Categorical aspects of quantum groups: multipliers and intrinsic groups

TL;DR

This paper explores the categorical relationships between quantum groups, their multipliers, and intrinsic groups, establishing functorial correspondences and revealing the intrinsic group as the maximal classical subgroup.

Contribution

It introduces a functorial framework connecting multipliers and intrinsic groups of quantum groups, and characterizes the intrinsic group as the maximal classical subgroup within this setting.

Findings

The multiplier algebra assignment forms a functor between categories.

The intrinsic group can be realized as a class of multipliers.

The intrinsic group is the maximal classical quantum subgroup and is closed in the strong Vaes sense.

Abstract

We show that the assignment of the (left) completely bounded multiplier algebra $M_{cb}^l(L^1(\mathbb G))$ to a locally compact quantum group $\mathbb G$, and the assignment of the intrinsic group, form functors between appropriate categories. Morphisms of locally compact quantum groups can be described by Hopf $*$-homomorphisms between universal $C^*$-algebras, by bicharacters, or by special sorts of coactions. We show that the whole theory of completely bounded multipliers can be lifted to the universal $C^*$-algebra level, and that then the different pictures of both multipliers (reduced, universal, and as centralisers) and morphisms interact in extremely natural ways. The intrinsic group of a quantum group can be realised as a class of multipliers, and so our techniques immediately apply. We also show how to think of the intrinsic group using the universal $C^*$-algebra picture, and then, again, show how the differing views on the intrinsic group interact naturally with morphisms. We show that the intrinsic group is the "maximal classical" quantum subgroup of a locally compact quantum group, show that it is even closed in the strong Vaes sense, and that the intrinsic group functor is an adjoint to the inclusion functor from locally compact groups to quantum groups.