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

TL;DR

The paper proves that any faithful isometric action of a compact quantum group on a smooth, compact, connected Riemannian manifold must be classical, showing such quantum groups are actually commutative and coincide with classical isometry groups.

Contribution

It establishes that non-commutative (quantum) isometry groups cannot act faithfully and isometrically on classical manifolds, confirming they are always classical groups.

Findings

Quantum isometry groups are always classical for smooth, compact, connected manifolds.

Any faithful isometric quantum group action reduces to a classical group action.

Quantum symmetries in this setting do not extend beyond classical symmetries.

Abstract

Suppose that a compact quantum group $\clq$ acts faithfully on a smooth, compact, connected manifold $M$, i.e. has a $C^*$ (co)-action $\alpha$ on $C(M)$, such that the action $\alpha$ is isometric in the sense of \cite{Goswami} for some Riemannian structure on $M$. We prove that $\clq$ must be commutative as a $C^{\ast}$ algebra i.e. $\clq\cong C(G)$ for some compact group $G$ acting smoothly on $M$. In particular, the quantum isometry group of $M$ (in the sense of \cite{Goswami}) coincides with $C(ISO(M))$.