A characteristic map for compact quantum groups

TL;DR

This paper establishes a connection between Hopf-cyclic cohomology of quantum enveloping algebras and twisted cyclic cohomology of quantum groups, illustrating this with the Podleś sphere example.

Contribution

It introduces a characteristic map linking Hopf-cyclic and twisted cyclic cohomology for compact quantum groups, with explicit examples.

Findings

The characteristic map exists for compact Lie groups and their quantum analogs.

The Schmüdgen-Wagner index cocycle is in the image of this map.

Application to the Podleś sphere demonstrates the map's relevance.

Abstract

We show that if $G$ is a compact Lie group and $\mathfrak{g}$ is its Lie algebra, then there is a map from the Hopf-cyclic cohomology of the quantum enveloping algebra $U_q(\mathfrak{g})$ to the twisted cyclic cohomology of quantum group algebra $\mathcal{O}(G_q)$. We also show that the Schm\"udgen-Wagner index cocycle associated with the volume form of the differential calculus on the standard Podle\'s sphere $\mathcal{O}(S^2_q)$ is in the image of this map.