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.

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 ${\displaystyle {}_{H}^{H}{\mathcal {YD}}}$ if

A braided bialgebra in ${\displaystyle {}_{H}^{H}{\mathcal {YD}}}$ is called a braided Hopf algebra, if there is a morphism ${\displaystyle S:R\to R}$ of Yetter–Drinfeld modules such that

where ${\displaystyle \Delta _{R}(r)=r^{(1)}\otimes r^{(2)}}$ in slightly modified Sweedler notation – a change of notation is performed in order to avoid confusion in Radford's biproduct below.

For any braided Hopf algebra R in ${\displaystyle {}_{H}^{H}{\mathcal {YD}}}$ there exists a natural Hopf algebra ${\displaystyle R\#H}$ 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, ${\displaystyle R\#H}$ is just ${\displaystyle R\otimes H}$. The algebra structure of ${\displaystyle R\#H}$ is given by

where ${\displaystyle r,r'\in R,h,h'\in H}$, ${\displaystyle \Delta (h)=h_{(1)}\otimes h_{(2)}}$ (Sweedler notation) is the coproduct of ${\displaystyle h\in H}$, and ${\displaystyle \cdot :H\otimes R\to R}$ is the left action of H on R. Further, the coproduct of ${\displaystyle R\#H}$ is determined by the formula