The Commutators of Classical Groups

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,, 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].

Generation of relative commutator subgroups in Chevalley groups. II

“…See also [1,8] for other interesting occurrences of the above commutator subgroups in the theory of Chevalley groups. One can find many further related results, applications and open problems in our surveys and conference papers [10,11,13,16].…”

“…So far, we have not even mentioned another extremely important line of research, which initially was our main motivation to focus on relative commutator subgroups. Namely, the study of subgroups normalized by a relative elementary subgroup; see [5,10,11,13,16]…”

Generation of relative commutator subgroups in Chevalley groups

“…We refer the interested reader to our papers cited above, and to our conference papers [9,10,11,18], where one can find many further details and unsolved problems concerning relative commutator subgroups.…”

“…E(n, A, IJ + JI), so that one has to list both t ij (ξζ) and t ij (ζξ), where 1 i = j n, ξ ∈ I, ζ ∈ J, as generators. Later, this result was generalized to unitary groups in [15] (see also [16] for background, a detailed overview, and somewhat finer proofs).…”

“…Costa and Keller studied the normal structure of Sp(4, R) and G 2 (R) in their remarkable papers [6,7] and discovered that in these cases relative subgroups should be parametrized by radices, rather than by individual ideals, or admissible pairs. We refer the interested reader to our papers cited above, and to our conference papers [13,16,17,19], where one can find many further details and unsolved problems concerning relative commutator subgroups.…”