Parametrized braid groups of Chevalley groups

Abstract

We introduce the notion of a braid group parametrized by a ring, which is defined by generators and relations and based on the geometric idea of painted braids. We show that the parametrized braid group is isomorphic to the semi-direct product of the Steinberg group (of the ring) with the classical braid group. The technical heart of the proof is the Pure Braid Lemma, which asserts that certain elements of the parametrized braid group commute with the pure braid group. More generally, we define, for any crystallographic root system, a braid group and a parametrized braid group with parameters in a commutative ring. The parametrized braid group is expected to be isomorphic to the semi-direct product of the corresponding Steinberg group with the braid group. The first part of the paper (described above) treats the case of the root system $A_n$; in the second part, we handle the root system {$D_n$}. Other cases will be treated in the sequel.