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.

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 if

• is a unital associative algebra, where the multiplication map and the unit are maps of Yetter–Drinfeld modules,
• is a coassociative coalgebra with counit , and both and are maps of Yetter–Drinfeld modules,
• the maps and are algebra maps in the category , where the algebra structure of is determined by the unit and the multiplication map

:

Here c is the canonical braiding in the Yetter–Drinfeld category .

A braided bialgebra in is called a braided Hopf algebra, if there is a morphism of Yetter–Drinfeld modules such that

: for all

where 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
• A super Hopf algebra is nothing but a braided Hopf algebra over the group algebra .
• The tensor algebra of a Yetter–Drinfeld module is always a braided Hopf algebra. The coproduct of is defined in such a way that the elements of V are primitive, that is

:

The counit then satisfies the equation for all

• The universal quotient of , that is still a braided Hopf algebra containing 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 there exists a natural Hopf algebra 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, is just . The algebra structure of is given by

:

where , (Sweedler notation) is the coproduct of , and is the left action of H on R. Further, the coproduct of is determined by the formula

:

Here denotes the coproduct of r in R, and is the left coaction of H on

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.