Symmetry classes of quantum quasigroups

The theory of groups has a twofold symmetry, sending a group to its opposite. Groups invariant under the symmetry are abelian. The theory of quasigroups has a richer, sixfold symmetry, obtained by permuting the multiplication with its two divisions. The Sixfold Way identifies the various classes of quasigroups which are invariant under the respective subgroups of the symmetry group of the theory.

Quantum quasigroups provide a self-dual framework to unify the study of quasigroups and Hopf algebras. The goal of this paper is to classify the symmetry classes of quantum quasigroups. Corresponding to the Sixfold Way for classical quasigroups, we are able to identify a Sevenfold Way for general classes exhibiting a symmetry, and initiate a study of a fuller symmetry which holds for linear quantum quasigroups.