# Braided Hopf algebra

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-Drinfel'd category of a Hopf algebra "H".

Definition

Let "H" be a Hopf algebra over a field "k", and assume that the antipode of "H" is bijective. A Yetter-Drinfel'd module "R" over "H" is called a braided bialgebra in the Yetter-Drinfel'd category $\left\{\right\}^H_Hmathcal\left\{YD\right\}$ if
* $\left(R,cdot ,eta \right)$ is a unital associative algebra, where the multiplication map $cdot :R imes R o R$ and the unit $eta :k o R$ are maps of Yetter-Drinfel'd modules,
* $\left(R,Delta ,varepsilon \right)$ is a coassociative coalgebra with counit $varepsilon$, and both $Delta$ and $varepsilon$ are maps of Yetter-Drinfel'd modules,
* the maps $Delta :R o Rotimes R$ and $varepsilon :R o k$ are algebra maps in the category $\left\{\right\}^H_Hmathcal\left\{YD\right\}$, where the algebra structure of $Rotimes R$ is determined by the unit $eta otimes eta\left(1\right) : k o Rotimes R$ and the multiplication
$\left(Rotimes R\right) imes \left(Rotimes R\right) o Rotimes R,quad \left(rotimes s,totimes u\right) mapsto sum _i rt_iotimes s_i u, quad ext\left\{and\right\}quad c\left(sotimes t\right)=sum _i t_iotimes s_i$.:Here "c" is the canonical braiding in the Yetter-Drinfel'd category $\left\{\right\}^H_Hmathcal\left\{YD\right\}$.

A braided bialgebra in $\left\{\right\}^H_Hmathcal\left\{YD\right\}$ is called a braided Hopf algebra, if there is a morphism $S:R o R$ of Yetter-Drinfel'd modules such that:: $S\left(r^\left\{\left(1\right)\right\}\right)r^\left\{\left(2\right)\right\}=r^\left\{\left(1\right)\right\}S\left(r^\left\{\left(2\right)\right\}\right)=eta\left(varepsilon \left(r\right)\right)$ for all $rin R$,

where $Delta _R\left(r\right)=r^\left\{\left(1\right)\right\}otimes r^\left\{\left(2\right)\right\}$ 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 $H=k\left(mathbb\left\{Z\right\}/2mathbb\left\{Z\right\}\right)$.
* The tensor algebra $TV$ of a Yetter-Drinfeld module $Vin \left\{\right\}^H_Hmathcal\left\{YD\right\}$ 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::$Delta \left(v\right)=1otimes v+votimes 1 quad ext\left\{for all\right\}quad vin V.$ :The counit $varepsilon :TV o k$ then satisfies the equation $varepsilon \left(v\right)=0$ for all $vin V .$
* Let $Vin \left\{\right\}^H_Hmathcal\left\{YD\right\}$. There exists a largest ideal of "TV" with the following properties.:: ::$Delta \left(I\right)subset Iotimes TV+TVotimes I.$:One has $Iin \left\{\right\}^H_Hmathcal\left\{YD\right\}$, and the quotient "TV/I" is a braided Hopf algebra in $\left\{\right\}^H_Hmathcal\left\{YD\right\}$. It is called the Nichols algebra of "V", named after the mathematician Warren Nichols, and is denoted by $mathfrak\left\{B\right\}\left(V\right)$.

For any braided Hopf algebra "R" in $\left\{\right\}^H_Hmathcal\left\{YD\right\}$ 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 $Rotimes H$. The algebra structure of $R# H$ is given by:: ,

where $r,r\text{'}in R,quad h,h\text{'}in H$, $Delta \left(h\right)=h_\left\{\left(1\right)\right\}otimes h_\left\{\left(2\right)\right\}$ (Sweedler notation) is the coproduct of $hin H$, and is the left action of "H" on "R". Further, the coproduct of $R# H$ is determined by the formula:: $Delta \left(r#h\right)=\left(r^\left\{\left(1\right)\right\}#r^\left\{\left(2\right)\right\}\left\{\right\}_\left\{\left(-1\right)\right\}h_\left\{\left(1\right)\right\}\right)otimes \left(r^\left\{\left(2\right)\right\}\left\{\right\}_\left\{\left(0\right)\right\}#h_\left\{\left(2\right)\right\}\right), quad rin R,hin H.$

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

References

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.

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