Twisted entire cyclic cohomology, J-L-O cocycles and equivariant spectral triples

TL;DR

This paper develops a framework for twisted entire cyclic cohomology and twisted Chern characters in noncommutative geometry, demonstrating their invariance under quantum group actions, extending classical concepts with new operator-theoretic tools.

Contribution

It introduces twisted entire cyclic cohomology and twisted Chern characters based on twisted spectral data, generalizing existing noncommutative geometric invariants.

Findings

Constructs twisted spectral data from equivariant spectral triples.

Defines twisted Chern character invariant under quantum group actions.

Shows invariance contrasts with non-twisted Chern characters.

Abstract

We study the "quantized calculus" corresponding to the algebraic ideas related to "twisted cyclic cohomology" introduced in [KMT]. With very similar definitions and techniques as those used in [jlo], we define and study "twisted entire cyclic cohomology" and the "twisted Chern character" associated with an appropriate operator-theoretic data called "twisted spectral data", which consists of a spectral triple in the conventional sense of noncommutative geometry ([Con]) and an additional positive operator having specified properties. Furthermore, it is shown that given a spectral triple (in the conventional sense) which is equivariant under the action of a compact matrix pseudogroup, it is possible to obtain a canonical twisted spectral data and hence the corresponding (twisted) Chern character, which will be invariant under the action of the pseudogroup, in contrast to the fact that the Chern character coming from the conventional noncommutative geometry need not be invariant under the above action.