Abstract:
I will begin this talk with an overview of Chris Plyley and David Riley's work on Hopf algebras with a $\mathbb{Z}_2$-graded algebra structure. (Note this is weaker than the super Hopf algebra concept, which requires the grading to respect the coalgebra).  Such a grading allows the authors to study Hopf algebras which act on algebras by both automorphisms and anti-automorphisms.  As a quasigroup theorist, my interests lie mainly in the paper's concluding section, in which Plyley and Riley apply oriented Hopf algebra actions to furnish an example of a Lie algebra graded by a nonassociative, quasigroup operation.  According to the authors, such "exotic" gradings were once conjectured to not exist.  I will also discuss prospects for broadening the class of quasigroups that can coact on algebras.