The Commutators of Classical Groups

Abstract

In his seminal paper, half a century ago, Hyman Bass established commutator formulas for a (stable) general linear group, which were the key step in defining the group K₁. Namely, he proved that for an associative ring A with identity,

\( E(A)=\left[E(A),E(A)\right]=\left[\mathrm{GL}(A),\mathrm{GL}(A)\right], \)

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 for classical groups, algebraic groups, and their analogs, and mostly in relation to subnormal subgroups of these groups. The basic classical theorems and methods developed for their proofs are associated with the names of the heroes of classical algebraic K-theory: Bak, Quillen, Milnor, Suslin, Swan, Vaserstein, and others.

One of the dominant techniques in establishing commutator type results is localization. In the present paper, some recent applications of localization 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), are described. Some auxiliary results and corollaries of the main results are also stated.

The paper provides a general overview of the subject and covers the current activities. It contains complete proofs borrowed from our previous papers and expositions of several main results to give the reader a self-contained source.