Flat connections and Brauer Type Algebras

We introduce a Brauer type algebra $B_G (\Upsilon)$ associated with every pseudo reflection group and every Coxeter group $G$. When $G$ is a Coxeter group of simply-laced type we show $B_G (\Upsilon)$ is isomorphic to the generalized Brauer algebra of simply-laced type introduced by Cohen, Gijsbers and Wales ({\it J. Algebra}, {\bf 280} (2005), 107-153). We also prove $B_G (\Upsilon)$ has a cellular structure and be semisimple for generic parameters when $G$ is a dihedral group or the type $H_3$ Coxeter group. Moreover, in the process of construction, we introduce a further generalization of Lawrence-Krammer representation to complex braid groups associated with all pseudo reflection groups.Comment: This paper is an improved version of my last one "Algebras associated with Pseudo Reflection Groups: A Generalization of Brauer Algebras", with many mistakes corrected and new contents adde