2020/03/04 by Emir, Kadir, Paseka, Jan
#16T05 #18D05 #18D10 #18G30 #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.2003.02058
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 ∂ \colon Hn → Hn-1 and i \colon Hn-1 → Hn of Hopf algebra morphisms, such that ∂ i = id. Given a Hopf algebra projection (∂ \colon I → H,i) in a braided monoidal category \mathfrakC, 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 \mathfrakC. 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.