Cleft and Galois extensions associated to a weak Hopf quasigroup

TL;DR

This paper introduces and compares the concepts of cleft and Galois extensions related to weak Hopf quasigroups, generalizing classical results for Hopf algebras and quasigroups, and establishing their equivalence under certain conditions.

Contribution

It defines cleft and Galois extensions for weak Hopf quasigroups and proves their equivalence, extending classical algebraic structures to a broader context.

Findings

Cleft and Galois extensions are equivalent under suitable conditions.

Classical results for (weak) Hopf algebras are recovered as special cases.

New definitions for extensions associated to Hopf quasigroups are established.

Abstract

In this paper we introduce the notions of cleft and Galois (with normal basis) extension associated to a weak Hopf quasigroup. We show that, under suitable conditions, both notions are equivalent. As a particular instance we recover the classical results for (weak) Hopf algebras. Moreover, taking into account that weak Hopf quasigroups generalize the notion of Hopf quasigroup, we obtain the definitions of cleft and Galois (with normal basis) extension associated to a Hopf quasigroup and we get the equivalence betwen these extensions in this setting.