We prove that, given any smooth action of a compact quantum group (in the sense of cite{rigidity}) on a compact smooth manifold satisfying some more natural conditions, one can get a Riemannian structure on the manifold for which the corresponding $C^infty(M)$-valued inner product on the space of one-forms is preserved by the action.
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. $clqcong C(G)$ 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 $C(ISO(M))$.
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.
Suppose that a compact quantum group Q acts faithfully and isomet- rically (in the sense of [10]) on a smooth compact, oriented, connected Riemannian manifold M . If the manifold is stably parallelizable then it is shown that the compact quantum group is necessarily commutative as a C ast algebra i.e. Q = C(G) for some compact group G. Using this, it is also proved that the quantum isometry group of Rieffel deformation of such manifold M must be a Rieffel-Wang deformation of C(ISO(M))
We formulate a definition of isometric action of a compact quantum group (CQG) on a compact metric space, generalizing Banicas 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 $f : X raro IR^n$ for some $n$ such that $d_0(f(x),f(y))=phi(d(x,y))$ (where $d_0$ is the Euclidean metric) for some homeomorphism $phi$ of $IR^+$. As concrete examples, we obtain Wangs quantum permutation group $cls_n^+$ and also the free wreath product of $IZ_2$ by $cls_n^+$ as the quantum isometry groups for certain compact connected metric spaces constructed by taking topological joins of intervals in cite{huang1}.
Let $G$ be one of the classical compact, simple, centre-less, connected Lie groups or rank $n$ with a maximal torus $T$, the Lie algebra $clg$ and let ${ E_i, F_i, H_i, i=1, ldots, n }$ be the standard set of generators corresponding to a basis of the root system. Consider the adjoint-orbit space $M={ {rm Ad}_g(H_1),~g in G }$, identified with the homogeneous space $G/L$ where $L={ g in G:~{rm Ad}_g(H_1)=H_1}$. We prove that the `coordinate functions ${ f_i, i=1, ldots, n }$, (where $f_i(g):=lambda_i({rm Ad}_g(H_1))$, ${ lambda_1, ldots, lambda_n}$ is basis of $clg^prime$) are `quadratically independent in the sense that they do not satisfy any nontrivial homogeneous quadratic relations among them. Using this, it is proved that there is no genuine compact quantum group which can act faithtully on $C(M)$ such that the action leaves invariant the linear span of the above cordinate functions. As a corollary, it is also shown that any compact quantum group having a faithful action on the noncommutative manifold obtained by Rieffel deformation of $M$ satisfying a similar `linearity condition must be a Rieffel-Wang type deformation of some compact group.
We prove that a compact quantum group with faithful Haar state which has a faithful action on a compact space must be a Kac algebra, with bounded antipode and the square of the antipode being identity. The main tool in proving this is the theory of ergodic quantum group action on $C^*$ algebras. Using the above fact, we also formulate a definition of isometric action of a compact quantum group on a compact metric space, generalizing the definition given by Banica for finite metric spaces, and prove for certain special class of metric spaces the existence of the universal object in the category of those compact quantum groups which act isometrically and are `bigger than the classical isometry group.
Given a spectral triple of compact type with a real structure in the sense of [Dabrowski L., J. Geom. Phys. 56 (2006), 86-107] (which is a modification of Connes original definition to accommodate examples coming from quantum group theory) and references therein, we prove that there is always a universal object in the category of compact quantum group acting by orientation preserving isometries (in the sense of [Bhowmick J., Goswami D., J. Funct. Anal. 257 (2009), 2530-2572]) and also preserving the real structure of the spectral triple. This gives a natural definition of quantum isometry group in the context of real spectral triples without fixing a choice of volume form as in [Bhowmick J., Goswami D., J. Funct. Anal. 257 (2009), 2530-2572].
We give a new sufficient condition on a spectral triple to ensure that the quantum group of orientation and volume preserving isometries defined in cite{qorient} has a $C^*$-action on the underlying $C^*$ algebra.
We prove that an arbitrary (not necessarily countably generated) Hilbert $G$-$cla$ module on a G-C^* algebra $cla$ admits an equivariant embedding into a trivial $G-cla$ module, provided G is a compact Lie group and its action on $cla$ is ergodic.
