# Nichols Algebras as Braided Hopf Algebras

Given a Hopf algebra $H$ and a Yetter--Drinfeld module $V$ over $H$, it is well-known that $V$ has an induced braided vector space structure, and so, one can consider it's Nichols algebra which is a braided Hopf algebra in the braided monoidal category of Yetter--Drinfeld modules over $H$.

The notion, however, of a Nichols algebra makes sense for any object in any braided monoidal category ${\cal C}$. Is it true in general that the Nichols algebra of any object in ${\cal C}$ will be a braided Hopf algebra in ${\cal C}$?

• It is in general not quite clear how one would define the Nichols algebra in a braided monoidal category $\mathcal{C}$. Maybe what one can make sense of is tensor Hopf algebras, but even here more assumptions are needed. We need at least biproducts in the category. The Nichols algebra is defined different ways, possibly as a quotient of the tensor algebra. Having an abelian category structure would be desirable. Certainly, more details are needed to make this a precise question. IMO, if whatever definition does not give a braided Hopf algebra, then it should not be called a Nichols algebra. Apr 5, 2017 at 20:34