
Zap. Nauchn. Sem. POMI, 2016, Volume 443, Pages 151–221
(Mi znsl6264)




The commutators of classical groups
R. Hazrat^{a}, N. Vavilov^{b}, Z. Zhang^{c} ^{a} Centre for Research in Mathematics, Western Sydney University,
Australia
^{b} Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia
^{c} Department of Mathematics, Beijing Institute of Technology, Beijing, China
Abstract:
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,
$$
E(A)=[E(A),E(A)]=[\operatorname{GL}(A),\operatorname{GL}(A)],
$$
where $\operatorname{GL}(A)$ is the stable general linear group and $E(A)$ is its elementary subgroup. Since then, various commutator formulas have been studied in stable and nonstable settings, and for a range of classical and algebraic likegroups, 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 algebraiclike groups, such as general linear groups, $\operatorname{GL}(n,A)$, unitary groups $\operatorname{GU}(2n,A,\Lambda)$ and Chevalley groups $G(\Phi,A)$. 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 selfcontained source. We have borrowed the proofs from our previous papers and expositions [38–50, 99, 100, 129–132].
Key words and phrases:
general linear groups, unitary groups, Chevalley groups, elementary subgroups, elementary generators, localisation, relative subgroups, conjugation calculus, commutator calculus, Noetherian reduction, the Quillen–Suslin lemma, localisationcompletion, commutator formulae, commutator width, nilpotency of $\mathrm K_1$, nilpotent filtration.
Funding Agency 
Grant Number 
Ministry of Education and Science of the Russian Federation 
6.38.74.2011 
Russian Science Foundation 
141100297 
The work of the second author on relative localisation in unitary groups and generation of relative commutator subgroups was supported by the State Financed research task 6.38.74.2011 at the Saint Petersburg State University
“Structure theory and geometry of algebraic groups and their applications in representation theory and algebraic $K$theory”. His work on the results on multiple commutator formula presented in sections 10, 11, and 13 was supported by the Russian Science Foundation project 141100297 “Decomposition of unipotents on reductive groups”. 
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UDC:
512 Received: 02.12.2015
Language: English
Citation:
R. Hazrat, N. Vavilov, Z. Zhang, “The commutators of classical groups”, Problems in the theory of representations of algebras and groups. Part 29, Zap. Nauchn. Sem. POMI, 443, POMI, St. Petersburg, 2016, 151–221
Citation in format AMSBIB
\Bibitem{HazVavZha16}
\by R.~Hazrat, N.~Vavilov, Z.~Zhang
\paper The commutators of classical groups
\inbook Problems in the theory of representations of algebras and groups. Part~29
\serial Zap. Nauchn. Sem. POMI
\yr 2016
\vol 443
\pages 151221
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6264}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=3507772}
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