We formulate a quantum group analogue of the group of orientation-preserving Riemannian isometries of a compact Riemannian spin manifold, more generally, of a (possibly R-twisted 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 μ (2) and S 2 μ,c are discussed.

In this follow-up of [4], where quantum isometry group of a noncommutative manifold has been defined, we explicitly compute such quantum groups for a number of classical as well as noncommutative manifolds including the spheres and the tori. It is also proved that the quantum isometry group of an isospectral deformation of a (classical or noncommutative) manifold is a suitable deformation of the quantum isometry group of the original (undeformed) manifold.

Abstract. Let Γ be a finitely generated discrete group. The standard spectral triple on the group C * -algebra C * (Γ) is shown to admit the quantum group of orientation preserving isometries. This leads to new examples of compact quantum groups. In particular the quantum isometry group of the C * -algebra of the free group on n-generators is computed and turns out to be a quantum group extension of the quantum permutation group A 2n of Wang. The quantum groups of orientation and real structure preserving isometries are also considered and construction of the Laplacian for the standard spectral triple on C * (Γ) discussed.From the early days of the theory of spectral triples a prominent role was played by an example of a spectral triple on a group C * -algebra C * (Γ), where Γ is a (discrete) group equipped with a fixed length function l : Γ → R + (it appears already in Connes' original paper [Co 2 ]). When Γ is finitely generated, properties of the natural word-length function on Γ and its associated spectral triple reflect deep combinatorial geometric aspects of Γ -so for example summability of the triple corresponds to the growth conditions, Rieffel regularity of the spectral triple is closely related to the Haagerup's Rapid Decay property ([OR]).Nowadays spectral triples are viewed as objects best suited to describe noncommutative (compact, Riemannian) manifolds. Thus, especially in the mathematical physics literature, the focus seems to be on investigating spectral triples on deformations of function algebras of classical spaces. In recent work of Goswami and the first named author ([Go 1 ], [BG 2 ], [Go 2 ]), the notion of a quantum isometry group of a spectral triple, understood as the universal compact quantum group acting on the corresponding noncommutative manifold in the way compatible with the spectral triple structure was introduced. It was shown that if the spectral triple is sufficiently well-behaved, the quantum isometry group exists, and in the classical situation the definition of 'compatible' actions given in [Go 1 ] describes precisely the relevant group actions preserving the Riemannian metric. Moreover Goswami and the first named author investigated quantum isometry groups of classical manifolds and their noncommutative deformations. In [BGS] together with Goswami we studied quantum isometry groups of spectral triples on AF C * -algebras introduced in [CI]. Already there the nature of the problem becomes more combinatorial and many connections with quantum permutation groups ([Wan], [BBC 1 ]), or more general quantum symmetry group of finite graphs ([Bic]) appear.The aim of this paper is the initiation of the study of quantum isometry groups of spectral triples on group C * -algebras. We begin by introducing basic notations Permanent address of the second named author:

We compute the quantum isometry group of the finite noncommutative geometry F describing the internal degrees of freedom in the Standard Model of particle physics. We show that this provides genuine quantum symmetries of the spectral triple corresponding to M × F , where M is a compact spin manifold. We also prove that the bosonic and fermionic part of the spectral action are preserved by these symmetries. IntroductionIn modern theoretical physics, symmetries play a fundamental role in determining the dynamics of a theory. In the two foremost examples, namely General Relativity and the Standard Model of elementary particles, the dynamics is dictated by invariance under diffeomorphisms and under local gauge transformations respectively. As a way to unify external (i.e. diffeomorphisms) and internal (i.e. local gauge) symmetries, Connes and Chamseddine proposed a model from Noncommutative Geometry [15] based on the product of the canonical commutative spectral triple of a compact Riemannian spin manifold M and a finite dimensional noncommutative one, describing an "internal" finite noncommutative space F [12,13,18,20]. In this picture, diffeomorphisms are realized as outer automorphisms of the algebra, while inner automorphisms correspond to the gauge transformations. Inner fluctuations of the Dirac operator are divided in two classes: the 1-forms coming from commutators with the Dirac operator of M give the gauge bosons, while the 1-forms coming from the Dirac operator of F give the Higgs field. The gravitational and bosonic part S b of the action is encoded in the spectrum of the gauged Dirac operator, which is invariant under isometries of the Hilbert space. The fermionic part S f is also defined in terms of the spectral data. The result is an Euclidean version of the Standard Model minimally coupled to gravity (cf. [20] and references therein).In his "Erlangen program", Klein linked the study of geometry with the analysis of its group of symmetries. Dealing with quantum geometries, it is natural to study quantum symmetries. The idea of using quantum group symmetries to understand the conceptual significance of the finite geometry F is mentioned in a final remark by Connes in [17]. Preliminary studies on the Hopf-algebra level appeared in [30,21,26]. Following Connes' suggestion, quantum automorphisms of finite-dimensional complex C * -algebras were introduced by Wang in [37,38] and later the quantum permutation groups of finite sets and graphs have been studied by a number of mathematicians, see e.g. [3,4,11,34]. These are compact quantum groups in the sense of Woronowicz [41]. The notion of compact quantum symmetries for "continuous" mathematical structures, like commutative and noncommutative manifolds (spectral triples), first appeared in [28], where quantum isometry groups were defined in terms of a Laplacian, followed by the definition of "quantum groups of orientation preserving isometries" based on the theory of spectral triples in [7], and on spectral triples with a real structure in [29]. Comp...

Abstract. Quantum isometry groups of spectral triples associated with approximately finite-dimensional C * -algebras are shown to arise as inductive limits of quantum symmetry groups of corresponding truncated Bratteli diagrams. This is used to determine explicitly the quantum isometry group of the natural spectral triple on the algebra of continuous functions on the middlethird Cantor set. It is also shown that the quantum symmetry groups of finite graphs or metric spaces coincide with the quantum isometry groups of the corresponding classical objects equipped with natural Laplacians.