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Search Results: 1 - 10 of 277 matches for " Ludwik Dabrowski "
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The Garden of Quantum Spheres
Ludwik Dabrowski
Physics , 2002,
Abstract: A list of known quantum spheres of dimension one, two and three is presented.
Geometry of Quantum Spheres
Ludwik Dabrowski
Mathematics , 2005, DOI: 10.1016/j.geomphys.2005.04.003
Abstract: Spectral triples on the q-deformed spheres of dimension two and three are reviewed.
The local index formula for quantum SU(2)
Ludwik Dabrowski
Mathematics , 2006,
Abstract: The local index formula of Connes--Moscovici for the isospectral noncommutative geometry recently constructed on quantum SU(2) is discussed. The cosphere bundle and the dimension spectrum as well as the local cyclic cocycles yielding the index formula, are presented.
Spinors and Theta Deformations
Ludwik Dabrowski
Mathematics , 2008, DOI: 10.1134/S106192080903008X
Abstract: The construction due to Connes and Landi of Dirac operators on theta-deformed manifolds is recalled, stressing the aspect of spin structure. The description of Connes and Dubois-Violette is extended to arbitrary spin structure.
Towards a noncommutative Brouwer fixed-point theorem
Ludwik Dabrowski
Mathematics , 2015,
Abstract: We present some results and conjectures on a generalization to the noncommutative setup of the Brouwer fixed-point theorem from the Borsuk-Ulam theorem perspective.
Canonical k-Minkowski Spacetime
Ludwik Dabrowski,Gherardo Piacitelli
Physics , 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.
Left Regular Representation of $sl_q(3)$: Reduction and Intertwiners
Ludwik Dabrowski,Preeti Parashar
Physics , 1994, DOI: 10.1088/0305-4470/28/10/014
Abstract: Reduction of the left regular representation of quantum algebra $sl_q(3)$ is studied and ~$q$-difference intertwining operators are constructed. The irreducible representations correspond to the spaces of local sections of certain line bundles over the q-flag manifold.
Product of real spectral triples
Ludwik Dabrowski,Giacomo Dossena
Physics , 2010, DOI: 10.1142/S021988781100597X
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.
Dirac operator on spinors and diffeomorphisms
Ludwik Dabrowski,Giacomo Dossena
Physics , 2012, DOI: 10.1088/0264-9381/30/1/015006
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.
Noncommutative circle bundles and new Dirac operators
Ludwik Dabrowski,Andrzej Sitarz
Mathematics , 2010, DOI: 10.1007/s00220-012-1550-8
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.
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