COMMUTATORS OF RELATIVE AND UNRELATIVE ELEMENTARY UNITARY GROUPS

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, Λ) be any form ring and let n ≥ 3. Bak’s hyperbolic unitary group GU(2n, A, Λ) is considered. Further, let (I, Γ) be a form ideal of (A, Λ). One can associate with the ideal (I, Γ) the corresponding true elementary subgroup FU(2n, I, Γ) and the relative elementary subgroup EU(2n, I, Γ) of GU(2n, A, Λ). Let (J, Δ) be another form ideal of (A, Λ). In the present paper an unexpected result is proved that the nonobvious type of generators for (Formula Presented) EU(2n, I, Γ), EU(2n, J, Δ), as constructed in the authors’ previous papers with Hazrat, are redundant and can be expressed as products of the obvious generators, the elementary conjugates Zij (ξ, c) = Tji(c)Tij (ξ)Tji(−c), and the elementary commutators Yij (a, b) = [Tij (a), Tji(b)], where a ∈ (I, Γ), b ∈ (J, Δ), c ∈ (A, Λ), and ξ ∈ (I, Γ) ◦ (J, Δ). It follows that FU(2n, I, Γ), [FU(2n, J, Δ)] = [EU(2n, I, Γ), EU(2n, J, Δ)]. 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 Yij (a, b) are central modulo EU(2n, (I, Γ) ◦ (J, Δ)) and behave as symbols.