The cyclic-homology Chern-Weil homomorphism for principal coactions

TL;DR

This paper develops a noncommutative version of the classical Chern-Weil homomorphism using cyclic homology, extending it to principal coactions and quantum groupoids, and proves its invariance under certain extensions.

Contribution

It introduces a cyclic-homology Chern-Weil homomorphism for principal coactions, extending classical concepts to noncommutative geometry and quantum groupoids.

Findings

Defined the cyclic-homology Chern-Weil homomorphism for principal coactions.

Proved isomorphism of cyclic homology groups under certain nilpotent extensions.

Utilized chain homotopy invariance in the context of quantum groupoids.

Abstract

We view the space of cotraces in the structural coalgebra of a principal coaction as a noncommutative counterpart of the classical Cartan model. Then we define the cyclic-homology Chern-Weil homomorphism by extending the Chern-Galois character from the characters of finite-dimensional comodules to arbitrary cotraces. To reduce the cyclic-homology Chern-Weil homomorphism to a tautological natural transformation, we replace the unital coaction-invariant subalgebra by its certain natural H-unital nilpotent extension (row extension), and prove that their cyclic-homology groups are isomorphic. In the proof, we use a chain homotopy invariance of complexes computing Hochschild, and hence cyclic homology, for arbitrary row extensions. In the context of the cyclic-homology Chern-Weil homomorphism, a row extension is provided by the Ehresmann-Schauenburg quantum groupoid with a nonstandard multiplication.