The commutators of classical groups

TL;DR

This paper reviews recent advances in the study of commutator formulas within classical algebraic groups, emphasizing localization techniques and their applications to higher and relative commutators.

Contribution

It provides a comprehensive overview of localization methods applied to commutator problems in classical groups, including new results and complete proofs for key theorems.

Findings

Localization techniques effectively analyze higher/relative commutators.

Complete proofs of main results are provided for self-containment.

Applications extend to groups like GL(n,A), unitary, and Chevalley 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, E(A)=[E(A),E(A)]=[GL(A),GL(A)] 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, 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 self-contained source. We have borrowed some of the proofs from our previous papers and expositions