New Deformations of Group Algebras of Coxeter Groups

The goal of this paper is to define new deformations of group algebras of Coxeter groups. Recall that a Coxeter group W is generated by elements si, i ∈ I modulo two kinds of relations – the involutivity relations si = 1 and the relations (sisj) mij = 1, where 2 ≤ mij = mji ≤ ∞; in presence of the involutivity relations these are equivalent to the braid relations sisjsi... = sjsisj... (mij factors). The traditional way to deform the group algebra C[W ] is to deform the involutivity relations to the Hecke relations (si − q)(si + q −1) = 0, and keep the braid relations unchanged. This yields the Hecke algebra Cq[W ] of W , which is classically known to be a flat deformation of C[W ]. On the contrary, the deformation A of C[W ] that we study in this paper is obtained by keeping the involutivity relations fixed, and deforming the braid relations to