Noncommutative index theory for mirror quantum spheres

Noncommutative Balls and Mirror Quantum Spheres

Noncommutative analogues of n-dimensional balls are defined by repeated application of the quantum double suspension to the classical low-dimensional spaces. In the ‘even-dimensional’ case they correspond to the Twisted Canonical Commutation Relations of Pusz and Woronowicz. Then quantum spheres are constructed as double manifolds of noncommutative balls. Both C-algebras and polynomial algebras...

Quantum Isometries and Noncommutative Spheres

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.

Quantum symmetry groups of noncommutative spheres

We show that the noncommutative spheres of Connes and Landi are quantum homogeneous spaces for certain compact quantum groups. We give a general construction of homogeneous spaces which support noncommutative spin geometries.

Anomalies and Noncommutative Index Theory

These lectures are devoted to a description of anomalies in quantum field theory from the point of view of noncommutative geometry and topology. We will in particular introduce the basic methods of cyclic cohomology and explain the noncommutative counterparts of the Atiyah-Singer index theorem.

Equivariant KK-theory and noncommutative index theory