Yetter-Drinfeld modules for group-cograded Hopf quasigroups

TL;DR

This paper introduces the concept of p-Yetter-Drinfeld quasimodules over group-cograded Hopf quasigroups and demonstrates their categorical properties, including braided structures under certain conditions.

Contribution

It extends the theory of Yetter-Drinfeld modules to group-cograded Hopf quasigroups and establishes their categorical structures.

Findings

The category of Yetter-Drinfeld quasimodules is a crossed category.

The subcategory of Yetter-Drinfeld modules is a braided crossed category.

Categorical structures depend on the antipode being bijective.

Abstract

Let $H$ be a crossed group-cograded Hopf quasigroup. We first introduce the notion of $p$-Yetter-Drinfeld quasimodule over $H$. If the antipode of $H$ is bijective, we show that the category $\mathscr Y\mathscr D\mathscr Q(H)$ of Yetter-Drinfeld quasimodules over $H$ is a crossed category, and the subcategory $\mathscr Y\mathscr D(H)$ of Yetter-Drinfeld modules is a braided crossed category.