On The Structure Of Cyclotomic Nazarov-Wenzl Algebras

In[18],Nazarov introduced a class of infinite dimensional associative algebras called the affine Wenzl algebras.In order to classify the finite dimensional irreducible modules for affine Wenzl algebras,Ariki,Mathas and Rui[3]introduced a class of finite dimensional associative algebras W_r,ncalled the cyclotomic Nazarov-Wenzl algebras.Under certain assumptions,they[3]have proved that W_r,nis cellular in the sense of[13].Using the representation theory of cellular algebras,Ariki,Mathas and Rui[3]have classified the irreducible W_r,n-modules over a field. Hence,they have constructed all of the finite dimensional irreducible modules for affine Wenzl algebras.In this paper,we will go on studying structures and the representations of cyclotomic Nazarov-Wenzl algebras.In particular,we will give recursive formulae for the Gram determinants associated to all cell modules for W_r,n.Using the representation theory of cellular algebras together with the previous recursive formulae,we can get a sufficient and necessary condition for W_r,nbeing semisimple over an arbitrary field F with char.F≠2.