Braided Hopf Crossed Modules Through Simplicial Structures

Abstract

Any simplicial Hopf algebra involves 2n different projections between the Hopf algebras Hn, Hn−1 for each n ≥ 1. The word projection, here meaning a tuple ∂ : Hn → Hn−1 and i : Hn−1 → Hn of Hopf algebra morphisms, such that ∂ i = id. Given a Hopf algebra projection (∂ : I → H, i) in a braided monoidal category 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 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.