Title: 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.