TY - JOUR

T1 - Commutators of relative and unrelative elementary unitary groups

AU - Vavilov, N.

AU - Zhang, Zuhong

N1 - N. Vavilov, Z. Zhang, “Commutators of relative and unrelative elementary unitary groups”, Алгебра и анализ, 34:1 (2022), 61–104

PY - 2022

Y1 - 2022

N2 - 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 unrelativised 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 GL(n, R) and for Chevalley groups over a commutative ring with 1, respectively. Namely, let (A,Λ) be any form ring and n ≥ 3. We consider Bak’s hyperbolic unitary group GU(2n, A,Λ). Further, let (I, Γ) be a form ideal of (A,Λ). One can associate with (I, Γ) the corresponding elementary subgroup FU(2n, I, Γ) and the relative elementary subgroup EU(2n, I, Γ) of GU(2n, A,Λ). Let (J, ∆) be another form ideal of (A,Λ). In the present paper we prove an unexpected result that the non-obvious type of generators for [ EU(2n, I, Γ),EU(2n, J, ∆)], as constructed in our previous papers with Hazrat, are redundant and can be expressed as products of the obvious generators, the elementary conjugates Zij (ab, c) = Tji(c)Tij (ab)Tji(−c) and Zij (ba, c), and the elementary commutators Yij (a, b) = [Tji(a), Tij (b)], where a ∈ (I, Γ), b ∈ (J, ∆), c ∈ (A,Λ). It follows that [ FU(2n, I, Γ), FU(2n, J, ∆)] = [ EU(2n, I, Γ),EU(2n, J, ∆)]. In fact, we establish much more precise generation results. In particular, even the elementary commutators Yij (a, b) should be taken for one long root position and one short root position. Moreover, Yij (a, b) are central modulo EU(2n,(I, Γ)◦(J, ∆)) 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.

AB - 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 unrelativised 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 GL(n, R) and for Chevalley groups over a commutative ring with 1, respectively. Namely, let (A,Λ) be any form ring and n ≥ 3. We consider Bak’s hyperbolic unitary group GU(2n, A,Λ). Further, let (I, Γ) be a form ideal of (A,Λ). One can associate with (I, Γ) the corresponding elementary subgroup FU(2n, I, Γ) and the relative elementary subgroup EU(2n, I, Γ) of GU(2n, A,Λ). Let (J, ∆) be another form ideal of (A,Λ). In the present paper we prove an unexpected result that the non-obvious type of generators for [ EU(2n, I, Γ),EU(2n, J, ∆)], as constructed in our previous papers with Hazrat, are redundant and can be expressed as products of the obvious generators, the elementary conjugates Zij (ab, c) = Tji(c)Tij (ab)Tji(−c) and Zij (ba, c), and the elementary commutators Yij (a, b) = [Tji(a), Tij (b)], where a ∈ (I, Γ), b ∈ (J, ∆), c ∈ (A,Λ). It follows that [ FU(2n, I, Γ), FU(2n, J, ∆)] = [ EU(2n, I, Γ),EU(2n, J, ∆)]. In fact, we establish much more precise generation results. In particular, even the elementary commutators Yij (a, b) should be taken for one long root position and one short root position. Moreover, Yij (a, b) are central modulo EU(2n,(I, Γ)◦(J, ∆)) 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.

KW - унитарные группы, элементарная подгруппа, коммутационные формулы, элементарные образующие, кратные коммутационные формулы, форменные кольца, форменные идеалы

KW - Bak's unitary groups

KW - elementary subgroups

KW - congruence subgroups

KW - standard commutator formula

KW - unrelativized commutator formula

KW - elementary generators

KW - multiple commutator formula

UR - http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=aa&paperid=1796&option_lang=rus

M3 - Article

VL - 34

SP - 61

EP - 104

JO - АЛГЕБРА И АНАЛИЗ

JF - АЛГЕБРА И АНАЛИЗ

SN - 0234-0852

IS - 1

ER -