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: Let $\Phi$ be a reduced irreducible root system of rank $\ge 2$, let $R$ be a commutative ring and let $I,J$ be two ideals of $R$. In the present paper we describe generators of the commutator groups of relative elementary subgroups $\big[E(\Phi,R,I),E(\Phi,R,J)\big]$ both as normal subgroups of the elementary Chevalley group $E(\Phi,R)$, and as groups. Namely, let $x_{\a}(\xi)$, $\a\in\Phi$, $\xi\in R$, be an elementary generator of $E(\Phi,R)$. As a normal subgroup of the absolute elementary group $E(\Phi,R)$, the relative elementary subgroup is generated by $x_{\a}(\xi)$, $\a\in\Phi$, $\xi\in I$. Classical results due to Michael Stein, Jacques Tits and Leonid Vaserstein assert that as a group $E(\Phi,R,I)$ is generated by $z_{\a}(\xi,\eta)$, where $\a\in\Phi$, $\xi\in I$, $\eta\in R$. In the present paper, we prove the following birelative analogues of these results. As a normal subgroup of $E(\Phi,R)$ the relative commutator subgroup $\big[E(\Phi,R,I),E(\Phi,R,J)\big]$ is generated by the following three types of generators: i) $\big[x_{\alpha}(\xi),z_{\alpha}(\zeta,\eta)\big]$, ii) $\big[x_{\alpha}(\xi),x_{-\alpha}(\zeta)\big]$, and iii) $x_{\alpha}(\xi\zeta)$, where $\alpha\in\Phi$, $\xi\in I$, $\zeta\in J$, $\eta\in R$. As a group, the generators are essentially the same, only that type iii) should be enlarged to iv) $z_{\alpha}(\xi\zeta,\eta)$. For classical groups, these results, with much more computational proofs, were established in previous papers by the authors. There is already an amazing application of these results, namely in the recent work of Alexei Stepanov on relative commutator width.
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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: This paper is the first part of a systematic survey on the structure of classical groups over general rings. We intend to cover various proofs of the main structure theorems, commutator formulas, finiteness and stability conditions, stability and prestability theorems, the nilpotency of K 1, the centrality of K 2, automorphisms and homomorphisms, etc. This first part covers background material such as one-sided inverse, elementary transformations, definitions of obvious subgroups, Bruhat and Gauß decompositions, relative subgroups, finitary phenomena, and transvections. Bibliography: 674 titles.
    Journal of Mathematical Sciences 188(5).


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