Generalized Commutator Formulas

Communications in Algebra (Impact Factor: 0.39). 04/2011; 39:1441-1454. DOI: 10.1080/00927871003738964

ABSTRACT Let A be an algebra which is a direct limit of module finite algebras over a commutative ring R with 1. Let I, J be two-sided ideals of A, GL n (A, I) the principal congruence subgroup of level I in GL n (A), and E n (A, I) the relative elementary subgroup of level I. Using Bak's localization-patching method, we prove the commutator formula which is a generalization of the standard commutator formular. This answers a problem posed by Stepanov and Vavilov.

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    ABSTRACT: In the current article we study structure of a Chevalley group $G(R)$ over a commutative ring $R$. We generalize and improve the following results: (1) standard, relative, and multi-relative commutator formulas; (2) nilpotent structure of [relative] $\K_1$; (3) bounded word length of commutators. To this end we enlarge the elementary group, construct a generic element for the extended elementary group, and use localization in the universal ring. The key step is a construction of a generic element for a principle congruence subgroup, corresponding to a principle ideal.
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    ABSTRACT: We revisit localisation and patching method in the setting of Chevalley groups. Introducing certain subgroups of relative elementary Chevalley groups, we develop relative versions of the conjugation calculus and the commutator calculus in Chevalley groups $G(\Phi,R)$, $\rk(\Phi)\geq 2$, which are both more general, and substantially easier than the ones available in the literature. For classical groups such relative commutator calculus has been recently developed by the authors in \cite{RZ,RNZ}. As an application we prove the mixed commutator formula, \[ \big [E(\Phi,R,\ma),C(\Phi,R,\mb)\big ]=\big [E(\Phi,R,\ma),E(\Phi,R,\mb)\big], \] for two ideals $\ma,\mb\unlhd R$. This answers a problem posed in a paper by Alexei Stepanov and the second author.
    Journal of Algebra 07/2013; 385:262-293. DOI:10.1016/j.jalgebra.2013.03.011 · 0.60 Impact Factor
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    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.


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