Israel Journal of Mathematics

, Volume 219, Issue 1, pp 287–330 | Cite as

Multiple commutator formulas for unitary groups

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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.

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Copyright information

© Hebrew University of Jerusalem 2017

Authors and Affiliations

  1. 1.Centre for Research in MathematicsWestern Sydney UniversityWestern SydneyAustralia
  2. 2.Department of Mathematics and MechanicsSt. Petersburg State UniversitySt. PetersburgRussia
  3. 3.Department of MathematicsBeijing Institute of TechnologyBeijingChina

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