We describe an approach to the noncommutative instantons on the 4-sphere based on quantum group theory. We quantize the Hopf bundle S 7 → S 4 making use of the concept of quantum coisotropic subgroups. The analysis of the semiclassical Poisson-Lie structure of U (4) shows that the diagonal SU (2) must be conjugated to be properly quantized. The quantum coisotropic subgroup we obtain is the standard SU q (2); it determines a new deformation of the 4-sphere Σ 4 q as the algebra of coinvariants in S 7 q . We show that the quantum vector bundle associated to the fundamental corepresentation of SU q (2) is finitely generated and projective and we compute the explicit projector. We give the unitary representations of Σ 4 q , we define two 0-summable Fredholm modules and we compute the Chern-Connes pairing between the projector and their characters. It comes out that even the zero class in cyclic homology is non trivial.
We develop a quantum duality principle for coisotropic subgroups of a (formal) Poisson group and its dual: namely, starting from a quantum coisotropic subgroup (for a quantization of a given Poisson group) we provide functorial recipes to produce quantizations of the dual coisotropic subgroup (in the dual formal Poisson group). By the natural link between subgroups and homogeneous spaces, we argue a quantum duality principle for Poisson homogeneous spaces which are Poisson quotients, i.e. have at least one zero-dimensional symplectic leaf. As an application, we provide an explicit quantization of the homogeneous SL_n^*-space of Stokes matrices, with the Poisson structure given by Dubrovin and Ugaglia
ABSTRACT. The g-Laguerre polynomials correspond to an indeterminate moment problem. For explicit discrete non-N-extremal measures corresponding to Ramanujan's i^i-summation, we complement the orthogonal g-Laguerre polynomials to an explicit orthogonal basis for the corresponding L 2 -space. The dual orthogonal system consists of so-called big q-Bessel functions, which can be obtained as a rigorous limit of the orthogonal system of big g-Jacobi polynomials. Interpretations on the SU(1,1) and 12(2) quantum groups are discussed.
It is shown that the quantum instanton bundle introduced in [Commun. Math. Phys. 226 (2002) 419] has a bijective canonical map and is, therefore, a coalgebra Galois extension.
ABSTRACT. Motivated by examples obtained from conformal deformations of spectral triples and a spectral triple construction on quantum cones, we propose a new twisted reality condition for the Dirac operator.
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