Quasigroupoids and weak Hopf quasigroups

Abstract: Quasigroupoids and weak Hopf quasigroups are non associative generalizations of groupoids and weak Hopf algebras. In this talk we will show that the category of finite quasigroupoids is equivalent to the one of pointed cosemisimple weak Hopf quasigroups over a given field K. As a consequence, we obtain a categorical equivalence between the categories of quasigroups, in the sense of Klim and Majid (i.e., loops with the inverse property), and the category of pointed cosemisimple Hopf quasigroups over K. On the other hand, in this talk we introduce the notion of exact factorization of a quasigroupoid and the notion of matched pair of quasigroupoids with common base. We prove that if (A,H) is a matched pair of quasigroupoids it is posible to construct a new quasigroupoid called the double cross product of A and H. Moreover, we show that, if a quasigroupoid B admits an exact factorization, there exists a matched pair of quasigroupoids (A,H) and an isomorphism of quasigroupoids between the double cross product of A and H and B. Finally, we show that every matched pair of quasigroupoids (A,H) induces, thanks to the quasigroupoid magma construction, a pair (K[A], K[H]) of weak Hopf quasigroups and a double crossed product of weak Hopf quasigroups isomorphic as weak Hopf quasigroups to the quasigroupoid magma of the double cross product gorupoid of A and H .

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar