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Multiple Commutator Formulas for Unitary Groups

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 $\GU(2n,\FormR)$, $n\ge 3$. 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 $(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 [\big[\EU(2n,I_0,Γ_0),&\GU(2n,I_1,Γ_1),\GU(2n, I_2,Γ_2),..., \GU(2n,I_m,Γ_m)\big]= &\big[\EU(2n,I_0,Γ_0),\EU(2n,I_1,Γ_1),\EU(2n,I_2,Γ_2),..., \EU(2n, I_m, Γ_m)\big],] 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.