01391nas a2200109 4500008004100000245005300041210005000094260001300144520105400157100001901211856005101230 2014 en d00aA modular spectral triple for κ-Minkowski space0 amodular spectral triple for κMinkowski space bElsevier3 aWe present a spectral triple for κ-Minkowski space in two dimensions. Starting from an algebra naturally associated to this space, a Hilbert space is built using a weight which is invariant under the κ-Poincaré algebra. The weight satisfies a KMS condition and its associated modular operator plays an important role in the construction. This forces us to introduce two ingredients which have a modular flavour: the first is a twisted commutator, used to obtain a boundedness condition for the Dirac operator, and the second is a weight replacing the usual operator trace, used to measure the growth of the resolvent of the Dirac operator. We show that, under some assumptions related to the symmetries and the classical limit, there is a unique Dirac operator and automorphism such that the twisted commutator is bounded. Then, using the weight mentioned above, we compute the spectral dimension associated to the spectral triple and find that is equal to the classical dimension. Finally we briefly discuss the introduction of a real structure.1 aMatassa, Marco uhttp://urania.sissa.it/xmlui/handle/1963/3489500656nas a2200121 4500008004100000245008200041210006900123260001000192520017100202653002900373100001900402856011300421 2014 en d00aNon-commutative integration for spectral triples associated to quantum groups0 aNoncommutative integration for spectral triples associated to qu bSISSA3 aThis thesis is dedicated to the study of non-commutative integration, in the sense of
spectral triples, for some non-commutative spaces associated to quantum groups.10aNon-commutative geometry1 aMatassa, Marco uhttps://www.math.sissa.it/publication/non-commutative-integration-spectral-triples-associated-quantum-groups00844nas a2200109 4500008004100000245005200041210005200093260002900145520049000174100001900664856005100683 2014 en d00aQuantum dimension and quantum projective spaces0 aQuantum dimension and quantum projective spaces bInstitute of Mathematics3 aWe show that the family of spectral triples for quantum projective spaces introduced by D'Andrea and Dbrowski, which have spectral dimension equal to zero, can be reconsidered as modular spectral triples by taking into account the action of the element K2por its inverse. The spectral dimension computed in this sense coincides with the dimension of the classical projective spaces. The connection with the well known notion of quantum dimension of quantum group theory is pointed out.1 aMatassa, Marco uhttp://urania.sissa.it/xmlui/handle/1963/3476402150nas a2200157 4500008004100000245008500041210006900126260003000195520152900225653003001754100003001784700002301814700001901837700002001856856011601876 2012 en d00aDeformed Lorentz symmetry and relative locality in a curved/expanding spacetime0 aDeformed Lorentz symmetry and relative locality in a curvedexpan bAmerican Physical Society3 aThe interest of part of the quantum-gravity community in the possibility of\r\nPlanck-scale-deformed Lorentz symmetry is also fueled by the opportunities for testing the relevant scenarios with analyses, from a signal-propagation perspective, of observations of bursts of particles from cosmological distances. In this respect the fact that so far the implications of deformed Lorentz symmetry have been investigated only for flat (Minkowskian) spacetimes represents a very significant limitation, since for propagation over cosmological distances the curvature/expansion of spacetime is evidently tangible. We here provide a significant step toward filling this gap by exhibiting an explicit example of Planck-scale-deformed relativistic symmetries of a spacetime with constant rate of expansion (deSitterian). Technically we obtain the first ever example of a relativistic theory of worldlines of particles with 3 nontrivial relativistic invariants: a large speed scale (\"speed-of-light scale\"), a large distance scale (inverse of the \"expansion-rate scale\"), and a large momentum scale (\"Planck scale\"). We address some of the challenges that had obstructed success for previous attempts by exploiting the recent understanding of the connection between deformed Lorentz symmetry and relativity of spacetime locality. We also offer a preliminary analysis of the differences between the scenario we here propose and the most studied scenario for broken (rather than deformed) Lorentz symmetry in expanding spacetimes.10aDoubly special relativity1 aAmelino-Camelia, Giovanni1 aMarciano, Antonino1 aMatassa, Marco1 aRosati, Giacomo uhttps://www.math.sissa.it/publication/deformed-lorentz-symmetry-and-relative-locality-curvedexpanding-spacetime