Multiple commutator formulas

Hazrat, Roozbeh; Zhang, Zuhong

doi:10.1007/s11856-012-0135-8

Cited by 19 publications

(47 citation statements)

References 16 publications

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“…The following theorem is the main result of the paper [51] by the first and the fourth authors. Initially, it was conceived as part of the answer to a problem proposed by the second and the third authors [118,120].…”

Section: Multiple Commutator Formulamentioning

confidence: 91%

“…Actually, the proof of this result in [48] replaces most of explicit fiddling with the Chevalley commutator formula and commutator identites by a reference to some obvious properties of parabolic subgroups, which makes it considerably less computational than the proofs of Theorem 2A and Theorem 2B in [47,51].…”

Section: Generators Of Relative Commutator Subgroups As Normal Subgroupsmentioning

confidence: 99%

“…As a matter of fact, it turned out to be of significant independent interest. The proof of the following result in [51] is based on a further enhancement of relative localization which we outline in the next section. In this theorem, the arrangement of brackets on the left-hand side may be arbitrary.…”

Section: Multiple Commutator Formulamentioning

confidence: 99%

“…For GL n and unitary groups, the details of these calculations can be found in [51] and [47], for Chevalley groups they are still unpublished.…”

Section: A Further Piece Of Commutator Calculusmentioning

confidence: 99%

“…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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confidence: 99%

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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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Section: Multiple Commutator Formulamentioning

confidence: 91%