Partial $*$-algebraic quantum groups and Drinfeld doubles of partial compact quantum groups

TL;DR

This paper introduces partial algebraic quantum groups, generalizing partial compact quantum groups, and demonstrates that their Drinfeld doubles can be constructed as partial $*$-algebraic quantum groups, expanding the framework of quantum group theory.

Contribution

It defines partial algebraic quantum groups as a new class within weak multiplier Hopf algebras and constructs Drinfeld doubles in this setting, extending previous concepts.

Findings

Partial algebraic quantum groups generalize partial compact quantum groups.

Drinfeld doubles of partial compact quantum groups can be realized as partial $*$-algebraic quantum groups.

The framework connects weak multiplier Hopf algebras with quantum group doubles.

Abstract

We introduce a notion of partial algebraic quantum group. This is an important special case of a weak multiplier Hopf algebra with integrals, as introduced in the work of Van Daele and Wang. At the same time, it generalizes the notion of partial compact quantum group as introduced by De Commer and Timmermann. As an application, we show that the Drinfeld double of a partial compact quantum group can be defined as a partial $*$-algebraic quantum group.