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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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Authors and Affiliations

  1. Centre for Research in Mathematics, Western Sydney University, Western Sydney, Australia

    R. Hazrat

  2. Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia

    N. Vavilov

  3. Department of Mathematics, Beijing Institute of Technology, Beijing, China

    Z. Zhang

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  1. R. Hazrat
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  2. N. Vavilov
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  3. Z. Zhang
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Correspondence to R. Hazrat.

Additional information

The second author started this research within the framework of the RFFI/Indian Academy cooperation project 10-01-92651 “Higher composition laws, algebraic K-theory and algebraic groups” (SPbGU–Tata Institute) and the RFFI/BRFFI cooperation project 10-01-90016 “The structure of forms of reductive groups, and behaviour of small unipotent elements in representations of algebraic groups” (SPbGU–Mathematics Institute of the Belorussian Academy). Currently his work is supported by the RFFI research projects 11-01-00756 (RGPU) and 12-01-00947 (POMI), by the State Financed research task 6.38.74.2011 at the Saint Petersburg State University “Structure theory and geometry of algebraic groups and their applications in representation theory and algebraic K-theory” and by the Presidential Grant 6.10.61.2012 for the leading scientific schools.

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Hazrat, R., Vavilov, N. & Zhang, Z. Multiple commutator formulas for unitary groups. Isr. J. Math. 219, 287–330 (2017). https://doi.org/10.1007/s11856-017-1481-3

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