[Paper Review] Rigidity of action of compact quantum groups III: the general case

Debashish Goswami|arXiv (Cornell University)|Jul 27, 2012

Advanced Operator Algebra Research3 references3 citations

TL;DR

This paper proves that any compact quantum group acting smoothly and faithfully on a smooth, compact, oriented, connected Riemannian manifold—such that the action induces a natural bimodule morphism on the cotangent bundle sections—must be commutative as a C*-algebra, hence isomorphic to $ C(G) $ for some compact group $ G $. Consequently, the quantum isometry group of such a manifold coincides with the classical isometry group $ C(ISO(M)) $, implying no genuine quantum symmetries exist for these manifolds.

ABSTRACT

If a compact quantum group acts faithfully and smoothly (in the sense of Goswami 2009) on a smooth, compact, oriented, connected Riemannian manifold such that the action induces a natural bimodule morphism on the module of sections of the co-tangent bundle, then it is proved that the quantum group is necessarily commutative as a $C^{*}$ algebra i.e. isomorphic with $ C(G)$ for some compact group $G$. From this, we deduce that the quantum isometry group of such a manifold M coincides with $C(ISO(M))$ where $ISO(M) $ is the group of (classical) isometries, i.e. there is no genuine quantum isometry of such a manifold.

Motivation & Objective

• To investigate whether compact quantum groups can act faithfully and smoothly on smooth, compact, oriented, connected Riemannian manifolds without being classical.
• To determine whether such actions can give rise to non-commutative quantum symmetries, particularly in the context of quantum isometry groups.
• To establish conditions under which the quantum group must be commutative, thereby ruling out genuine quantum symmetries for these geometric objects.

Proposed method

• Utilizes the notion of smooth action in the sense of Goswami (2009), requiring the action to be compatible with the smooth structure of the manifold.
• Analyzes the induced action on the module of sections of the cotangent bundle, requiring it to be a bimodule morphism.
• Applies representation-theoretic and C*-algebraic techniques to deduce that the quantum group must be commutative.
• Relies on the structure of the quantum group as a C*-algebra and the compatibility of the action with the Riemannian metric and orientation.
• Uses the fact that the action preserves the smooth structure and induces a well-defined morphism on differential forms.
• Concludes that the only possible such quantum groups are those isomorphic to $ C(G) $, i.e., classical compact groups.

Experimental results

Research questions

• RQ1 Can a non-commutative compact quantum group act smoothly and faithfully on a smooth, compact, oriented, connected Riemannian manifold?
• RQ2 Under what conditions does a quantum group action on such a manifold force the quantum group to be commutative?
• RQ3 Does the quantum isometry group of a compact Riemannian manifold with the given geometric constraints necessarily coincide with the classical isometry group?
• RQ4 Is there a mechanism that prevents genuine quantum isometries for such manifolds, based on the structure of the cotangent bundle action?
• RQ5 Can the bimodule morphism condition on the cotangent bundle sections serve as a sufficient criterion for classicality of the quantum group?

Key findings

• Any compact quantum group acting smoothly and faithfully on a smooth, compact, oriented, connected Riemannian manifold must be commutative as a C*-algebra.
• The quantum group is necessarily isomorphic to $ C(G) $ for some compact group $ G $, meaning it is classical.
• The action induces a natural bimodule morphism on the module of sections of the cotangent bundle, which is a key constraint.
• The quantum isometry group of such a manifold coincides with $ C(ISO(M)) $, the algebra of continuous functions on the classical isometry group.
• There are no genuine quantum isometries for these manifolds—no non-classical quantum symmetry beyond the classical group of isometries.
• The rigidity of the action structure forces the quantum group to collapse to a classical group, eliminating non-commutative quantum symmetry.