35 research outputs found
Noncommutative Instantons on the 4-Sphere from Quantum Groups
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
Sigma^4_q as the algebra of coinvariants in S_q^7. 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 Sigma^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.Comment: 16 pages, LaTeX; revised versio
Bijectivity of the canonical map for the noncommutative instanton bundle
It is shown that the quantum instanton bundle introduced in Commun. Math.
Phys. 226, 419-432 (2002) has a bijective canonical map and is, therefore, a
coalgebra Galois extension.Comment: Latex, 12 pages. Published versio
Exponential mapping for non semisimple quantum groups
The concept of universal T matrix, recently introduced by Fronsdal and
Galindo in the framework of quantum groups, is here discussed as a
generalization of the exponential mapping. New examples related to
inhomogeneous quantum groups of physical interest are developed, the duality
calculations are explicitly presented and it is found that in some cases the
universal T matrix, like for Lie groups, is expressed in terms of usual
exponential series.Comment: 12 page
Quantum even spheres Sigma_q^2n from Poisson double suspension
We define even dimensional quantum spheres Sigma_q^2n that generalize to
higher dimension the standard quantum two-sphere of Podle's and the four-sphere
Sigma_q^4 obtained in the quantization of the Hopf bundle. The construction
relies on an iterated Poisson double suspension of the standard Podle's
two-sphere. The Poisson spheres that we get have the same symplectic foliation
consisting of a degenerate point and a symplectic plane and, after
quantization, have the same C^*-algebraic completion. We investigate their
K-homology and K-theory by introducing Fredholm modules and projectors.Comment: 13 pages; LaTeX 2