Noncommutative Finite-Dimensional Manifolds. I. Spherical Manifolds and Related Examples

@article{Connes2001NoncommutativeFM,
  title={Noncommutative Finite-Dimensional Manifolds. I. Spherical Manifolds and Related Examples},
  author={Alain Connes and Michel Dubois-Violette},
  journal={Communications in Mathematical Physics},
  year={2001},
  volume={230},
  pages={539-579}
}
Abstract: We exhibit large classes of examples of noncommutative finite-dimensional manifolds which are (non-formal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative three-dimensional spherical manifolds, a noncommutative version of the sphere S3 defined by basic K-theoretic equations. We find a 3-parameter family of deformations of the standard 3-sphere S3 and a corresponding 3-parameter deformation of the 4-dimensional Euclidean… 
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References

SHOWING 1-10 OF 66 REFERENCES
Noncommutative Manifolds, the Instanton Algebra¶and Isospectral Deformations
Abstract: We give new examples of noncommutative manifolds that are less standard than the NC-torus or Moyal deformations of ℝn. They arise naturally from basic considerations of noncommutative
Projective Modules over Higher-Dimensional Non-Commutative Tori
The non-commutative tori provide probably the most accessible interesting examples of non-commutative differentiable manifolds. We can identify an ordinary n-torus Tn with its algebra, C(Tn), of
A Short survey of noncommutative geometry
We give a survey of selected topics in noncommutative geometry, with some emphasis on those directly related to physics, including our recent work with Dirk Kreimer on renormalization and the
Gravity coupled with matter and the foundation of non-commutative geometry
We first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length elementds. Its unitary representations correspond
Twisting of Quantum Differentials and¶the Planck Scale Hopf Algebra
Abstract: We show that the crossed modules and bicovariant differential calculi on two Hopf algebras related by a cocycle twist are in 1-1 correspondence. In particular, for quantum groups which are
Regularity of the four dimensional Sklyanin algebra
The notation of a (non-commutative) regular, graded algebra is introduced in (AS). The results of that paper, combined with those in (ATV1, 2), give a complete description of the regular graded rings
Remarques sur le caractère algébrique du procédé pseudo-différentiel et de certaines de ses extensions
Let a be an associative algebra, and (D i ) i a family of commuting derivations. We show that the pseudo-differential process leads to associative algebraic structures for vector subspaces a⊂a. There
Compact matrix pseudogroups
The compact matrix pseudogroup is a non-commutative compact space endowed with a group structure. The precise definition is given and a number of examples is presented. Among them we have compact
On the theory of quantum groups
By using the results of S. L. Woronowicz, we show that for the twisted version of the classical compact matrix groups, the Hopf algebraAh of representative elements is isomorphic as a co-algebra to
DERIVATIONS ET CALCUL DIFFERENTIEL NON COMMUTATIF. II
We study canonical operation of the Lie algebra Der(#7B-A) of derivations of an algebra #7B-A with a unit in the graded differential algebra Ω(#7B-A). We introduce different graded differential
...
1
2
3
4
5
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