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 vectorspace in that category.

The notion should not be confused with quasitriangular Hopf algebra.

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 ${\displaystyle {}_{H}^{H}{\mathcal {YD}}}$ if

A braided bialgebra in ${\displaystyle {}_{H}^{H}{\mathcal {YD}}}$ is called a braided Hopf algebra, if there is a morphism ${\displaystyle S:R\to R}$ of Yetter–Drinfeld modules such that

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