Hopf quivers and Nichols algebras in positive characteristic

Abstract

We apply a combinatorial formula of the first author and Rosso, for products in Hopf quiver algebras, to determine the structure of Nichols algebras. We illustrate this technique by explicitly constructing new examples of Nichols algebras in positive characteristic. We further describe the corresponding Radford biproducts and some liftings of these biproducts, which are new finite dimensional pointed Hopf algebras.

(Weak) Incidence Bialgebras of Monoidal Categories

“…Cibils and Rosso [7] (see also [8]) showed that a graded Hopf algebra structure on kQ endows Q 0 with a group and kQ 1 with a kQ 0 -Hopf bimodule structure. This in turn gives rise to what Cibils and Rosso call ramification data.…”

Weak Hopf Quivers, Clifford Monoids and Weak Hopf Algebras

“…Further classification of the algebra of the Hopf quiver and its dual version with its corresponding self-dual Hopf modules structures are given by Huang et al [14]. Cibils et al [6] obtained the structures of Nichols algebras in positive characteristic from the Hopf quiver algebras. We generalize this notion by giving a definition of the ramification data for a Clifford monoid.…”

On the structure and cohomology ring of connected Hopf algebras

“…In [27], Scherotzke classified finitedimensional pointed rank one Hopf algebras in positive characteristic which are generated by group-like and skew-primitive elements. Many Hopf algebras in positive characteristic also come from Nichols algebras which are of much interest, see [7,15].…”