Birman-Murakami-Wenzl type algebras for arbitrary Coxeter systems

Abstract

In this paper we first present a Birman-Murakami-Wenzl type algebra for every Coxeter system of rank 2 (corresponding to dihedral groups). We prove they have semisimple for generic parameters, and having natural cellular structures. And classcify their irreducible representations. Among them there is one serving as a generalization of the Lawrence-Krammer representation with quite neat shape and the "correct" dimension. We conjecture they are isomorphic to the generalized Lawrence-Krammer representaions defined by I.Marin as monodromy of certain KZ connections. We prove these representations are irreducible for generic parameters, and find a quite neat invariant bilinear form on them. Based on above constructions for rank 2, we introduce a Birman-Murakami-Wenzl type algebra for an arbitrary Coxeter system. For every Coxeter system, the introduced algebra is a quotient of group algebra of the Artin group (associated with this Coxeter system), having the corresponding Hecke algebra as a quotient. The simple generators of the Artin group have degree 3 annihiating polynomials in this algebra.