[Paper Review] Monoidal equivalence for locally compact quantum groups

Kenny De Commer|arXiv (Cornell University)|Apr 15, 2008

Advanced Operator Algebra Research31 references10 citations

TL;DR

This paper establishes monoidal equivalence between locally compact quantum groups via Galois objects and coactions, proving that twisting a quantum group by a unitary 2-cocycle yields another locally compact quantum group. It constructs dual quantum groups and equivalence bimodules that induce monoidal equivalences of their representation categories.

ABSTRACT

In this article we investigate the notion of monoidal equivalence for locally compact quantum groups, using the notion of a (bi-)Galois object. If (M,∆) is the von Neumann algebraic realization of a locally compact quantum group, a Galois object consists of a von Neumann algebra N together with an ergodic integrable coaction α of M on N, such that C ⊆ α(N) ⊆ N ⋊ M is the basic construction (in a canonical way). We show that we can associate to (N,α) another locally compact quantum group (P,∆P) and coaction γ of P on N, such that (N,γ) is a Galois object and such that γ and α commute. In fact, these objects are the building blocks of a special kind of quantum groupoid. We show that there are equivalence bimodules for all realizations of the duals of (M,∆) and (P,∆P), and that these provide monoidal equivalences between the categories of their ∗-representations. By these investigations, we are able to prove the following statement: any twisting of a locally compact quantum group by a unitary 2-cocycle is again a locally compact quantum group. We are also able to obtain a specific example of the following phenomenon: one can twist a compact quantum group by a 2-cocycle, and end up with a locally compact quantum group which is no longer compact.

Motivation & Objective

• To define and characterize monoidal equivalence for locally compact quantum groups using Galois objects and coactions.
• To construct a dual quantum group from a given Galois object and coaction, ensuring commutativity of coactions.
• To establish equivalence bimodules between the representation categories of dual quantum groups.
• To prove that twisting a locally compact quantum group by a unitary 2-cocycle results in another locally compact quantum group.
• To demonstrate that a compact quantum group can be twisted into a non-compact locally compact quantum group.

Proposed method

• Utilizes the von Neumann algebraic framework for locally compact quantum groups, with (M,∆) as the base quantum group.
• Defines a Galois object (N,α) as a von Neumann algebra N with an ergodic, integrable coaction α of M such that C ⊆ α(N) ⊆ N ⋊ M forms the basic construction.
• Constructs a new quantum group (P,∆P) and a coaction γ of P on N such that (N,γ) is a Galois object and γ commutes with α.
• Uses the structure of quantum groupoids built from these commuting coactions to derive equivalence bimodules.
• Applies the bimodules to establish monoidal equivalences between the categories of ∗-representations of the duals of (M,∆) and (P,∆P).
• Applies the framework to show that unitary 2-cocycle twists preserve the locally compact quantum group structure.

Experimental results

Research questions

• RQ1 Can monoidal equivalence between locally compact quantum groups be characterized via Galois objects and commuting coactions?
• RQ2 How can a dual quantum group be constructed from a given Galois object and coaction?
• RQ3 Do equivalence bimodules exist between the representation categories of dual quantum groups arising from Galois objects?
• RQ4 Is the operation of twisting a locally compact quantum group by a unitary 2-cocycle closed within the class of locally compact quantum groups?
• RQ5 Can a compact quantum group be twisted into a non-compact locally compact quantum group via a 2-cocycle?

Key findings

• Monoidal equivalence between the representation categories of dual quantum groups is established via equivalence bimodules derived from Galois objects.
• Any locally compact quantum group can be twisted by a unitary 2-cocycle to yield another locally compact quantum group.
• The construction yields a new quantum group (P,∆P) and a commuting coaction γ on the Galois object N, forming a quantum groupoid structure.
• A compact quantum group can be twisted by a 2-cocycle to produce a locally compact quantum group that is no longer compact.
• The basic construction in the sense of Jones is canonically realized in the inclusion C ⊆ α(N) ⊆ N ⋊ M for Galois objects.
• The framework provides a systematic method to generate monoidal equivalences between representation categories of dual quantum groups.