Multiple Commutator Formulas for Unitary Groups

Abstract

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 $(I,\Gamma)$ of the form ring $(\FormR)$ we denote by $\EU(2n,I,\Gamma)$ and $\GU(2n,I,\Gamma)$ the relative elementary group and the principal congruence subgroup of level $(I,\Gamma)$, respectively. Now, let $(I_i,\Gamma_i) $, $i=0,...,m$, be form ideals of the form ring $(A,\Lambda)$. The main result of the present paper is the following multiple commutator formula [\big[\EU(2n,I_0,\Gamma_0),&\GU(2n,I_1,\Gamma_1),\GU(2n, I_2,\Gamma_2),..., \GU(2n,I_m,\Gamma_m)\big]= &\big[\EU(2n,I_0,\Gamma_0),\EU(2n,I_1,\Gamma_1),\EU(2n,I_2,\Gamma_2),..., \EU(2n, I_m, \Gamma_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.