# 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 $_{H}^{H}{\mathcal {YD}}$ if
* $(R,\cdot ,\eta )$ is a unital associative algebra, where the multiplication map $\cdot :R\times R\to R$ and the unit $\eta :k\to R$ are maps of Yetter–Drinfeld modules,
* $(R,\Delta ,\varepsilon )$ is a coassociative coalgebra with counit $\varepsilon$, and both $\Delta$ and $\varepsilon$ are maps of Yetter–Drinfeld modules,
* the maps $\Delta :R\to R\otimes R$ and $\varepsilon :R\to k$ are algebra maps in the category $_{H}^{H}{\mathcal {YD}}$, where the algebra structure of $R\otimes R$ is determined by the unit $\eta \otimes \eta (1):k\to R\otimes R$ and the multiplication map
       $(R\otimes R)\times (R\otimes R)\to R\otimes R,\quad (r\otimes s,t\otimes u)\mapsto \sum _{i}rt_{i}\otimes s_{i}u,\quad {\text{and}}\quad c(s\otimes t)=\sum _{i}t_{i}\otimes s_{i}.$
    Here c is the canonical braiding in the Yetter–Drinfeld category $_{H}^{H}{\mathcal {YD}}$.
A braided bialgebra in $_{H}^{H}{\mathcal {YD}}$ is called a braided Hopf algebra, if there is a morphism $S:R\to R$ of Yetter–Drinfeld modules such that
       $S(r^{(1)})r^{(2)}=r^{(1)}S(r^{(2)})=\eta (\varepsilon (r))$ for all $r\in R,$
where $\Delta _{R}(r)=r^{(1)}\otimes r^{(2)}$ in slightly modified Sweedler notation – a change of notation is performed in order to avoid confusion in Radford's biproduct below.

## Examples

* Any Hopf algebra is also a braided Hopf algebra over $H=k$
* A super Hopf algebra is nothing but a braided Hopf algebra over the group algebra $H=k[\mathbb {Z} /2\mathbb {Z} ]$.
* The tensor algebra $TV$ of a Yetter–Drinfeld module $V\in_{H}^{H}{\mathcal {YD}}$ is always a braided Hopf algebra. The coproduct $\Delta$ of $TV$ is defined in such a way that the elements of V are primitive, that is
       $\Delta (v)=1\otimes v+v\otimes 1\quad {\text{for all}}\quad v\in V.$
    The counit $\varepsilon :TV\to k$ then satisfies the equation $\varepsilon (v)=0$ for all $v\in V.$
* The universal quotient of $TV$, that is still a braided Hopf algebra containing $V$ 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 $_{H}^{H}{\mathcal {YD}}$ 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
       $(r\#h)(r'\#h')=r(h_{(1)}{\boldsymbol {.}}r')\#h_{(2)}h',$
where $r,r'\in R,\quad h,h'\in H$, $\Delta (h)=h_{(1)}\otimes h_{(2)}$ (Sweedler notation) is the coproduct of $h\in H$, and ${\boldsymbol {.}}:H\otimes R\to R$ is the left action of H on R. Further, the coproduct of $R\#H$ is determined by the formula
       $\Delta (r\#h)=(r^{(1)}\#r^{(2)}_{(-1)}h_{(1)})\otimes (r^{(2)}_{(0)}\#h_{(2)}),\quad r\in R,h\in H.$
Here $\Delta _{R}(r)=r^{(1)}\otimes r^{(2)}$ denotes the coproduct of r in R, and $\delta (r^{(2)})=r^{(2)}_{(-1)}\otimes r^{(2)}_{(0)}$ is the left coaction of H on $r^{(2)}\in R.$