Cross Products by Braided Groups and Bosonization

Abstract

Let H be a braided-cocommutative Hopf algebra in a braided monoidal category C and B a Hopf algebra in C on which H acts. We construct a cross product Hopf algebra B[formula] H in C . As an application we show that every B in a certain class can be converted to an equivalent ordinary Hopf algebra by a process of bosonization. The class includes, for example, all super-Hopf algebras. The constructions respect any quasitriangular structures on B. As a corollary, we show that if ( H, R ) is an ordinary quasitriangular Hopf algebra then the smash product by the adjoint action of H on itself. H Ad[formula] H, can be given the structure of a Hopf algebra. We prove a similar result for any Hopf algebra to which a quasitriangular Hopf algebra maps.