Braided 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

• ${\displaystyle (R,\cdot ,\eta )}$ is a unital associative algebra, where the multiplication map ${\displaystyle \cdot :R\times R\to R}$ and the unit ${\displaystyle \eta :k\to R}$ are maps of Yetter–Drinfeld modules,
• ${\displaystyle (R,\Delta ,\varepsilon )}$ is a coassociative coalgebra with counit ${\displaystyle \varepsilon }$, and both ${\displaystyle \Delta }$ and ${\displaystyle \varepsilon }$ are maps of Yetter–Drinfeld modules,
• the maps ${\displaystyle \Delta :R\to R\otimes R}$ and ${\displaystyle \varepsilon :R\to k}$ are algebra maps in the category ${\displaystyle {}_{H}^{H}{\mathcal {YD}}}$, where the algebra structure of ${\displaystyle R\otimes R}$ is determined by the unit ${\displaystyle \eta \otimes \eta (1):k\to R\otimes R}$ and the multiplication map

${\displaystyle (R\otimes R)\times (R\otimes R)\to R\otimes R,\quad (r\otimes s,t\otimes u)\mapsto \sum _{i}rt_{i}\otimes s_{i}u,\quad {\text{and}}\quad c(s\otimes t)=\sum _{i}t_{i}\otimes s_{i}.}$

Here c is the canonical braiding in the Yetter–Drinfeld category ${\displaystyle {}_{H}^{H}{\mathcal {YD}}}$.

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

${\displaystyle S(r^{(1)})r^{(2)}=r^{(1)}S(r^{(2)})=\eta (\varepsilon (r))}$ for all ${\displaystyle r\in R,}$

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.

• Any Hopf algebra is also a braided Hopf algebra over ${\displaystyle H=k}$
• A super Hopf algebra is nothing but a braided Hopf algebra over the group algebra ${\displaystyle H=k[\mathbb {Z} /2\mathbb {Z} ]}$.
• The tensor algebra ${\displaystyle TV}$ of a Yetter–Drinfeld module ${\displaystyle V\in {}_{H}^{H}{\mathcal {YD}}}$ is always a braided Hopf algebra. The coproduct ${\displaystyle \Delta }$ of ${\displaystyle TV}$ is defined in such a way that the elements of V are primitive, that is

${\displaystyle \Delta (v)=1\otimes v+v\otimes 1\quad {\text{for all}}\quad v\in V.}$

The counit ${\displaystyle \varepsilon :TV\to k}$ then satisfies the equation ${\displaystyle \varepsilon (v)=0}$ for all ${\displaystyle v\in V.}$

• The universal quotient of ${\displaystyle TV}$, that is still a braided Hopf algebra containing ${\displaystyle V}$ 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.

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

${\displaystyle (r\#h)(r'\#h')=r(h_{(1)}{\boldsymbol {.}}r')\#h_{(2)}h',}$

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

${\displaystyle \Delta (r\#h)=(r^{(1)}\#r^{(2)}{}_{(-1)}h_{(1)})\otimes (r^{(2)}{}_{(0)}\#h_{(2)}),\quad r\in R,h\in H.}$

Here ${\displaystyle \Delta _{R}(r)=r^{(1)}\otimes r^{(2)}}$ denotes the coproduct of r in R, and ${\displaystyle \delta (r^{(2)})=r^{(2)}{}_{(-1)}\otimes r^{(2)}{}_{(0)}}$ is the left coaction of H on ${\displaystyle r^{(2)}\in R.}$

• 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.