Nonassociative constructions from inverse property quasigroups

Abstract

The notion of a Hopf algebra has been generalized many times by weakening certain properties; we introduce Hopf quasigroups which weaken the associativity of the algebra. Hopf quasigroups are coalgebras with a nonassociative product satisfying certain conditions with the antipode re ecting the properties of classical inverse property quasigroups. The de nitions and properties of Hopf quasigroups are dualized to obtain a theory of Hopf coquasigroups, or `algebraic quasigroups'. In this setting we are able to study the coordinate algebra over a quasigroup, and in particular the 7-sphere. One particular class of Hopf quasigroups is obtained by taking a bicrossproduct of a subgroup and a set of coset representatives, in much the same way that Hopf algebras are obtained from matched pairs of groups. Through this construction the bicrossproduct can also be given the structure of a quasi-Hopf algebra. We adapt the theory of Hopf algebras to Hopf (co)quasigroups, de ning integrals and Fourier transformations on these objects. This leads to the expected properties of separable and Frobenius Hopf coquasigroups and notions of (co)semisimplicity.