The commutators of classical groups

R. Hazrat ; N. Vavilov ; Z. Zhang

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 29, Tome 443 (2016), pp. 151-221

Résumé

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 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, $\operatorname{GL}(n,A)$, 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 self-contained source. We have borrowed the proofs from our previous papers and expositions [38–50, 99, 100, 129–132].