Commutators of relative and unrelative elementary unitary groups

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 forGL(n,R)GL(n,R)and for Chevalley groups over a commutative ring with 1, respectively. Namely, let(A,Λ)(A,\Lambda )be any form ring and letn≥3n\ge 3. Bak’s hyperbolic unitary groupGU(2n,A,Λ)GU(2n,A,\Lambda )is considered. Further, let(I,Γ)(I,\Gamma )be a form ideal of(A,Λ)(A,\Lambda ). One can associate with the ideal(I,Γ)(I,\Gamma )the corresponding true elementary subgroupFU(2n,I,Γ)FU(2n,I,\Gamma )and the relative elementary subgroupEU(2n,I,Γ)EU(2n,I,\Gamma )ofGU(2n,A,Λ)GU(2n,A,\Lambda ). Let(J,Δ)(J,\Delta )be another form ideal of(A,Λ)(A,\Lambda ). In the present paper an unexpected result is proved that the nonobvious type of generators for[EU(2n,I,Γ),EU(2n,J,Δ)]\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 conjugatesZij(ξ,c)=Tji(c)Tij(ξ)Tji(−c)Z_{ij}(\xi ,c)=T_{ji}(c)T_{ij}(\xi )T_{ji}(-c), and the elementary commutatorsYij(a,b)=[Tij(a),Tji(b)]Y_{ij}(a,b)=[T_{ij}(a),T_{ji}(b)], wherea∈(I,Γ)a\in (I,\Gamma ),b∈(J,Δ)b\in (J,\Delta ),c∈(A,Λ)c\in (A,\Lambda ), andξ∈(I,Γ)∘(J,Δ)\xi \in (I,\Gamma )\circ (J,\Delta ). It follows that[FU(2n,I,Γ),FU(2n,J,Δ)]=[EU(2n,I,Γ),EU(2n,J,Δ)]\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 commutatorsYij(a,b)Y_{ij}(a,b)should be taken for one long root position and one short root position. Moreover, theYij(a,b)Y_{ij}(a,b)are central moduloEU(2n,(I,Γ)∘(J,Δ))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.