Fundamental theorem of Poisson $(A,H)$-Hopf module for weak Hopf algebras

TL;DR

This paper generalizes the fundamental theorem of Poisson Hopf modules to weak Hopf algebras with bijective antipodes, expanding the theoretical framework and exploring projectivity in the related module category.

Contribution

It extends the fundamental theorem of Poisson Hopf modules to weak Hopf algebras and investigates their relative projectivity.

Findings

Generalization of the fundamental theorem to weak Hopf algebras

Establishment of relative projectivity in Poisson Hopf modules

Enhanced understanding of module category structures

Abstract

Let $H$ be a weak Hopf algebra with a bijective antipode and $A$ an $H$-comodule Poisson algebra. In this paper, we mainly generalize the fundamental theorem of Poisson Hopf modules to the case of weak Hopf algebras. Besides we will deduce the relative projectivity in the category of Poisson Hopf module.