Braided Hopf Crossed Modules Through Simplicial Structures

Abstract

Any simplicial Hopf algebra involves $2n$ different projections between the Hopf algebras $H_n,H_{n-1}$ for each $n\ge 1$. The word projection, here meaning a tuple $\partial :H_n\to H_{n-1}$ and $i:H_{n-1}\to H_n$ of Hopf algebra morphisms, such that $\partial i=\mathrm{id}$. Given a Hopf algebra projection $(\partial :I\to H,i)$ in a braided monoidal category $\mathfrak{C}$, one can obtain a new Hopf algebra structure living in the category of Yetter-Drinfeld modules over $H$, due to Radford's theorem. The underlying set of this Hopf algebra is obtained by an equalizer which only defines a sub-algebra (not a sub-coalgebra) of $I$ in $\mathfrak{C}$. In fact, this is a braided Hopf algebra since the category of Yetter-Drinfeld modules over a Hopf algebra with an invertible antipode is braided monoidal. To apply Radford's theorem in a simplicial Hopf algebra successively, we require some extra functorial properties of Yetter-Drinfeld modules. Furthermore, this allows us to model Majid's braided Hopf crossed module notion from the perspective of a simplicial structure.

30 pages, preliminary version, comments are welcome

References in corpus (1)

• Strict quantum 2-groups