Abstract: The reality condition for spectral triples is a noncommutative generalization of the charge conjugation for Dirac spinors, but not always it is satisfied on interesting examples. Motivated by conformal deformations of spectral triples and a spectral triple construction on quantum cones, we propose a new twisted reality condition.

Abstract: t is well-known that the Poisson-Lie group dual of a compact semi-simple Lie group G with its standard Poisson-Lie structure can be identified with the solvable part AN of the Iwasawa decomposition of its complexification. In the setting of quantum groups, this entails that the dual of the standard q-deformation of G can be seen as a q-deformation of AN. While this statement has been made rigorous by Drinfeld within the setting of formal series Hopf algebras, a corresponding general statement in the operator algebraic setting seems lacking at the moment. In this talk, we will discuss in detail the simplest case of G equal to SU(2). We show how the function algebras on the duals of the quantum SU(2) groups fit naturally into a continuous field of C*-algebras with classical limit the function algebra on the group AN, in this case the group of special upper triangular 2 by 2-matrices with positive diagonal. The global C*-algebra of sections can be described as a crossed product of a classical space by a partial automorphism (in the sense of Exel). We next show compatibility with the coproduct structure. This is joint work with M. Floré.

Abstract: Our main theorem is that the pullback of an associated noncommutative vector bundle induced by an equivariant map of quantum principal bundles is a noncommutative vector bundle associated via the same finite-dimensional representation of the structural quantum group. On the level of K_0-groups of vector bundles, we realize the induced map by the pullback of explicit matrix idempotents. Finally, we construct quantum quaternionic projective spaces together with noncommutative tautological quaternionic line bundles and their duals. As a key application of the main theorem, we show that these bundles are stably non-trivial as noncommutative complex vector bundles.