Partition C*-algebras II - Links to Compact Matrix Quantum Groups

TL;DR

This paper explores how partition C*-algebras can often be endowed with Hopf algebra and compact matrix quantum group structures, providing algebraic criteria for their relation to known quantum groups and extending the theory of easy quantum groups.

Contribution

It establishes algebraic conditions under which partition C*-algebras can be associated with quantum group structures, bypassing Tannaka-Krein duality and expanding the classification of quantum groups.

Findings

Many partition C*-algebras correspond to quantum subgroups of Wang's free orthogonal quantum group.

Even with generalized categories of partitions, the resulting quantum groups often remain within the Banica-Speicher class.

Discussion of potential non-unitary Banica-Speicher quantum groups.

Abstract

In a recent article, we gave a definition of partition C*-algebras. These are universal C*-algebras based on algebraic relations which are induced from partitions of sets. In this follow up article, we show that often we can associate a Hopf algebra structure to partition C*-algebras, and also a compact matrix quantum group structure. This follows the lines of Banica and Speicher's approach to quantum groups; however, we access them in a more algebraic way circumventing Tannaka-Krein duality. We give criteria when these quantum groups are quantum subgroups of Wang's free orthogonal quantum group. As a consequence, we see that even if we start with (generalized) categories of partitions which do not contain the pair partitions, in many cases we do not go beyond the class of Banica-Speicher quantum groups (aka easy quantum groups). However, we also discuss possible non-unitary Banica-Speicher quantum groups.