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Journals and Conferences
Using principles of quantum symmetries we derive the algebraic part of the real spectral triple data for the standard Podleś quantum sphere: the equivariant representation, chiral grading γ, reality structure J and the Dirac operator, which has bounded commutators with the elements of the algebra and satisfies the first order condition. Mathematics Subject… (More)
We introduce noncommutative algebras Aq of quantum 4-spheres S 4 q , with q ∈ R, defined via a suspension of the quantum group SUq(2), and a quantum instanton bundle described by a selfadjoint idempotent e ∈ Mat 4(Aq), e 2 = e = e. Contrary to what happens for the classical case or for the noncommutative instanton constructed in , the first Chern-Connes… (More)
We construct spectral triples on all Podleś quantum spheres S2 qt. These noncommutative geometries are equivariant for a left action of Uq(su(2)) and are regular, even and of metric dimension 2. They are all isospectral to the undeformed round geometry of the sphere S2. There is also an equivariant real structure for which both the commutant property and… (More)
We reformulate the concept of connection on a Hopf-Galois extension B⊆ P in order to apply it in computing the Chern-Connes pairing between the cyclic cohomology HC(B) and K0(B). This reformulation allows us to show that a Hopf-Galois extension admitting a strong connection is projective and left faithfully flat. It also enables us to conclude that a strong… (More)
A list of known quantum spheres of dimension one, two and three is presented. M.S.C.: 81R60, 81R50, 20G42, 58B34, 58B32, 17B37.
Spectral triples on the q-deformed spheres of dimension two and three are reviewed. M.S.C.: 81R60, 81R50, 20G42, 58B34, 58B32, 17B37.
We study σ-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 CPq−1. Dedicated to… (More)
We translate the concept of the join of topological spaces to the language of C∗-algebras, replace the C∗-algebra of functions on the interval [0, 1] with evaluation maps at 0 and 1 by a unital C∗-algebra C with appropriate two surjections, and introduce the notion of the fusion of unital C∗-algebras. An appropriate modification of this construction yields… (More)
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 Podleś sphere we construct Uq(su(2))equivariant Dirac operator and real structure which satisfy these modified properties.
We introduce non-linear σ-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 σ-models. In particular we construct a σ-model instanton with topological charge equal… (More)