Actions, quotients and lattices of locally compact quantum groups

TL;DR

This paper investigates property (T) in locally compact quantum groups, establishing permanence results, characterizing invariant weights, and introducing quantum lattices, thereby extending classical group properties to the quantum setting.

Contribution

It introduces a notion of lattice in LCQGs, proves property (T) lifts from lattices to groups, and characterizes invariant weights and states in quantum homogeneous spaces.

Findings

Property (T) is preserved under certain quantum group extensions.

Invariant weights on quantum homogeneous spaces are characterized.

Property (T) lifts from lattices to ambient quantum groups.

Abstract

We prove a number of property (T) permanence results for locally compact quantum groups under exact sequences and the presence of invariant states, analogous to their classical versions. Along the way we characterize the existence of invariant weights on quantum homogeneous spaces of quotient type, and relate invariant states for LCQG actions on von Neumann algebras to invariant vectors in canonical unitary implementations, providing an application to amenability. Finally, we introduce a notion of lattice in a locally compact quantum group, noting examples provided by Drinfeld doubles of compact quantum groups. We show that property (T) lifts from a lattice to the ambient LCQG, just as it does classically, thus obtaining new examples of non-classical, non-compact, non-discrete LCQGs with property (T).