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 Zap. Nauchn. Sem. POMI, 2014, Volume 421, Pages 166–213 (Mi znsl5758)

The yoga of commutators: further applications

R. Hazrata, A. V. Stepanovbc, N. A. Vavilovb, Z. Zhangd

a University of Western Sydney, Australia
b St. Petersburg State University, Universitetsky pr. 28, Peterhof, 198504 St. Petersburg, Russia
c St. Petersburg Electrotechnical University
d Beijing Institute of Technology, China

Abstract: In the present paper we describe some recent applications of localisation methods to the study of commutators in the groups of points of algebraic and algebraic-like groups, such as $\mathrm{GL}(n,R)$, Bak's unitary groups $\mathrm{GU}(2l,R,\Lambda)$ and Chevalley groups $G(\Phi,R)$. In particular, we announce multiple relative commutator formula and general multiple relative commutator formula, as well as results on the bounded width of relative commutators in 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.

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

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English version:
Journal of Mathematical Sciences (New York), 2014, 200:6, 742–768

Document Type: Article
UDC: 512.54
Language: English

Citation: R. Hazrat, A. V. Stepanov, N. A. Vavilov, Z. Zhang, “The yoga of commutators: further applications”, Representation theory, dynamical systems, combinatorial methods. Part XXIII, Zap. Nauchn. Sem. POMI, 421, POMI, St. Petersburg, 2014, 166–213; J. Math. Sci. (N. Y.), 200:6 (2014), 742–768

Citation in format AMSBIB
\Bibitem{HazSteVav14}
\by R.~Hazrat, A.~V.~Stepanov, N.~A.~Vavilov, Z.~Zhang
\paper The yoga of commutators: further applications
\inbook Representation theory, dynamical systems, combinatorial methods. Part~XXIII
\serial Zap. Nauchn. Sem. POMI
\yr 2014
\vol 421
\pages 166--213
\publ POMI
\mathnet{http://mi.mathnet.ru/znsl5758}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2014
\vol 200
\issue 6
\pages 742--768
\crossref{https://doi.org/10.1007/s10958-014-1967-z}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84940231943}


• http://mi.mathnet.ru/eng/znsl5758
• http://mi.mathnet.ru/eng/znsl/v421/p166

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This publication is cited in the following articles:
1. J. Math. Sci. (N. Y.), 209:6 (2015), 922–934
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