The near centre of Doi-Hopf module categoryA M (H) C

Abstract

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.