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.

%B Noncommutative Analysis, Operator Theory and Applications %I Springer International Publishing %C Cham %P 1–25 %@ 978-3-319-29116-1 %G eng %U https://doi.org/10.1007/978-3-319-29116-1_1 %R 10.1007/978-3-319-29116-1_1 %0 Journal Article %D 2014 %T Quantum gauge symmetries in noncommutative geometry %A Jyotishman Bhowmick %A Francesco D'Andrea %A Biswarup Krishna Das %A Ludwik Dabrowski %X We 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). %I European Mathematical Society Publishing House %G en %U http://urania.sissa.it/xmlui/handle/1963/34897 %1 35182 %2 Mathematics %4 1 %$ Submitted by gfeltrin@sissa.it (gfeltrin@sissa.it) on 2015-11-04T16:59:52Z No. of bitstreams: 1 preprint2014.pdf: 652206 bytes, checksum: d452b46b1c285be861389dcab56e1f6e (MD5) %R 10.4171/JNCG/161 %0 Journal Article %J Commun. Math. Phys. 307:101-131, 2011 %D 2011 %T Quantum Isometries of the finite noncommutative geometry of the Standard Model %A Jyotishman Bhowmick %A Francesco D'Andrea %A Ludwik Dabrowski %X 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 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. %B Commun. Math. Phys. 307:101-131, 2011 %I Springer %G en %U http://hdl.handle.net/1963/4906 %1 4688 %2 Mathematics %3 Mathematical Physics %4 -1 %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2011-10-25T07:42:04Z\\nNo. of bitstreams: 1\\n1009.2850v3.pdf: 383426 bytes, checksum: 9d28d3070f7e7c39ec9486f40fd4f13b (MD5) %R 10.1007/s00220-011-1301-2 %0 Journal Article %J Comm. Math. Phys. 295 (2010) 731-790 %D 2010 %T Dirac Operators on Quantum Projective Spaces %A Francesco D'Andrea %A Ludwik Dabrowski %X We 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 0After 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. %B J. Math. Phys. 47 (2006) 062105 %G en_US %U http://hdl.handle.net/1963/2131 %1 2112 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-09-18T11:48:16Z\\nNo. of bitstreams: 1\\n0503012v4.pdf: 270606 bytes, checksum: b96bcb17609481fbf7d542681a2c0cfa (MD5) %R 10.1063/1.2204808