@article {doi:10.1063/1.5026311,
	title = {Spectral triples on the Jiang-Su algebra},
	journal = {Journal of Mathematical Physics},
	volume = {59},
	number = {5},
	year = {2018},
	pages = {053507},
	doi = {10.1063/1.5026311},
	url = {https://doi.org/10.1063/1.5026311},
	author = {Jacopo Bassi and Ludwik Dabrowski}
}
@article {2014,
	title = {Dirac operators on noncommutative principal circle bundles},
	number = {International Journal of Geometric Methods in Modern Physics;volume 11; issue 1; article number 1450012;},
	year = {2014},
	publisher = {World Scientific Publishing},
	abstract = {We study spectral triples over noncommutative principal U(1)-bundles of arbitrary dimension and a compatibility condition between the connection and the Dirac operator on the total space and on the base space of the bundle. Examples of low-dimensional noncommutative tori are analyzed in more detail and all connections found that are compatible with an admissible Dirac operator. Conversely, a family of new Dirac operators on the noncommutative tori, which arise from the base-space Dirac operator and a suitable connection is exhibited. These examples are extended to the theta-deformed principal U(1)-bundle S 3 θ {\textrightarrow} S2.},
	doi = {10.1142/S0219887814500121},
	url = {http://urania.sissa.it/xmlui/handle/1963/35125},
	author = {Andrzej Sitarz and Alessandro Zucca and Ludwik Dabrowski}
}
@article {2014,
	title = {Quantum gauge symmetries in noncommutative geometry},
	number = {Journal of Noncommutative Geometry;volume 8; issue 2; pages 433-471;},
	year = {2014},
	publisher = {European Mathematical Society Publishing House},
	abstract = {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).},
	doi = {10.4171/JNCG/161},
	url = {http://urania.sissa.it/xmlui/handle/1963/34897},
	author = {Jyotishman Bhowmick and Francesco D{\textquoteright}Andrea and Biswarup Krishna Das and Ludwik Dabrowski}
}
@article {2013,
	title = {Curved noncommutative torus and Gauss--Bonnet},
	journal = {Journal of Mathematical Physics. Volume 54, Issue 1, 22 January 2013, Article number 013518},
	number = {arXiv:1204.0420v1;},
	year = {2013},
	note = {The article is composed of 13 pages and is recorded in PDF format},
	publisher = {American Institute of Physics},
	abstract = {We study perturbations of the flat geometry of the noncommutative
two-dimensional torus T^2_\theta (with irrational \theta). They are described
by spectral triples (A_\theta, \H, D), with the Dirac operator D, which is a
differential operator with coefficients in the commutant of the (smooth)
algebra A_\theta of T_\theta. We show, up to the second order in perturbation,
that the zeta-function at 0 vanishes and so the Gauss-Bonnet theorem holds. We
also calculate first two terms of the perturbative expansion of the
corresponding local scalar curvature.},
	keywords = {Geometry},
	doi = {10.1063/1.4776202},
	url = {http://hdl.handle.net/1963/7376},
	author = {Ludwik Dabrowski and Andrzej Sitarz}
}
@article {2013,
	title = {Dirac operator on spinors and diffeomorphisms},
	journal = {Classical and Quantum Gravity. Volume 30, Issue 1, 7 January 2013, Article number 015006},
	number = {arXiv:1209.2021v1;},
	year = {2013},
	note = {This article is composed of 13 pages and is recorded in PDF format},
	publisher = {IOP Publishing},
	abstract = {The issue of general covariance of spinors and related objects is
reconsidered. Given an oriented manifold $M$, to each spin structure $\sigma$
and Riemannian metric $g$ there is associated a space $S_{\sigma, g}$ of spinor
fields on $M$ and a Hilbert space $\HH_{\sigma, g}= L^2(S_{\sigma,
g},\vol{M}{g})$ of $L^2$-spinors of $S_{\sigma, g}$. The group $\diff{M}$ of
orientation-preserving diffeomorphisms of $M$ acts both on $g$ (by pullback)
and on $[\sigma]$ (by a suitably defined pullback $f^*\sigma$). Any $f\in
\diff{M}$ lifts in exactly two ways to a unitary operator $U$ from
$\HH_{\sigma, g} $ to $\HH_{f^*\sigma,f^*g}$. The canonically defined Dirac
operator is shown to be equivariant with respect to the action of $U$, so in
particular its spectrum is invariant under the diffeomorphisms.},
	keywords = {gravity},
	doi = {10.1088/0264-9381/30/1/015006},
	url = {http://hdl.handle.net/1963/7377},
	author = {Ludwik Dabrowski and Giacomo Dossena}
}
@article {2013,
	title = {Noncommutative circle bundles and new Dirac operators},
	journal = {Communications in Mathematical Physics. Volume 318, Issue 1, 2013, Pages 111-130},
	number = {arXiv:1012.3055v2;},
	year = {2013},
	note = {This article is composed of 25 pages and is recorded in PDF format},
	publisher = {Springer},
	abstract = {We study spectral triples over noncommutative principal U(1) bundles. Basing
on the classical situation and the abstract algebraic approach, we propose an
operatorial definition for a connection and compatibility between the
connection and the Dirac operator on the total space and on the base space of
the bundle. We analyze in details the example of the noncommutative three-torus
viewed as a U(1) bundle over the noncommutative two-torus and find all
connections compatible with an admissible Dirac operator. Conversely, we find a
family of new Dirac operators on the noncommutative tori, which arise from the
base-space Dirac operator and a suitable connection.},
	keywords = {Quantum principal bundles},
	doi = {10.1007/s00220-012-1550-8},
	url = {http://hdl.handle.net/1963/7384},
	author = {Ludwik Dabrowski and Andrzej Sitarz}
}
@article {2011,
	title = {Poincar{\'e} covariance and κ-Minkowski spacetime},
	journal = {Physics Letters A 375 (2011) 3496-3498},
	number = {SISSA;43/2010/FM},
	year = {2011},
	publisher = {Elsevier},
	abstract = {A fully Poincar{\'e} covariant model is constructed out of the k-Minkowski spacetime. Covariance is implemented by a unitary representation of the Poincar{\'e} group, and thus complies with the original Wigner approach to quantum symmetries. This provides yet another example (besides the DFR model), where Poincar{\'e} covariance is realised {\'a} la Wigner in the presence of two characteristic dimensionful parameters: the light speed and the Planck length. In other words, a Doubly Special Relativity (DSR) framework may well be realised without deforming the meaning of \\\"Poincar{\'e} covariance\\\".},
	doi = {10.1016/j.physleta.2011.08.011},
	url = {http://hdl.handle.net/1963/3893},
	author = {Ludwik Dabrowski and Gherardo Piacitelli}
}
@article {2011,
	title = {Product of real spectral triples},
	journal = {International Journal of Geometric Methods in Modern Physics 8 (2011) 1833-1848},
	number = {arXiv:1011.4456;},
	year = {2011},
	note = {Based on the talk given at the conference \\\"Noncommutative Geometry and Quantum Physics, Vietri sul Mare, Aug 31 - Sept 5, 2009\\\"},
	publisher = {World Scientific},
	abstract = {We construct the product of real spectral triples of arbitrary finite dimension (and arbitrary parity) taking into account the fact that in the even case there are two possible real structures, in the odd case there are two inequivalent representations of the gamma matrices (Clifford algebra), and in the even-even case there are two natural candidates for the Dirac operator of the product triple.},
	doi = {10.1142/S021988781100597X},
	url = {http://hdl.handle.net/1963/5510},
	author = {Ludwik Dabrowski and Giacomo Dossena}
}
@article {2011,
	title = {Quantum Isometries of the finite noncommutative geometry of the Standard  Model},
	journal = {Commun. Math. Phys. 307:101-131, 2011},
	number = {arXiv:1009.2850;},
	year = {2011},
	publisher = {Springer},
	abstract = {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.},
	doi = {10.1007/s00220-011-1301-2},
	url = {http://hdl.handle.net/1963/4906},
	author = {Jyotishman Bhowmick and Francesco D{\textquoteright}Andrea and Ludwik Dabrowski}
}
@article {2010,
	title = {Canonical k-Minkowski Spacetime},
	number = {SISSA;26/2010/FM},
	year = {2010},
	abstract = {A complete classification of the regular representations of the relations [T,X_j] = (i/k)X_j, j=1,...,d, is given. The quantisation of RxR^d canonically (in the sense of Weyl) associated with the universal representation of the above relations is intrinsically \\\"radial\\\", this meaning that it only involves the time variable and the distance from the origin; angle variables remain classical. The time axis through the origin is a spectral singularity of the model: in the large scale limit it is topologically disjoint from the rest. The symbolic calculus is developed; in particular there is a trace functional on symbols. For suitable choices of states localised very close to the origin, the uncertainties of all spacetime coordinates can be made simultaneously small at wish. On the contrary, uncertainty relations become important at \\\"large\\\" distances: Planck scale effects should be visible at LHC energies, if processes are spread in a region of size 1mm (order of peak nominal beam size) around the origin of spacetime.},
	url = {http://hdl.handle.net/1963/3863},
	author = {Gherardo Piacitelli and Ludwik Dabrowski}
}
@article {2010,
	title = {Dirac Operators on Quantum Projective Spaces},
	journal = {Comm. Math. Phys. 295 (2010) 731-790},
	number = {SISSA;17/2009/FM},
	year = {2010},
	note = {Quantum Algebra},
	abstract = {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 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.},
	doi = {10.1007/s00220-010-0989-8},
	url = {http://hdl.handle.net/1963/3606},
	author = {Francesco D{\textquoteright}Andrea and Ludwik Dabrowski}
}
@article {2010,
	title = {Lorentz Covariant k-Minkowski Spacetime},
	journal = {Phys. Rev. D 81 (2010) 125024},
	number = {SISSA;04/2010/FM},
	year = {2010},
	abstract = {In recent years, different views on the interpretation of Lorentz covariance of non commuting coordinates were discussed. Here, by a general procedure, we construct the minimal canonical central covariantisation of the k-Minkowski spacetime. We then show that, though the usual k-Minkowski spacetime is covariant under deformed (or twisted) Lorentz action, the resulting framework is equivalent to taking a non covariant restriction of the covariantised model. We conclude with some general comments on the approach of deformed covariance.},
	doi = {10.1103/PhysRevD.81.125024},
	url = {http://hdl.handle.net/1963/3829},
	author = {Ludwik Dabrowski and Michal Godlinski and Gherardo Piacitelli}
}
@article {2008,
	title = {The Isospectral Dirac Operator on the 4-dimensional Orthogonal Quantum Sphere},
	journal = {Comm. Math. Phys. 279 (2008) 77-116},
	number = {SISSA;95/2007/MP},
	year = {2008},
	abstract = {Equivariance 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 {\textquoteleft}instanton\\\' projection. A real structure which satisfies all required properties modulo a suitable ideal of {\textquoteleft}infinitesimals\\\' is also introduced.},
	doi = {10.1007/s00220-008-0420-x},
	url = {http://hdl.handle.net/1963/2567},
	author = {Francesco D{\textquoteright}Andrea and Ludwik Dabrowski and Giovanni Landi}
}
@article {2008,
	title = {The Noncommutative Geometry of the Quantum Projective Plane},
	journal = {Rev. Math. Phys. 20 (2008) 979-1006},
	number = {SISSA;100/2007/MP},
	year = {2008},
	abstract = {We 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)).},
	doi = {10.1142/S0129055X08003493},
	url = {http://hdl.handle.net/1963/2548},
	author = {Francesco D{\textquoteright}Andrea and Ludwik Dabrowski and Giovanni Landi}
}
@article {2007,
	title = {Dirac operators on all Podles quantum spheres},
	journal = {J. Noncomm. Geom. 1 (2007) 213-239},
	number = {arXiv.org;math/0606480v2},
	year = {2007},
	abstract = {We 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.},
	doi = {10.4171/JNCG/5},
	url = {http://hdl.handle.net/1963/2177},
	author = {Francesco D{\textquoteright}Andrea and Ludwik Dabrowski and Giovanni Landi and Elmar Wagner}
}
@article {2006,
	title = {Local Index Formula on the Equatorial Podles Sphere},
	number = {SISSA;85/2005/FM},
	year = {2006},
	abstract = {We discuss spectral properties of the equatorial Podles sphere. As a preparation we also study the {\textquoteleft}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....},
	doi = {10.1007/s11005-005-0047-1},
	url = {http://hdl.handle.net/1963/1782},
	author = {Francesco D{\textquoteright}Andrea and Ludwik Dabrowski}
}
@article {2005,
	title = {The Dirac operator on SU_q(2)},
	journal = {Commun. Math. Phys. 259 (2005) 729-759},
	number = {arXiv:math/0411609;},
	year = {2005},
	note = {v2: minor changes},
	publisher = {Springer},
	abstract = {We construct a 3^+ summable spectral triple (A(SU_q(2)),H,D) over the quantum group SU_q(2) which is equivariant with respect to a left and a right action of U_q(su(2)). The geometry is isospectral to the classical case since the
spectrum of the operator D is the same as that of the usual Dirac operator on
the 3-dimensional round sphere. The presence of an equivariant real structure J
demands a modification in the axiomatic framework of spectral geometry, whereby the commutant and first-order properties need be satisfied only modulo infinitesimals of arbitrary high order.},
	doi = {10.1007/s00220-005-1383-9},
	url = {http://hdl.handle.net/1963/4425},
	author = {Ludwik Dabrowski and Giovanni Landi and Andrzej Sitarz and Walter van Suijlekom and Joseph C. Varilly}
}
@article {2005,
	title = {The local index formula for SUq(2)},
	journal = {K-Theory 35 (2005) 375-394},
	number = {SISSA;01/2005/FM},
	year = {2005},
	abstract = {We discuss the local index formula of Connes-Moscovici for the isospectral noncommutative geometry that we have recently constructed on quantum SU(2). We work out the cosphere bundle and the dimension spectrum as well as the local cyclic cocycles yielding the index formula.},
	doi = {10.1007/s10977-005-3116-4},
	url = {http://hdl.handle.net/1963/1713},
	author = {Walter van Suijlekom and Ludwik Dabrowski and Giovanni Landi and Andrzej Sitarz and Joseph C. Varilly}
}
@article {2005,
	title = {The spectral geometry of the equatorial Podles sphere},
	journal = {C. R. Math. 340 (2005) 819-822},
	number = {SISSA;71/2004/FM},
	year = {2005},
	abstract = {We propose a slight modification of the properties of a spectral geometry a la Connes, which allows for some of the algebraic relations to be satisfied only modulo compact operators. On the equatorial Podles sphere we construct suq2-equivariant Dirac operator and real structure which satisfy these modified properties.},
	doi = {10.1016/j.crma.2005.04.003},
	url = {http://hdl.handle.net/1963/2275},
	author = {Ludwik Dabrowski and Giovanni Landi and Mario Paschke and Andrzej Sitarz}
}
@article {2003,
	title = {Non-linear sigma-models in noncommutative geometry: fields with values in finite spaces},
	journal = {Mod. Phys. Lett. A 18 (2003) 2371-2379},
	number = {arXiv.org;math/0309143v1},
	year = {2003},
	publisher = {World Scientific},
	abstract = {We study sigma-models on noncommutative spaces, notably on noncommutative tori. We construct instanton solutions carrying a nontrivial topological charge q and satisfying a Belavin-Polyakov bound. The moduli space of these instantons is conjectured to consists of an ordinary torus endowed with a complex structure times a projective space $CP^{q-1}$.},
	doi = {10.1142/S0217732303012593},
	url = {http://hdl.handle.net/1963/3215},
	author = {Ludwik Dabrowski and Thomas Krajewski and Giovanni Landi}
}
@article {2003,
	title = {Quantum spin coverings and statistics},
	journal = {J. Phys. A 36 (2003), no. 13, 3829-3840},
	number = {SISSA;97/2002/FM},
	year = {2003},
	publisher = {IOP Publishing},
	abstract = {SL_q(2) at odd roots of unity q^l =1 is studied as a quantum cover of the complex rotation group SO(3,C), in terms of the associated Hopf algebras of (quantum) polynomial functions. We work out the irreducible corepresentations, the decomposition of their tensor products and a coquasitriangular structure, with the associated braiding (or statistics). As an example, the case l=3 is discussed in detail.},
	doi = {10.1088/0305-4470/36/13/314},
	url = {http://hdl.handle.net/1963/1667},
	author = {Ludwik Dabrowski and Cesare Reina}
}
@article {2002,
	title = {Instanton algebras and quantum 4-spheres},
	journal = {Differential Geom. Appl. 16 (2002) 277-284},
	number = {arXiv.org;math/0101177v2},
	year = {2002},
	publisher = {Elsevier},
	abstract = {We study some generalized instanton algebras which are required to describe {\textquoteleft}instantonic complex rank 2 bundles\\\'. The spaces on which the bundles are defined are not prescribed from the beginning but rather are obtained from some natural requirements on the instantons. They turn out to be quantum 4-spheres $S^4_q$, with $q\\\\in\\\\IC$, and the instantons are described by self-adjoint idempotents e. We shall also clarify some issues related to the vanishing of the first Chern-Connes class $ch_1(e)$ and on the use of the second Chern-Connes class $ch_2(e)$ as a volume form.},
	doi = {10.1016/S0926-2245(02)00066-9},
	url = {http://hdl.handle.net/1963/3134},
	author = {Ludwik Dabrowski and Giovanni Landi}
}
@article {2001,
	title = {Dirac operator on the standard Podles quantum sphere},
	journal = {Noncommutative geometry and quantum groups (Warsaw 2001),49,Banach Center Publ., 61, Polish Acad.Sci., Warsaw,2003},
	number = {SISSA;98/2002/FM},
	year = {2001},
	publisher = {SISSA Library},
	url = {http://hdl.handle.net/1963/1668},
	author = {Ludwik Dabrowski and Andrzej Sitarz}
}
@article {2001,
	title = {Instantons on the Quantum 4-Spheres S^4_q},
	journal = {Comm. Math. Phys. 221 (2001) 161-168},
	number = {arXiv.org;math/0012103v2},
	year = {2001},
	publisher = {Springer},
	abstract = {We introduce noncommutative algebras $A_q$ of quantum 4-spheres $S^4_q$, with $q\\\\in\\\\IR$, defined via a suspension of the quantum group $SU_q(2)$, and a quantum instanton bundle described by a selfadjoint idempotent $e\\\\in \\\\Mat_4(A_q)$, $e^2=e=e^*$. Contrary to what happens for the classical case or for the noncommutative instanton constructed in Connes-Landi, the first Chern-Connes class $ch_1(e)$ does not vanish thus signaling a dimension drop. The second Chern-Connes class $ch_2(e)$ does not vanish as well and the couple $(ch_1(e), ch_2(e))$ defines a cycle in the $(b,B)$ bicomplex of cyclic homology.},
	doi = {10.1007/PL00005572},
	url = {http://hdl.handle.net/1963/3135},
	author = {Ludwik Dabrowski and Giovanni Landi and Tetsuya Masuda}
}
@article {2000,
	title = {A(SLq(2)) at roots of unity is a free module over A(SL(2))},
	journal = {Lett. Math. Phys., 2000, 52, 339},
	number = {SISSA;42/00/FM},
	year = {2000},
	publisher = {SISSA Library},
	doi = {10.1023/A:1007601131002},
	url = {http://hdl.handle.net/1963/1500},
	author = {Ludwik Dabrowski and Cesare Reina and Alessandro Zampa}
}
@article {2000,
	title = {A Remark on One-Dimensional Many-Body Problems with Point Interactions},
	journal = {Int. J. Mod. Phys. B 14 (2000) 721-727},
	number = {arXiv.org;quant-ph/0001089v1},
	year = {2000},
	publisher = {World Scientific},
	abstract = {The integrability of one dimensional quantum mechanical many-body problems with general contact interactions is extensively studied. It is shown that besides the pure (repulsive or attractive) $\\\\delta$-function interaction there is another singular point interactions which gives rise to a new one-parameter family of integrable quantum mechanical many-body systems. The bound states and scattering matrices are calculated for both bosonic and fermionic statistics.},
	doi = {10.1142/S0217979200000601},
	url = {http://hdl.handle.net/1963/3214},
	author = {Sergio Albeverio and Ludwik Dabrowski and Shao-Ming Fei}
}
@article {2000,
	title = {Some Properties of Non-linear sigma-Models in Noncommutative Geometry},
	journal = {Int. J. Mod. Phys. B 14 (2000) 2367-2382},
	number = {SISSA;158/99/FM},
	year = {2000},
	publisher = {SISSA Library},
	abstract = {We introduce non-linear $\\\\sigma$-models in the framework of noncommutative geometry with special emphasis on models defined on the noncommutative torus. We choose as target spaces the two point space and the circle and illustrate some characteristic features of the corresponding $\\\\sigma$-models. In particular we construct a $\\\\sigma$-model instanton with topological charge equal to 1. We also define and investigate some properties of a noncommutative analogue of the Wess-Zumino-Witten model.},
	doi = {10.1142/S0217979200001898},
	url = {http://hdl.handle.net/1963/1373},
	author = {Ludwik Dabrowski and Thomas Krajewski and Giovanni Landi}
}
@mastersthesis {1984,
	title = {Spin Structures and Global Conformal Transformations},
	year = {1984},
	school = {SISSA},
	keywords = {Spinors},
	url = {http://hdl.handle.net/1963/5854},
	author = {Ludwik Dabrowski}
}
