2017/08/16 by Chen, Zhi
#16T99 #F.4.3 #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1708.04833
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.