Cyclotomic Birman-Murakami-Wenzl Algebras

In this paper, we give a definition of the cyclotomic Birman-Murakami-Wenzl algebras B_r,n, which is equivalent to that given by Haering-Oldenburg in [HO]. By using Ariki-Mathas-Rui's method on cyclotomic Nazarov-Wenzl algebras in [AMR], we give an equivalent condition for B_r,2 being free with rank 3r². Such a condition is called the u-admissible condition. Under this condition, through constructing the seminormal representations of B_r,n, together with Weddurburn-Artin Theorem about semisimple algebras, we prove that B_r,n is free over a commutative ring R with rank rⁿ(2n - 1)!!. In fact, B_r,n is a "weakly" cellular algebra. Further, we classify the irreducible B_r,n-modules over an arbitrary field. When r is odd, we construct another "weakly" cellular basis too.