Multiple Commutator Formulas for Unitary Groups

Let $(\text{FormR})$ 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 $\text{GU}(2n,\text{FormR})$, $n\ge 3$. For a form ideal $(I,\mathrm{\Gamma })$ of the form ring $(\text{FormR})$ we denote by $\text{EU}(2n,I,\mathrm{\Gamma })$ and $\text{GU}(2n,I,\mathrm{\Gamma })$ the relative elementary group and the principal congruence subgroup of level $(I,\mathrm{\Gamma })$, respectively. Now, let $(I_i,{\mathrm{\Gamma }}_i)$, $i=0,\mathrm{.}\mathrm{.}\mathrm{.},m$, be form ideals of the form ring $(A,\mathrm{\Lambda })$. 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.