We construct a noncommutative deformation of odd-dimensional spheres that preserves the natural partition of the (2N + 1)-dimensional sphere into (N + 1)many solid tori. This generalizes the case N = 1 referred to as the Heegaard quantum sphere. Our twisted odd-dimensional quantum sphere C∗-algebras are given as multipullback C∗-algebras. We prove that they are isomorphic to the universal C∗-al...

We report on some recent work on deformation of spaces, notably deformation of spheres, describing two classes of examples. The first class of examples consists of noncommutative manifolds associated with the so called θ-deformations which were introduced in [17] out of a simple analysis in terms of cycles in the (b,B)-complex of cyclic homology. These examples have non-trivial global features ...

Associated to the standard SUq(n) R-matrices, we introduce quantum spheres S q , projective quantum spaces CP n−1 q , and quantum Grassmann manifolds Gk(C n q ). These algebras are shown to be homogeneous spaces of standard quantum groups and are also quantum principle bundles in the sense of T. Brzeziński and S. Majid [1].

We introduce and study two new examples of noncommutative spheres: the half-liberated sphere, and the free sphere. Together with the usual sphere, these two spheres have the property that the corresponding quantum isometry group is “easy”, in the representation theory sense. We present as well some general comments on the axiomatization problem, and on the “untwisted” and “non-easy” case.

The (left coalgebra) line bundle associated to the quantum Hopf fibration of any quantum two-sphere is shown to be a finitely generated projective module. The corresponding projector is constructed and its monopole charge is computed. It is shown that the Dirac q-monopole connection on any quantum two-sphere induces the Grassmannian connection built with this projector.