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Register a new userQuantum Group of Orientation preserving Riemannian Isometries
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We formulate a quantum group analogue of the group of orinetation-preserving Riemannian isometries of a compact Riemannian spin manifold, more generally, of a (possibly $R$-twisted in the sense of a paper of one of the authors, and of compact type) spectral triple. The main advantage of this formulation, which is directly in terms of the Dirac operator, is that it does not need the existence of any `good Laplacian as in our previous works on quantum isometry groups. Several interesting examples, including those coming from Rieffel-type deformation as well as the equivariant spectral triples on $SU_mu(2)$ and $S^2_{mu 0}$ are dicussed.
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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.
We formulate a quantum generalization of the notion of the group of Riemannian isometries for a compact Riemannian manifold, by introducing a natural notion of smooth and isometric action by a compact quantum group on a classical or noncommutative ma
nifold described by spectral triples, and then proving the existence of a universal object (called the quantum isometry group) in the category of compact quantum groups acting smoothly and isometrically on a given (possibly noncommutative) manifold satisfying certain regularity assumptions. In fact, we identify the quantum isometry group with the universal object in a bigger category, namely the category of `quantum families of smooth isometries, defined along the line of Woronowicz and Soltan. We also construct a spectral triple on the Hilbert space of forms on a noncommutative manifold which is equivariant with respect to a natural unitary representation of the quantum isometry group. We give explicit description of quantum isometry groups of commutative and noncommutative tori, and in this context, obtain the quantum double torus defined in cite{hajac} as the universal quantum group of holomorphic isometries of the noncommutative torus.
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 concise proof that large classes of optimal (constant curvature or Einstein) pseudo-Riemannian metrics are maximally symmetric within their conformal class.
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.
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