Weak Partial Representations

TL;DR

This paper introduces the concept of partial representations for weak Hopf algebras, constructs a universal algebra that encapsulates these representations, and explores their algebraic structures and applications.

Contribution

It defines partial representations of weak Hopf algebras, constructs the universal algebra $H_{par}^w$, and demonstrates its structure as a Hopf algebroid and quantum inverse semigroup.

Findings

$H_{par}^w$ factorizes partial representations via algebra morphisms

$H_{par}^w$ is isomorphic to a partial smash product

$H_{par}^w$ has a Hopf algebroid and quantum inverse semigroup structure

Abstract

We introduce the notion of partial representation of a weak Hopf algebra. We present the universal algebra $H_{par}^w$, which factorizes these partial representations by algebra morphisms. Also, it is shown that $\Hp$ is isomorphic to a partial smash product, that it has the structure of a Hopf algebroid and also that it can be endowed with a quantum inverse semigroup structure. Moreover, it is shown that the algebra objects in the module category over $H_{par}^w$ correspond to symmetrical partial module algebras.