Multiple commutator formulas for unitary groups

Abstract

Let (A,Λ) be a formring such that A is quasi-finite R-algebra (i.e., a direct limit of module finite algebras) with identity. We consider the hyperbolic Bak’s unitary groups GU(2n, A, Λ), n ≥ 3. For a form ideal (I, Γ) of the form ring (A, Λ) we denote by EU(2n, I, Γ) and GU(2n, I, Γ) the relative elementary group and the principal congruence subgroup of level (I, Γ), respectively. Now, let (I i , Γ i ), i = 0,...,m, be form ideals of the form ring (A, Λ). The main result of the present paper is the following multiple commutator formula: [EU(2n, I 0, Γ 0),GU(2n, I 1, Γ 1), GU(2n, I 2, Γ 2),..., GU(2n, I m , Γ m )] =[EU(2n, I 0, Γ 0), EU(2n, I 1, Γ 1), EU(2n, I 2, Γ 2),..., EU(2n, I m , Γ m )], which is a broad generalization of the standard commutator formulas. This result contains all previous results on commutator formulas for classicallike groups over commutative and finite-dimensional rings.