Quantum quasigroups and loops

Quantum quasigroups and quantum loops are self-dual objects providing a general framework for the nonassociative extension of quantum group techniques. Bialgebra reducts of Hopf algebras are quantum loops, while sufficient conditions are given for quantum loop structure to augment to a Hopf algebra. The Moufang–Hopf algebras of Benkart et al., the Hopf quasigroups and coquasigroups of Klim–Majid, and the coassociative H-bialgebras of Pérez-Izquierdo (for instance, the universal enveloping algebras of Sabinin algebras), all form quantum loops. Other quantum quasigroups offer natural nonassociative extensions of Hopf algebra constructions: quasigroup algebras, dual quasigroup algebras, and quantum couples of groups with quasigroups. Further examples include an algebra of rooted binary trees, and an algebra of skein polynomials.