: https://doi.org/10.1090/spmj/1745

Published electronically: December 16, 2022

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Abstract:

In the present paper, which is an outgrowth of the authors’ joint work with Anthony Bak and Roozbeh Hazrat on the unitary commutator calculus [9, 27, 30, 31], generators are found for the mixed commutator subgroups of relative elementary groups and unrelativized versions of commutator formulas are obtained in the setting of Bak’s unitary groups. It is a direct sequel of the papers [71, 76, 78, 79] and [77, 80], where 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 let $n\ge 3$. Bak’s hyperbolic unitary group $GU(2n,A,\Lambda )$ is considered. Further, let $(I,\Gamma )$ be a form ideal of $(A,\Lambda )$. One can associate with the ideal $(I,\Gamma )$ the corresponding true 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 an unexpected result is proved that the nonobvious type of generators for $\big [EU(2n,I,\Gamma ),EU(2n,J,\Delta )\big ]$, as constructed in the authors’ 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 $a\in (I,\Gamma )$, $b\in (J,\Delta )$, $c\in (A,\Lambda )$, and $\xi \in (I,\Gamma )\circ (J,\Delta )$. 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, much more precise generation results are established. 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 $EU(2n,(I,\Gamma )\circ (J,\Delta ))$ and behave as symbols. This makes it possible to generalize and unify many previous results, including the multiple elementary commutator formula, and dramatically simplify their proofs.

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