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.

JF - Noncommutative Analysis, Operator Theory and Applications PB - Springer International Publishing CY - Cham SN - 978-3-319-29116-1 UR - https://doi.org/10.1007/978-3-319-29116-1_1 ER - TY - JOUR T1 - Quantum gauge symmetries in noncommutative geometry Y1 - 2014 A1 - Jyotishman Bhowmick A1 - Francesco D'Andrea A1 - Biswarup Krishna Das A1 - Ludwik Dabrowski AB - 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). PB - European Mathematical Society Publishing House UR - http://urania.sissa.it/xmlui/handle/1963/34897 U1 - 35182 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Quantum Isometries of the finite noncommutative geometry of the Standard Model JF - Commun. Math. Phys. 307:101-131, 2011 Y1 - 2011 A1 - Jyotishman Bhowmick A1 - Francesco D'Andrea A1 - Ludwik Dabrowski AB - 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. PB - Springer UR - http://hdl.handle.net/1963/4906 U1 - 4688 U2 - Mathematics U3 - Mathematical Physics U4 - -1 ER - TY - JOUR T1 - Dirac Operators on Quantum Projective Spaces JF - Comm. Math. Phys. 295 (2010) 731-790 Y1 - 2010 A1 - Francesco D'Andrea A1 - Ludwik Dabrowski AB - 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. UR - http://hdl.handle.net/1963/2131 U1 - 2112 U2 - Mathematics U3 - Mathematical Physics ER -