Hopf-dihedral (co)homology and $L$-theory

TL;DR

This paper extends the Connes-Moscovici characteristic map to dihedral cases for Hopf *-algebras and uses it to identify new non-trivial $L$-theory classes, including a novel class for the Podleś sphere.

Contribution

It introduces a dihedral extension of the characteristic map and demonstrates its application in detecting new $L$-theory classes in Hopf-symmetric *-algebras.

Findings

Extended the characteristic map to dihedral Hopf *-algebras.

Detected a new $L$-class of the Podleś sphere.

Provided a method to identify non-trivial $L$-theory classes.

Abstract

We develop an appropriate dihedral extension of the Connes-Moscovici characteristic map for Hopf *-algebras. We then observe that one can use this extension together with the dihedral Chern character to detect non-trivial $L$-theory classes of a *-algebra that carry a Hopf symmetry over a Hopf *-algebra. Using our machinery we detect a previously unknown $L$-class of the standard Podle\'s sphere.