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The 27-dimensional Hopf algebra A(F), defined by the exact sequence of quantum groups A(SL(2, C))
We introduce noncommutative algebras A q of quantum 4-spheres S 4 q , with q ∈ R, defined via a suspension of the quantum group SU q (2), and a quantum instanton bundle described by a selfadjoint idempotent e ∈ Mat 4 (A q), e 2 = e = e *. Contrary to what happens for the classical case or for the noncommutative instanton constructed in , the first… (More)
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)
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… (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 2n (B) and K 0 (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… (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 mod-ulo compact operators. On the equatorial Podle´s sphere we construct U q (su(2))-equivariant Dirac operator and real structure which satisfy these modified properties .
We discuss the local index formula of Connes–Moscovici for the isospectral noncom-mutative 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.
We construct spectral triples on all Podle´s quantum spheres S 2 qt. These noncom-mutative 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 sphere S 2. There is also an equivariant real structure for which both the commutant property… (More)
The exact sequence of " coordinate-ring " Hopf algebras A(SL(2, C)) F r → A(SL q (2)) → A(F) determined by the Frobenius map F r, and the same way obtained exact sequence of (quantum) Borel subgroups, are studied when q is a cubic root of unity. An A(SL(2, C))-linear splitting of A(SL q (2)) making A(SL(2, C)) a direct summand of A(SL q (2)) is constructed… (More)
Properties of metrics and pairs consisting of left and right connections are studied on the bimodules of differential 1-forms. Those bimodules are obtained from the derivation based calculus of an algebra of matrix valued functions, and an SL q (2, C)-covariant calculus of the quantum plane plane at a generic q and the cubic root of unity. It is shown that,… (More)