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 {}^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.:: Isubset igoplus _{n=2}^infty T^nV,:: 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:: (r# h)(r'#h')=r(h_{(1)}oldsymbol{.}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 hin H , and oldsymbol{.}:Hotimes R o 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 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.


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