Non-existence of faithful isometric action of compact quantum groups on compact, connected Riemannian manifolds

Abstract

Suppose that a compact quantum group Q acts faithfully on a smooth, compact, connected manifold M , i.e. has a C * (co)-action α on C(M ), such that the action α is isometric in the sense of [10] for some Riemannian structure on M . We prove that Q must be commutative as a C * algebra i.e. Q ∼ = C(G) for some compact group G acting smoothly on M . In particular, the quantum isometry group of M (in the sense of [10]) coincides with C(ISO(M )).

Quantum Rigidity of Negatively Curved Manifolds

“…Structure-preserving actions of quantum groups on Riemannian manifolds were defined earlier in [4], and one striking phenomenon discovered in [13] is that compact, connected Riemannian manifolds have no truly quantum symmetries: any structure-preserving action by a quantum group on such a manifold factors through an isometric action of an ordinary compact group (we also say that the action is classical).…”

“…They are only conjecturally equivalent, with the global notion being weaker. Taking a cue from [13], one can then ask the same type of rigidity question: are all isometric quantum actions on the underlying metric space of a compact connected Riemannian manifold classical?…”

Actions of compact quantum groups

“…The above phenomenon of quantum groups acting on classical spaces is however much rarer than that of classical groups acting on quantum spaces, as shown by the work of D. Goswami and collaborators, see in particular [17].…”

Maximal torus theory for compact quantum groups

“…In view of [22], where the rigidity conjecture was proved in the general case, for any non-classical subgroup G ⊂ U + N , all this is obsolete. However, and here comes our point, we believe that the original philosophy in [2] can applied to a wider range of rigidity questions, where some of the tools from [22] might not be available. For a study of the half-classical case, based on [5], [11], we refer to the recent article [1].…”