Extension Of Tensor Algebra And Schur-Weyl Duality Of Type B

In this paper,on the basis of reviewing the existing Schur-Weyl duality proof of type A and referring to the tensor algebraic construction of Itoh,we obtain the Schur-Weyl duality of complex reflection group S_n,rand derive the Schur-Weyl duality of type B.In this paper,we first give a natural tensor algebra,and based on this tensor algebra,we give the multiplication of two operators（in the form of“left multiplication”and“differential”of tensor products）and the complex reflection group S_n,ron this tensor algebra.Next,we calculate the commutativity of the two operators,the relation with the given semisimple Lie algebra g on V,and give the properties of the definition of Euler operators（constructed by the two operators）.Then we give the span space A generated by two operators and verify the commutativity of the right multiplication of S_n,r.Then by means of the double centralization subtheorem,we verify that the right-acting centrators of the tensor Spaces A and S_n,rgenerated by two kinds of operators are each other.Then the tensor space generated by the two operators is verified to be equivalent to the universal envelope algebra of g.