Existence and examples of quantum isometry groups for a class of compact metric spaces

Abstract

We formulate a definition of isometric action of a compact quantum group (CQG) on a compact metric space, generalizing Banica's definition for finite metric spaces. For metric spaces (X, d) which can be isometrically embedded in some Euclidean space, we prove the existence of a universal object in the category of the compact quantum groups acting isometrically on (X, d). In fact, our existence theorem applies to a larger class, namely for any compact metric space (X, d) which admits a one-to-one continuous map f : X → R n for some n such that d 0 (f (x), f (y)) = φ(d(x, y)) (where d 0 is the Euclidean metric) for some homeomorphism φ of R + .As concrete examples, we obtain Wang's quantum permutation group S + n and also the free wreath product of Z 2 by S + n as the quantum isometry groups for certain compact connected metric spaces constructed by taking topological joins of intervals in [13].

Quantum Rigidity of Negatively Curved Manifolds

“…The underlying metric space of a compact, connected Riemannian manifold with negative sectional curvature has no truly quantum symmetries. This is conjectured to be true for arbitrary compact connected Riemannian manifolds in [12] (see the discussion at the end of Section 4 in that paper). The curvature condition is crucial in the proof of the theorem, but along the way we develop some geometric techniques that one might hope are of some interest in their own right and could be useful in studying quantum isometries of more general manifolds.…”

“…As mentioned in the introduction, it is conjectured in [12] that all isometric actions on compact connected Riemannian manifolds are classical. The curvature condition in the present paper was certainly very helpful in achieving this goal, as for example Theorem 6.1 is no longer true in its absence.…”

“…[12,19,23] or the survey [17]). We will generally omit the word 'compact' and simply refer to these objects as 'quantum groups'.…”

“…One is now confronted with two possible notions of quantum isometry: the global one of [12] and the infinitesimal one of [4]. They are only conjecturally equivalent, with the global notion being weaker.…”

“…Banica first introduced in [3] the notion of isometric action of a quantum group on a finite metric space, and the concept was later extended to arbitrary compact metric spaces in [12]. In rough terms (to be made precise in Sect.…”