Braided Hopf algebra
In mathematics, a braided Hopf algebra is a Hopf algebra in a braided monoidal category. The most common braided Hopf algebras are objects in a Yetter–Drinfeld category of a Hopf algebra H, particularly the Nichols algebra of a braided vector space in that category. The notion should not be confused with quasitriangular Hopf algebra. Definition 

Let H be a Hopf algebra over a field k, and assume that the antipode of H is bijective. A Yetter–Drinfeld module R over H is called a braided bialgebra in the Yetter–Drinfeld category  if
  is a unital associative algebra, where the multiplication map  and the unit  are maps of Yetter–Drinfeld modules,
  is a coassociative coalgebra with counit , and both  and  are maps of Yetter–Drinfeld modules,
 the maps  and  are algebra maps in the category , where the algebra structure of  is determined by the unit  and the multiplication map
 
Here c is the canonical braiding in the Yetter–Drinfeld category . A braided bialgebra in  is called a braided Hopf algebra, if there is a morphism  of Yetter–Drinfeld modules such that
  for all 

where  in slightly modified Sweedler notation – a change of notation is performed in order to avoid confusion in Radford's biproduct below. Examples 

 Any Hopf algebra is also a braided Hopf algebra over 
 A super Hopf algebra is nothing but a braided Hopf algebra over the group algebra . The tensor algebra  of a Yetter–Drinfeld module  is always a braided Hopf algebra. The coproduct  of  is defined in such a way that the elements of V are primitive, that is
                                                                              
The counit  then satisfies the equation  for all 
 The universal quotient of , that is still a braided Hopf algebra containing  as primitive elements is called the Nichols algebra. They take the role of quantum Borel algebras in the classification of pointed Hopf algebras, analogously to the classical Lie algebra case. Radford's biproduct 

For any braided Hopf algebra R in  there exists a natural Hopf algebra  which contains R as a subalgebra and H as a Hopf subalgebra. It is called  Radford's biproduct, named after its discoverer, the Hopf algebraist David Radford. It was rediscovered by Shahn Majid, who called it bosonization. As a vector space,  is just . The algebra structure of  is given by
 

where ,  (Sweedler notation) is the coproduct of , and  is the left action of H on R. Further, the coproduct of  is determined by the formula
 

Here  denotes the coproduct of r in R, and  is the left coaction of H on

References 

 Andruskiewitsch, Nicolás and Schneider, Hans-Jürgen, Pointed Hopf algebras,  New directions in Hopf algebras,  1–68, Math. Sci. Res. Inst. Publ., 43, Cambridge Univ. Press, Cambridge, 2002. Hopf algebras