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 $\{\}^H\_Hmathcal\{YD\}$ if
* $(R,cdot\; ,eta\; )$ 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,
* $(R,Delta\; ,varepsilon\; )$ 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 $\{\}^H\_Hmathcal\{YD\}$, where the algebra structure of $Rotimes\; R$ is determined by the unit $eta\; otimes\; eta(1)\; :\; k\; o\; Rotimes\; R$ and the multiplication
$(Rotimes\; R)\; imes\; (Rotimes\; R)\; o\; Rotimes\; R,quad\; (rotimes\; s,totimes\; u)\; mapsto\; sum\; \_i\; rt\_iotimes\; s\_i\; u,\; quad\; ext\{and\}quad\; c(sotimes\; t)=sum\; \_i\; t\_iotimes\; s\_i$.:Here "c" is the canonical braiding in the Yetter-Drinfel'd category $\{\}^H\_Hmathcal\{YD\}$.

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

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

Radford's biproduct

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

where $r,r\text{'}in\; R,quad\; h,h\text{'}in\; H$, $Delta\; (h)=h\_\{(1)\}otimes\; h\_\{(2)\}$ (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:: $Delta\; (r\#h)=(r^\{(1)\}\#r^\{(2)\}\{\}\_\{(-1)\}h\_\{(1)\})otimes\; (r^\{(2)\}\{\}\_\{(0)\}\#h\_\{(2)\}),\; quad\; rin\; R,hin\; H.$

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 $r^\{(2)\}in\; R$.

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.