Equational Quantum Quasigroups

As a self-dual framework to unify the study of quasigroups and Hopf algebras, quantum quasigroups are defined using a quantum analogue of the combinatorial approach to classical quasigroups, merely requiring invertibility of the left and right composites. In this paper, quantum quasigroups are redefined with a quantum analogue of the equational approach to classical quasigroups. Here, the left and right composites of auxiliary quantum quasigroups participate in diagrams whose commutativity witnesses the required invertibilities. Whenever the original and two auxiliary quantum quasigroups appear on an equal footing, the triality symmetry of the language of equational quasigroups is replicated. In particular, the problem arises as to when this triality emerges in the Hopf algebra context.