TITLE: Is there a q-analog to the braid group?
QUESTION [24 upvotes]: The braid group $B_n$ on $n$ strands fits into a short exact sequence of groups:
$$ 1 \longrightarrow P_n \longrightarrow B_n \longrightarrow S_n \longrightarrow 1,$$
where $S_n$ is the symmetric group on the strands, and $P_n$ is the normal subgroup of braids that do not permute the strands.
Since symmetric groups are, in some sense, ``general linear groups over the field with one element,'' perhaps there is some corresponding short exact sequence ending with $GL_n(F_q)$ that specializes to the exact sequence above as $q \rightarrow 1$.
In the spirit of the exact sequence above, is there a $q$-analog to the braid group?

REPLY [4 votes]: There is a nice paper by Etingof and Rains about a non-standard q-deformations of S_n:
http://xxx.lanl.gov/abs/math/0409261
They deform not quadratic relations but the braid relations.
It seems to be relevant to the discussion.