The cyclic-homology Chern-Weil homomorphism for principal coactions (1712.09860v1)

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.