Linear groups over general rings. I. Generalities

Vavilov, N. A.; Stepanov, A. P.

doi:10.1007/s10958-013-1146-7

Cited by 9 publications

(4 citation statements)

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“…Nevertheless, the earliest reference, where we could trace this result, was the paper by Leonid Vaserstein and Andrei Suslin [121]. We follow the proof given in [10,Lemma 4.8] (see also in [132,Theorem 11]). Lemma 3.…”

Section: General Linear Groupsmentioning

confidence: 93%

The Commutators of Classical Groups

2017

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In his seminal paper, half a century ago, Hyman Bass established a commutator formula in the setting of (stable) general linear group which was the key step in defining the K 1 group. Namely, he proved that for an associative ring A with identity,, where GL(A) is the stable general linear group and E(A) is its elementary subgroup. Since then, various commutator formulas have been studied in stable and non-stable settings, and for a range of classical and algebraic like-groups, mostly in relation to subnormal subgroups of these groups. The major classical theorems and methods developed include some of the splendid results of the heroes of classical algebraic K-theory; Bak, Quillen, Milnor, Suslin, Swan and Vaserstein, among others.One of the dominant techniques in establishing commutator type results is localisation. In this note we describe some recent applications of localisation methods to the study (higher/relative) commutators in the groups of points of algebraic and algebraic-like groups, such as general linear groups, GL(n, A), unitary groups GU(2n, A, Λ) and Chevalley groups G(Φ, A). We also state some of the intermediate results as well as some corollaries of these results.This note provides a general overview of the subject and covers the current activities. It contains complete proofs of several main results to give the reader a self-contained source. We have borrowed the proofs from our previous papers and expositions [39]-[51],[100, 101],[129]-[132].

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Section: General Linear Groupsmentioning

confidence: 93%

The Commutators of Classical Groups

2017

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“…For GL n , we use the notation from [35,93,121]. In this context, the ring R does not have to be commutative.…”

Section: The Groupsmentioning

confidence: 99%

The Yoga of Commutators: Further Applications

et al. 2014

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In the present paper, we describe some recent applications of localization methods to the study of commutators in the groups of points of algebraic and algebraic-like groups, such as GL(n, R), Bak's unitary groups GU(2n, R, Λ), and Chevalley groups G(Φ, R). In particular, we announce the multiple relative commutator formula and the general multiple relative commutator formula, as well as results on the bounded width of relative commutators in the elementary generators. We also state some of the intermediate results, as well as some corollaries of these results. At the end of the paper we attach an updated list of unsolved problems in the field. Bibliography: 132 titles.The present paper is a direct continuation of [39]. Its goal is to announce some major advances we achieved in 2010-2013, after the publication of [39], with the use of the new localization methods described therein. Namely, in [39] we described the methods themselves, and stated three typical applications to the study of commutators in algebraic-like groups:• relative standard commutator formulas [45,46,50,118,120];• universal length bound for commutators [42,86,92,94];• nilpotent structure of relative K 1 [9, 11, 36, 43].In the present paper, we briefly outline some further fragments of commutator calculus and state three fresh applications of these methods to the study of commutators in algebraic-like groups:• multiple commutator formula [47,49,50];• general multiple commutator formula [41,49];• relative commutator length [40,42,90].In particular, this solves several of the problems stated in [39]. Moreover, we describe two further recent relative versions of localization methods themselves:• relative Quillen-Suslin principle [7];• relative localization-completion [41].Also, we state some subsidiary results, which are not directly based on localization methods, but rather are pure group theory. Nevertheless, we believe they are fun in themselves and could be useful in further studies of commutators in algebraic-like groups. Here are three typical results of this kind for the relative elementary subgroups, which we discuss in the present paper:• generation of relative commutator subgroups [47][48][49]51].

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“…Vyacheslav Kopeiko and Andrei Suslin [29], [52], Giovanni Taddei, Leonid Vaserstein, You Hong, Anthony Bak, and the present authors generalised these results to other groups. In [6], [8] - [10], [16], [19], [22], [23], [30], [48], [64], [70] one can find different proofs of these results, many further references, and a detailed discussion of their role in the structure theory.…”

Section: Commutator Calculusmentioning

confidence: 99%

The yoga of commutators

Hazrat,

Stepanov,

Vavilov

et al. 2011

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Linear groups over general rings. I. Generalities

Cited by 9 publications

References 412 publications

The Commutators of Classical Groups

The Commutators of Classical Groups

The Yoga of Commutators: Further Applications

The yoga of commutators

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