### Parametrized braid groups of Chevalley groups, by Jean-Louis Loday and Michael R. Stein

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.

Jean-Louis Loday <loday@math.u-strasbg.fr>

Michael R. Stein <mike@math.northwestern.edu>