Quantum Inverse Semigroups

TL;DR

This paper introduces quantum inverse semigroups as a linearized generalization of inverse semigroups, exploring their connections to Hopf algebras, weak Hopf algebras, and Hopf categories, and establishing new examples related to Hopf algebroids.

Contribution

It defines quantum inverse semigroups, extends the classical concept, and links them to various algebraic structures like Hopf algebras and Hopf algebroids.

Findings

Several new examples of quantum inverse semigroups are presented.

Connections between quantum inverse semigroups and Hopf algebroids are established.

A generalized notion of local bisections for Hopf algebroids is introduced.

Abstract

In this work, the notion of a quantum inverse semigroup is introduced as a linearized generalization of inverse semigroups. Beyond the algebra of an inverse semigroup, which is the natural example of a quantum inverse semigroup, several other examples of this new structure are presented in different contexts, those are related to Hopf algebras, weak Hopf algebras, partial actions and Hopf categories. Finally, a generalized notion of local bisections are defined for commutative Hopf algebroids over a commutative base algebra giving rise to new examples of quantum inverse semigroups associated to Hopf algebroids in the same sense that inverse semigroups are related to groupoids.