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What are the relations between braided Hopf algebra and usual Hopf algebra?

Acoording to wikipedia, a braided Hopf algebra is defined as follows.

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 ${\displaystyle (R,\cdot ,\eta )}$ 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 $\Delta$ 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 ${}_{H}^{H}{\mathcal {YD}}$, 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 \begin{align} (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, \end{align} and $$ c(s\otimes t)=\sum _{i}t_{i}\otimes s_{i}. (R\otimes R)\times (R\otimes R)\to R\otimes R,\quad (r\otimes s,t\otimes u)\mapsto \sum _{i}rt_{i}\otimes s_{i}u, $$ and $$ c(s\otimes t)=\sum _{i}t_{i}\otimes s_{i}. $$ 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 $$ {\displaystyle S(r^{(1)})r^{(2)}=r^{(1)}S(r^{(2)})=\eta (\varepsilon (r))} {\displaystyle S(r^{(1)})r^{(2)}=r^{(1)}S(r^{(2)})=\eta (\varepsilon (r))}$$ for all ${\displaystyle r\in R,}$ where $${\displaystyle \Delta _{R}(r)=r^{(1)}\otimes r^{(2)}}$$ is the Sweedler notation.

The usual Hopf algebra is a braided Hopf algebra when the braiding is permutation. Is this correct? Thank you very much.

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  • $\begingroup$ this should be migrated to mathoverflow (imo). $\endgroup$ – KonKan Sep 25 '16 at 6:02
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Actually, Wikipedia defines a braided Hopf algebra as:

"...a Hopf algebra in a braided monoidal category"

and describes the "braided bialgebras in the Yetter–Drinfeld category ${}_{H}^{H}{\mathcal {YD}}$" mentioned in your post, as "the most common" example.

However, the notion of braided Hopf algebra (i.e.: a Hopf algebra in a braided monoidal category) is wider and includes several other equally -imo- common examples: If $G$ is a finite abelian group and $\theta$ is a bicharacter on $G$, then the Universal enveloping algebras (UEA) of the $g$-graded, $\theta$-colored Lie algebras are also such examples: they are braided groups or $\theta$-braided Hopf algebras. This is equivalent to saying that they are Hopf algebras in the braided monoidal Category $_{\mathbb{C}G}\mathcal{M}$ of representations of the quasitriangular group Hopf algebra $\mathbb{C}G$. (Note that here $\mathbb{C}G$ is considered "equipped" with its non-trivial quasitriangular structure). Some of the -historically- original examples of quantum groups, also fall into this or generalizations of this idea. For a review of the related definitions for braided hopf algebras you can see at this article (see Proposition 3.2).

Now regarding your second question, on the relation of braided Hopf algebras with ordinary Hopf algebras: Generally, given a Hopf algebra $A$, in the braided monoidal Category $_{H}\mathcal{M}$ of the representations of the quasitriangular Hopf algebra $H$, there is a general method for constructing a smash product $A\star H$ algebra, in such a way that: $A\star H$ is an ordinary Hopf algebra and moreover there is an equivalence of Categories between the category of representations of $A\star H$ and the category of (braided) representations of $A$. This is known in the bibliography as Majid's bosonization or Radford's biproduct. You can see at this article (see section 2, p.111-113) for a review of these methods.

Finally, regarding your last question: an ordinary Hopf algebra $H$, is a braided Hopf algebra (trivially), where the (symmetric) braiding is the usual twist map. This is equivalent to saying that, $H$ is a Hopf algebra in the symmetric monoidal category $_{\mathbb{C}G}\mathcal{M}$ of representations of the cocommutative group Hopf algebra $\mathbb{C}G$. Now, $\mathbb{C}G$ is considered "equipped" with its trivial quasitriangular structure (i.e. its cocommutativity), induced by the trivial $R$-matrix $R=1\otimes 1$.

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