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–Drinfeld category of a Hopf algebra H, particularly the Nichols algebra of a braided vector space in that category. The notion should not be confused with quasitriangular Hopf algebra.

## Contents

* 1 Definition
* 2 Examples
* 3 Radford's biproduct
* 4 References

## Definition

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 HH𝒴𝒟 if

* (R,⋅,η) is a unital associative algebra, where the multiplication map ⋅:R×R->R and the unit η:k->R are maps of Yetter–Drinfeld modules,
* (R,Δ,ε) is a coassociative coalgebra with counit ε, and both Δ and ε are maps of Yetter–Drinfeld modules,
* the maps Δ:R->R⊗R and ε:R->k are algebra maps in the category HH𝒴𝒟, where the algebra structure of R⊗R is determined by the unit η⊗η(1):k->R⊗R and the multiplication map

(R⊗R)×(R⊗R)->R⊗R,(r⊗s,t⊗u)↦∑irti⊗siu,andc(s⊗t)=∑iti⊗si.
Here c is the canonical braiding in the Yetter–Drinfeld category HH𝒴𝒟.

A braided bialgebra in HH𝒴𝒟 is called a braided Hopf algebra, if there is a morphism S:R->R of Yetter–Drinfeld modules such that

S(r(1))r(2)=r(1)S(r(2))=η(ε(r)) for all r∈R,

where ΔR(r)=r(1)⊗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[ℤ/2ℤ].
* The tensor algebra TV of a Yetter–Drinfeld module V∈HH𝒴𝒟 is always a braided Hopf algebra. The coproduct Δ of TV is defined in such a way that the elements of V are primitive, that is

Δ(v)=1⊗v+v⊗1for allv∈V.
The counit ε:TV->k then satisfies the equation ε(v)=0 for all v∈V.

* The universal quotient of TV, that is still a braided Hopf algebra containing 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.

## Radford's biproduct

For any braided Hopf algebra R in HH𝒴𝒟 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 R⊗H. The algebra structure of R#H is given by

(r#h)(r′#h′)=r(h(1).r′)#h(2)h′,

where r,r′∈R,h,h′∈H, Δ(h)=h(1)⊗h(2) (Sweedler notation) is the coproduct of h∈H, and .:H⊗R->R is the left action of H on R. Further, the coproduct of R#H is determined by the formula

Δ(r#h)=(r(1)#r(2)(−1)h(1))⊗(r(2)(0)#h(2)),r∈R,h∈H.

Here ΔR(r)=r(1)⊗r(2) denotes the coproduct of r in R, and δ(r(2))=r(2)(−1)⊗r(2)(0) is the left coaction of H on r(2)∈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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