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Zap. Nauchn. Sem. POMI, 2011, Volume 387, Pages 53–82 (Mi znsl4096)  

This article is cited in 5 scientific papers (total in 5 papers)

The yoga of commutators

R. Hazrata, A. Stepanovb, N. Vavilovc, Z. Zhangd

a Queen's University Belfast, U.K.
b С.-Петербургский государственный электротехнический университет, Санкт-Петербург, Россия
c С.-Петербургский государственный университет, Санкт-Петербург, Россия
d Beijing Institute of Technology, China

Abstract: In the present paper we discuss some recent versions of localization methods for calculations in the groups of points of algebraic-like and classical-like groups. Namely, we describe relative localization, universal localization, and enhanced versions of localization-completion. Apart from the general strategic description of these methods, we state some typical technical results of the conjugation calculus and the commutator calculus. Also, we state several recent results obtained therewith, such as relative standard commutator formulae, bounded width of commutators, with respect to the elementary generators, and nilpotent filtrations of congruence subgroups. Overall, this shows that localization methods can be much more efficient, than expected.

Key words and phrases: unitary groups, Chevalley groups, elementary subgroups, elementary generators, localization, relative subgroups, conjugation calculus, commutator calculus, Noetherian reduction, Quillen–Suslin lemma, localization-completion, commutator formulae, commutator width, nilpotency of $\mathrm K_1$, nilpotent filtration.

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English version:
Journal of Mathematical Sciences (New York), 2011, 179:6, 662–678

Document Type: Article
UDC: 512.54
Received: 27.11.2010
Language: English

Citation: R. Hazrat, A. Stepanov, N. Vavilov, Z. Zhang, “The yoga of commutators”, Representation theory, dynamical systems, combinatorial methods. Part XIX, Zap. Nauchn. Sem. POMI, 387, POMI, St. Petersburg, 2011, 53–82; J. Math. Sci. (N. Y.), 179:6 (2011), 662–678

Citation in format AMSBIB
\by R.~Hazrat, A.~Stepanov, N.~Vavilov, Z.~Zhang
\paper The yoga of commutators
\inbook Representation theory, dynamical systems, combinatorial methods. Part~XIX
\serial Zap. Nauchn. Sem. POMI
\yr 2011
\vol 387
\pages 53--82
\publ POMI
\publaddr St.~Petersburg
\jour J. Math. Sci. (N. Y.)
\yr 2011
\vol 179
\issue 6
\pages 662--678

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    This publication is cited in the following articles:
    1. N. A. Vavilov, A. V. Stepanov, “Lineinye gruppy nad obschimi koltsami I. Obschie mesta”, Voprosy teorii predstavlenii algebr i grupp. 22, Zap. nauchn. sem. POMI, 394, POMI, SPb., 2011, 33–139  mathnet  mathscinet; N. A. Vavilov, A. V. Stepanov, “Linear groups over general rings. I. Generalities”, J. Math. Sci. (N. Y.), 188:5 (2013), 490–550  crossref
    2. J. Math. Sci. (N. Y.), 200:6 (2014), 742–768  mathnet  crossref
    3. J. Math. Sci. (N. Y.), 209:6 (2015), 922–934  mathnet  crossref
    4. R. Hazrat, N. Vavilov, Z. Zhang, “The commutators of classical groups”, Voprosy teorii predstavlenii algebr i grupp. 29, Zap. nauchn. sem. POMI, 443, POMI, SPb., 2016, 151–221  mathnet  mathscinet; J. Math. Sci. (N. Y.), 222:4 (2017), 466–515  crossref
    5. R. Basu, “Local-global principle for general quadratic and general Hermitian groups and the nilpotence of $\mathrm{KH}_1$”, Voprosy teorii predstavlenii algebr i grupp. 30, Zap. nauchn. sem. POMI, 452, POMI, SPb., 2016, 5–31  mathnet  mathscinet
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