In this paper, we give a recursive formula to compute the Gram determinant associated to each cell module of the cyclotomic BMW algebras B r,n over an integral domain. As a by-product, we determine explicitly when B r,n is semisimple over a field. This generalizes our previous result on Birman-Murakami-

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a b s t r a c tAriki, Mathas and Rui [S. Ariki, A. Mathas, H. Rui, Cyclotomic Nazarov-Wenzl algebras, Nagoya Math. J. 182 (2006) 47-134 (special volume in honor of Professor G. Lusztig)] introduced a class of finite dimensional algebras W r,n , called the cyclotomic Nazarov-Wenzl algebras which are associative algebras over a commutative ring R generated by {S i , E i , X j | 1 ≤ i < n and 1 ≤ j ≤ n} satisfying the defining relations given in this paper. In particular,n s are quotients of affine Wenzl algebras in [M. Nazarov, Young's orthogonal form for Brauer's centralizer algebra, J. Algebra 182 (1996) 664-693]. It has been proved in the first cited reference above that W r,n is cellular in the sense of [J.J. Graham, G.I. Lehrer, Cellular algebras, Invent. Math. 123 (1996) 1-34]. Using the representation theory of cellular algebras, Ariki, Mathas and Ruihave classified the irreducible W r,n -modules under the assumption ω 0 = 0 in their abovecited work. In this paper, we are going to classify the irreducible W r,n -modules under the assumption ω 0 = 0. We will compute the Gram determinant associated to each cell module for W r,n no matter whether ω 0 is zero or not. At the end of this paper, we use our formulae for Gram determinants to determine the semisimplicity of W r,n for arbitrary parameters over an arbitrary field F with char F = 2.

In this paper, we compute all Gram determinants associated to all cell modules of Birman-Wenzl algebras. As a by-product, we give a necessary and sufficient condition for Birman-Wenzl algebras being semisimple over an arbitrary field.

In [H. Rui, A criterion on the semisimple Brauer algebras, J. Combin. Theory Ser. A 111 (2005) 78-88], the first author gave an algorithm for determining the pairs (n, δ) such that the Brauer algebra B n (δ) over a field F is semisimple. Such an algorithm involves a subset Z(n) ⊂ Z. In this note, we give an explicit description about Z(n). Using [H. Rui, A criterion on the semisimple Brauer algebras, J. Combin. Theory Ser. A 111 (2005) 78-88, 1.3] we verify Enyang's conjecture given in [J. Enyang, Specht modules and semisimplicity criteria for Brauer and Birman-Murakami-Wenzl algebras, preprint, 2005, 12.2].