00912nas a2200229   4500008004100000020002200041245004000063210004000103260004400143300001100187520024800198100002100446700002400467700002000491700001800511700002000529700002200549700001900571700002000590700002400610856004800634       2016                        eng d  a978-3-319-29116-100aPimsner Algebras and Circle Bundles0 aPimsner Algebras and Circle Bundles  aChambSpringer International Publishing  a1–253 a<p>We report on the connections between noncommutative principal circle bundles, Pimsner algebras and strongly graded algebras. We illustrate several results with examples of quantum weighted projective and lens spaces and θ-deformations.</p>1 aArici, Francesca1 aD'Andrea, Francesco1 aLandi, Giovanni1 aAlpay, Daniel1 aCipriani, Fabio1 aColombo, Fabrizio1 aGuido, Daniele1 aSabadini, Irene1 aSauvageot, Jean-Luc  uhttps://doi.org/10.1007/978-3-319-29116-1_101462nas a2200145   4500008004100000245005600041210005600097260005100153520096300204100002501167700002401192700002701216700002201243856005101265       2014                        en  d00aQuantum gauge symmetries in noncommutative geometry0 aQuantum gauge symmetries in noncommutative geometry  bEuropean Mathematical Society Publishing House3 aWe discuss generalizations of the notion of i) the group of unitary elements of a (real or complex) finite-dimensional C*-algebra, ii) gauge transformations and iii) (real) automorphisms in the framework of compact quantum group theory and spectral triples. The quantum analogue of these groups are defined as universal (initial) objects in some natural categories. After proving the existence of the universal objects, we discuss several examples that are of interest to physics, as they appear in the noncommutative geometry approach to particle physics: in particular, the C*-algebras M n(R), Mn(C) and Mn(H), describing the finite noncommutative space of the Einstein-Yang-Mills systems, and the algebras A F = C H M3 (C) and Aev = H H M4(C), that appear in Chamseddine-Connes derivation of the Standard Model of particle physics coupled to gravity. As a byproduct, we identify a "free" version of the symplectic group Sp.n/ (quaternionic unitary group).1 aBhowmick, Jyotishman1 aD'Andrea, Francesco1 aDas, Biswarup, Krishna1 aDabrowski, Ludwik  uhttp://urania.sissa.it/xmlui/handle/1963/3489700860nas a2200133   4500008004100000245008400041210006900125260001300194520041200207100002500619700002400644700002200668856003600690       2011                        en  d00aQuantum Isometries of the finite noncommutative geometry of the Standard  Model0 aQuantum Isometries of the finite noncommutative geometry of the   bSpringer3 aWe 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 x 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.1 aBhowmick, Jyotishman1 aD'Andrea, Francesco1 aDabrowski, Ludwik  uhttp://hdl.handle.net/1963/490600681nas a2200109   4500008004300000245004900043210004900092520034800141100002400489700002200513856003600535       2010                        en_Ud  00aDirac Operators on Quantum Projective Spaces0 aDirac Operators on Quantum Projective Spaces3 aWe construct a family of self-adjoint operators D_N which have compact resolvent and bounded commutators with the coordinate algebra of the quantum projective space CP_q(l), for any l>1 and 0<q<1. They provide 0^+ dimensional equivariant even spectral triples. If l is odd and N=(l+1)/2, the spectral triple is real with KO-dimension 2l mod 8.1 aD'Andrea, Francesco1 aDabrowski, Ludwik  uhttp://hdl.handle.net/1963/360601110nas a2200121   4500008004300000245008200043210006900125520069200194100002400886700002200910700002000932856003600952       2008                        en_Ud  00aThe Isospectral Dirac Operator on the 4-dimensional Orthogonal Quantum Sphere0 aIsospectral Dirac Operator on the 4dimensional Orthogonal Quantu3 aEquivariance under the action of Uq(so(5)) is used to compute the left regular and (chiral) spinorial representations of the algebra of the quantum Euclidean 4-sphere S^4_q. These representations are the constituents of a spectral triple on this sphere with a Dirac operator which is isospectral to the canonical one of the spin structure of the round undeformed four-sphere and which gives metric dimension four for the noncommutative geometry. Non-triviality of the geometry is proved by pairing the associated Fredholm module with an `instanton\\\' projection. A real structure which satisfies all required properties modulo a suitable ideal of `infinitesimals\\\' is also introduced.1 aD'Andrea, Francesco1 aDabrowski, Ludwik1 aLandi, Giovanni  uhttp://hdl.handle.net/1963/256700576nas a2200121   4500008004300000245006400043210006000107520018500167100002400352700002200376700002000398856003600418       2008                        en_Ud  00aThe Noncommutative Geometry of the Quantum Projective Plane0 aNoncommutative Geometry of the Quantum Projective Plane3 aWe study the spectral geometry of the quantum projective plane CP^2_q. In particular, we construct a Dirac operator which gives a 0^+ summable triple, equivariant under U_q(su(3)).1 aD'Andrea, Francesco1 aDabrowski, Ludwik1 aLandi, Giovanni  uhttp://hdl.handle.net/1963/254800855nas a2200133   4500008004300000245005000043210005000093520045800143100002400601700002200625700002000647700001800667856003600685       2007                        en_Ud  00aDirac operators on all Podles quantum spheres0 aDirac operators on all Podles quantum spheres3 aWe construct spectral triples on all Podles quantum spheres. These noncommutative geometries are equivariant for a left action of $U_q(su(2))$ and are regular, even and of metric dimension 2. They are all isospectral to the undeformed round geometry of the 2-sphere. There is also an equivariant real structure for which both the commutant property and the first order condition for the Dirac operators are valid up to infinitesimals of arbitrary order.1 aD'Andrea, Francesco1 aDabrowski, Ludwik1 aLandi, Giovanni1 aWagner, Elmar  uhttp://hdl.handle.net/1963/217701143nas a2200121   4500008004100000245005700041210005700098260001000155520076800165653002800933100002400961856003600985       2007                        en  d00aNoncommutative geometry and quantum group symmetries0 aNoncommutative geometry and quantum group symmetries  bSISSA3 aIt is a widespread belief that mathematics originates from the desire to understand (and eventually to formalize) some aspects of the real world. Quoting [Man07],   «we are doing mathematics in order to understand, create, and handle things, and perhaps this understanding is mathematics»  . Let me thus begin with a brief discussion of the physical ideas that motivated the development of Noncommutative Geometry and Quantum Group Theory - the areas of mathematics to which this dissertation belongs. Some physicists believe, and Einstein himself expressed this view in [Ein98a], that physics progresses in stages: there is no `final\\\' theory of Nature, but simply a sequence of theories which provide more and more accurate descriptions of the real world...10aNoncommutative geometry1 aD'Andrea, Francesco  uhttp://hdl.handle.net/1963/526901941nas a2200133   4500008004300000245006300043210005900106520150700165100002501672700003001697700002001727700002401747856003601771       2006                        en_Ud  00aA cyclic integral on k-Minkowski noncommutative space-time0 acyclic integral on kMinkowski noncommutative spacetime3 aWe examine some alternative possibilities for an action functional for $\\\\kappa$-Minkowski noncommutative spacetime, with an approach which should be applicable to other spacetimes with coordinate-dependent commutators of the spacetime coordinates ($[x_\\\\mu,x_\\\\nu]=f_{\\\\mu,\\\\nu}(x)$). Early works on $\\\\kappa$-Minkowski focused on $\\\\kappa$-Poincar\\\\\\\'e covariance and the dependence of the action functional on the choice of Weyl map, renouncing to invariance under cyclic permutations of the factors composing the argument of the action functional. A recent paper (hep-th/0307149), by Dimitrijevic, Jonke, Moller, Tsouchnika, Wess and Wohlgenannt, focused on a specific choice of Weyl map and, setting aside the issue of $\\\\kappa$-Poincar\\\\\\\'e covariance of the action functional, introduced in implicit form a cyclicity-inducing measure. We provide an explicit formula for (and derivation of) a choice of measure which indeed ensures cyclicity of the action functional, and we show that the same choice of measure is applicable to all the most used choices of Weyl map. We find that this ``cyclicity-inducing measure\\\'\\\' is not covariant under $\\\\kappa$-Poincar\\\\\\\'e transformations. We also notice that the cyclicity-inducing measure can be straightforwardly derived using a map which connects the $\\\\kappa$-Minkowski spacetime coordinates and the spacetime coordinates of a ``canonical\\\'\\\' noncommutative spacetime, with coordinate-independent commutators.1 aAgostini, Alessandra1 aAmelino-Camelia, Giovanni1 aArzano, Michele1 aD'Andrea, Francesco  uhttp://hdl.handle.net/1963/215800649nas a2200109   4500008004300000245005600043210005600099520030200155100002400457700002200481856003600503       2006                        en_Ud  00aLocal Index Formula on the Equatorial Podles Sphere0 aLocal Index Formula on the Equatorial Podles Sphere3 aWe discuss spectral properties of the equatorial Podles sphere. As a preparation we also study the `degenerate\\\' (i.e. $q=0$) case (related to the quantum disk). We consider two different spectral triples: one related to the Fock representation of the Toeplitz algebra and the isopectral one....1 aD'Andrea, Francesco1 aDabrowski, Ludwik  uhttp://hdl.handle.net/1963/178201217nas a2200097   4500008004300000245004300043210004200086520093100128100002401059856003601083       2006                        en_Ud  00aSpectral geometry of k-Minkowski space0 aSpectral geometry of kMinkowski space3 a<p>After recalling Snyder's idea of using vector fields over a smooth manifold as "coordinates on a noncommutative space", we discuss a two dimensional toy-model whose "dual" noncommutative coordinates form a Lie algebra: this is the well known $\kappa$-Minkowski space. We show how to improve Snyder's idea using the tools of quantum groups and noncommutative geometry. We find a natural representation of the coordinate algebra of $\kappa$-Minkowski as linear operators on an Hilbert space study its "spectral properties" and discuss how to obtain a Dirac operator for this space. We describe two Dirac operators. The first is associated with a spectral triple. We prove that the cyclic integral of M. Dimitrijevic et al. can be obtained as Dixmier trace associated to this triple. The second Dirac operator is equivariant for the action of the quantum Euclidean group, but it has unbounded commutators with the algebra.</p>1 aD'Andrea, Francesco  uhttp://hdl.handle.net/1963/2131