Factorizations of Hopf quasigroups

TL;DR

This paper introduces the concept of factorization in Hopf quasigroups and characterizes when a Hopf quasigroup can be expressed as a double cross product of two others, under certain conditions.

Contribution

It defines factorization in Hopf quasigroups and establishes an equivalence with double cross product structures when antipodes are isomorphisms.

Findings

Factorization in Hopf quasigroups is characterized by double cross products.

A Hopf quasigroup admits a factorization iff it is isomorphic to a double cross product.

The antipodes being isomorphisms is a key condition for the main result.

Abstract

In this paper we introduce the notion of factorization in the Hopf quasigroup setting and we prove that, if $A$ and $H$ are Hopf quasigroups such that their antipodes are isomorphisms, a Hopf quasigroup $X$ admits a factorization as $X=AH$ iff $X$ is isomorphic to a double cross product $A\bowtie H$ as Hopf quasigroups.