Equivariant representation theory for proper actions on discrete spaces

2025/08/20 by Rollier, Lukas
#FOS: Mathematics #Operator Algebras (math.OA) #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.2508.14991

Abstract

Starting from any proper action of any locally compact quantum group on any discrete quantum space, we show that its equivariant representation theory yields a concrete unitary 2-category of finite type Hilbert bimodules over the discrete quantum space, from which the quantum group and its action may be completely reconstructed as in a previous article by the author. In particular, this shows that any locally compact quantum group acting properly on a discrete quantum space must be an algebraic quantum group.

Citations

• Equivariant Tannaka-Krein reconstruction and quantum automorphism groups of discrete structures
• Approximation properties for dynamical W*-correspondences
• Groups with compact open subgroups and multiplier Hopf ^*-algebras
• The unitary implementation of a locally compact quantum group action
• Locally compact quantum groups in the von Neumann algebraic setting
• Universal C*-algebraic quantum groups arising from algebraic quantum groups
• On One-Parameter Unitary Groups in Hilbert Space

Related

• Equivariant Tannaka-Krein reconstruction and quantum automorphism groups of discrete structures
• Equivariant covering spaces of quantum homogeneous spaces
• Quantum group actions on rings and equivariant K-theory
• Equivariant Poincaré duality for quantum group actions
• Amenable fusion algebraic actions of discrete quantum groups on compact quantum spaces