Comonoidal W*-Morita equivalence for von Neumann bialgebras

TL;DR

This paper develops a theory connecting Galois co-objects and comonoidal W*-Morita equivalence for von Neumann bialgebras, showing invariance of quantum group properties and introducing projective corepresentations for constructing equivalences.

Contribution

It introduces Galois co-objects and comonoidal W*-Morita equivalence for von Neumann bialgebras, establishing invariance of quantum group properties and proposing a new construction method.

Findings

Invariance of quantum group properties under Morita equivalence

Introduction of projective corepresentations for von Neumann bialgebras

Construction of Galois co-objects via corepresentations

Abstract

A theory of Galois co-objects for von Neumann bialgebras is introduced. This concept is closely related to the notion of comonoidal W*-Morita equivalence between von Neumann bialgebras, which is a Morita equivalence taking the comultiplication structure into account. We show that the property of `being a von Neumann algebraic quantum group' (i.e. `having invariant weights') is preserved under this equivalence relation. We also introduce the notion of a projective corepresentation for a von Neumann bialgebra, and show how it leads to a construction method for Galois co-objects and comonoidal W*-Morita equivalences.