2012/05/31 by Roozbeh Hazrat, Hazrat, Roozbeh, N. A. Vavilov +4
Mathematics · #Algebraic structures and combinatorial models #Finite Group Theory Research #Rings, Modules, and Algebras #math.RA #msc:20G35
paper · pdf · doi:10.48550/arxiv.1205.6866
arXiv admin note: text overlap with arXiv:0911.5510
arxiv created 2012/05/31 · arxiv updated 2012/07/30
Let (\FormR) be a form ring 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,\FormR), n≥ 3. For a form ideal (I,Γ) of the form ring (\FormR) 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 (Ii,Γi) , 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,I0,Γ0),&\GU(2n,I1,Γ1),\GU(2n, I2,Γ2),..., \GU(2n,Im,Γm)]= &[\EU(2n,I0,Γ0),\EU(2n,I1,Γ1),\EU(2n,I2,Γ2),..., \EU(2n, Im, Γm)],] which is a broad generalization of the standard commutator formulas. This result contains all previous results on commutator formulas for classical like-groups over commutative and finite-dimensional rings.