Quantum $\mathrm{SO}(3)$ groups and quantum group actions on $M_2$

Sołtan, Piotr M.

doi:10.4171/jncg/48

Cited by 34 publications

(28 citation statements)

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“…It may be mentioned here that in a very recent article [19], P.M. Soltan has characterized SO μ (3) as the universal compact quantum group acting on the finite dimensional C * -algebra M 2 (C) such that the action preserves a functional ω μ defined in [19]. In the classical case, we have three equivalent descriptions of SO(3): (a) as a quotient of SU (2), (b) as the group of (orientation preserving) isometries of S 2 , and (c) as the automorphism group of M 2 .…”

Section: Introductionmentioning

confidence: 93%

Quantum isometry groups of the Podles spheres

Bhowmick

Goswami

2010

Journal of Functional Analysis

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Section: Introductionmentioning

confidence: 93%

Quantum isometry groups of the Podles spheres

Bhowmick

Goswami

2010

Journal of Functional Analysis

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“…then a pair (M 2 (C), ) is called a compact quantum group [33,28]. It is known [33] that for given any compact quantum group there exists a unique Haar state w.r.t.…”

Section: Preliminariesmentioning

confidence: 99%

“…Moreover, we note that the considered quadratic operators are related to quantum groups introduced in [33]. Certain class of quantum groups on M 2 (C) was investigated in [28].…”

Section: Introductionmentioning

confidence: 99%

On quantum quadratic operators of M2(C) and their dynamics

Mukhamedov

Akın

Temir

et al. 2011

Journal of Mathematical Analysis and Applications

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“…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].…”

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confidence: 99%

Quantum Isometries of the Finite Noncommutative Geometry of the Standard Model

Bhowmick

D’Andrea

Da̧browski

2011

Commun. Math. Phys.

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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...

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Quantum $\mathrm{SO}(3)$ groups and quantum group actions on $M_2$

Cited by 34 publications

References 16 publications

Quantum isometry groups of the Podles spheres

Quantum isometry groups of the Podles spheres

On quantum quadratic operators of M2(C) and their dynamics

Quantum Isometries of the Finite Noncommutative Geometry of the Standard Model

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