Globalization and the biactegory of partial modules

TL;DR

This paper explores the structure of partial modules over Hopf algebras, showing they form a biactegory over global modules and analyzing their dilation and enrichment properties.

Contribution

It establishes that partial modules form a biactegory over global modules and links their dilation to Hom-objects in this enriched setting.

Findings

Partial modules form a biactegory over global modules.

Dilation of partial modules relates to Hom-objects from the monoidal unit.

Standard dilation for finite-dimensional pointed Hopf algebras is isomorphic to a specific Hom-object.

Abstract

We show that the category of partial modules over a Hopf algebra $H$ is a biactegory (a bimodule category) over the category of global $H$-modules. The corresponding enrichment of partial modules over global modules is described, and the close relation between the dilation of partial modules and Hom-objects arising from this enrichment is investigated. In particular, for finite-dimensional pointed Hopf algebras, the standard dilation of a partial module $M$ is isomorphic to the Hom-object from the monoidal unit to $M$.