A new characterization of Kac-type discrete quantum groups

TL;DR

This paper characterizes discrete quantum groups and those of Kac type within the framework of symmetric monoidal dagger categories, providing new algebraic descriptions relevant to quantum logic and information theory.

Contribution

It introduces novel categorical characterizations of discrete quantum groups and Kac-type quantum groups using inversion relations and functions, confirming a conjecture and integrating them into category-internal algebra.

Findings

Discrete quantum groups characterized by an inversion relation

Kac-type quantum groups characterized by an inversion function

Confirms a conjecture about the structure of Kac-type quantum groups

Abstract

We obtain two related characterizations of discrete quantum groups and discrete quantum groups of Kac type as allegorical group objects in the symmetric monoidal dagger category of quantum sets and relations, of interest to quantum predicate logic and quantum information theory. Specifically, we characterize discrete quantum groups by the existence of an inversion relation and discrete quantum groups of Kac type by the existence of an inversion function. This confirms a conjectured description of discrete quantum groups of Kac type and brings them within the purview of category-internal universal algebra.