A categorical interpretation of Yetter-Drinfel’d modules

Abstract

It is known that any strict tensor category (C,⊗,I) can determine a strict braided tensor categoryZ(C), the centre ofC. WhenA is a finite Hopf algebra, Drinfel’d has proved thatZ( AM) is equivalent toD(A)M as a braided tensor category, whereAM is the left A-module category, andD(A) is the Drinfel’d double ofA. This is the categorical interpretation ofD(A). Z( AM) is proved to be equivalent to the Yetter-Drinfel’d module category,AYD A as a braided tensor category for any Hopf algebraA. Furthermore, for right A-comodule categoryM A, Z(MA) is proved to be equivalent to the Yetter-Drinfel’d module categoryAY DA as a braided tensor category. But,in the two cases, the Yetter-Drinfel’d module categoryAY DA has different braided tensor structures.