The near centre of Doi-Hopf module category _AM (H)~C

It is known that any strict tensor category (C?I) determines a braided tensor categoryZ(C), the centre ofC. WhenA is a finite dimension Hopf algebra, Drinfel’d has proved thatZ(A M) is equivalent to D(A) M as a braided tensor category, whereA M is the left A-module category andD(A) is the Drinfel’d double ofA. For a braided tensor category, the braidC U,v is a natural isomorphism for any pair of object (U,V) in. If weakening the natural isomorphism of the braidC U,V to a natural transformation, thenC U,V is a prebraid and the category with a prebraid is called a prebraided tensor category. Similarly it can be proved that any strict tensor category determines a prebraided tensor category Z～ (C), the near centre of. An interesting prebraided tensor structure of the Yetter-Drinfel’d category C*A YD C*A given, whereC # A is the smash product bialgebra ofC andA. And it is proved that the near centre of Doi-Hopf module A M(H) C is equivalent to the Yetter-Drinfel’ d C*A YD C*A as prebraided tensor categories. As corollaries, the prebraided tensor structures of the Yetter-Drinfel’d category A YD A , the centres of module category and comodule category are given.

It is known that any strict tensor category （ C, （?）, I） determines a braided tensor category Z（C）, the centre of C. When A is a finite dimension Hopf algebra, Drinfel’d has proved that Z（AM） is equivalent to (（D（A））M as a braided tensor category, where AM is the left A-module category and D （A） is the Drinfel’d double of A. For a braided tensor category C, the braid CU, V is a natural isomorphism for any pair of object （U, V） in C. If weakening the natural isomorphism of the braid CU, V to a natural transformation, then CU, V is a prebraid and the category C with a prebraid is called a prebraided tensor category. Similarly it can be proved that any strict tensor category C determines a prebraided tensor category Z’ （C）, the near centre of C. An interesting prebraided tensor structure of the Yetter-Drinfel’d category C#AYDC#A is given, where C # A is the smash product bialgebra of C and A. And it is proved that the near centre of Doi-Hopf module AM（H）C is equivalent to the