4 research outputs found
On the Absence of Continuous Symmetries for Noncommutative 3-Spheres
A large class of noncommutative spherical manifolds was obtained recently
from cohomology considerations. A one-parameter family of twisted 3-spheres was
discovered by Connes and Landi, and later generalized to a three-parameter
family by Connes and Dubois-Violette. The spheres of Connes and Landi were
shown to be homogeneous spaces for certain compact quantum groups. Here we
investigate whether or not this property can be extended to the noncommutative
three-spheres of Connes and Dubois-Violette. Upon restricting to quantum groups
which are continuous deformations of Spin(4) and SO(4) with standard
co-actions, our results suggest that this is not the case.Comment: 15 pages, no figure
The beat of a fuzzy drum: fuzzy Bessel functions for the disc
The fuzzy disc is a matrix approximation of the functions on a disc which
preserves rotational symmetry. In this paper we introduce a basis for the
algebra of functions on the fuzzy disc in terms of the eigenfunctions of a
properly defined fuzzy Laplacian. In the commutative limit they tend to the
eigenfunctions of the ordinary Laplacian on the disc, i.e. Bessel functions of
the first kind, thus deserving the name of fuzzy Bessel functions.Comment: 30 pages, 8 figure
q-Quaternions and q-deformed su(2) instantons
We construct (anti)instanton solutions of a would-be q-deformed su(2)
Yang-Mills theory on the quantum Euclidean space R_q^4 [the SO_q(4)-covariant
noncommutative space] by reinterpreting the function algebra on the latter as a
q-quaternion bialgebra. Since the (anti)selfduality equations are covariant
under the quantum group of deformed rotations, translations and scale change,
by applying the latter we can generate new solutions from the one centered at
the origin and with unit size. We also construct multi-instanton solutions. As
they depend on noncommuting parameters playing the roles of `sizes' and
`coordinates of the centers' of the instantons, this indicates that the moduli
space of a complete theory will be a noncommutative manifold. Similarly, gauge
transformations should be allowed to depend on additional noncommutative
parameters.Comment: Latex file, 39 pages. Final version appeared in JM