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 A braided bialgebra in is called a braided Hopf algebra, if there is a morphism S:R\to R of Yetter–Drinfeld modules such that where in slightly modified Sweedler notation – a change of notation is performed in order to avoid confusion in Radford's biproduct below.

Examples

Radford's biproduct

For any braided Hopf algebra R in there exists a natural Hopf algebra R# H 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, R# H is just R\otimes H. The algebra structure of R# H is given by where, (Sweedler notation) is the coproduct of h\in H, and is the left action of H on R. Further, the coproduct of R# H is determined by the formula Here denotes the coproduct of r in R, and is the left coaction of H on