Conformal transformations and equivariance in unbounded KK-theory

TL;DR

This paper extends unbounded KK-theory to include conformal and quantum group symmetries, providing new unbounded representatives of KK-classes with a novel perturbation approach.

Contribution

It introduces a framework for conformally equivariant KK-theory with unbounded representatives and a new multiplicative perturbation method for constructing KK-classes.

Findings

Constructed unbounded representatives of the gamma-element for Lorentz groups.

Demonstrated conformal equivariance of the Podleś sphere spectral triple.

Developed a new perturbation theory for KK-class representation.

Abstract

We extend unbounded Kasparov theory to encompass conformal group and quantum group equivariance. This new framework allows us to treat conformal actions on both manifolds and noncommutative spaces. As examples, we present unbounded representatives of Kasparov's $\gamma$-element for the real and complex Lorentz groups and display the conformal $SL_q(2)$-equivariance of the standard spectral triple of the Podle\'s sphere. In pursuing descent for conformally equivariant cycles, we are led to a new framework for representing Kasparov classes. Our new representatives are unbounded, possess a dynamical quality, and also include known twisted spectral triples. We define an equivalence relation on these new representatives whose classes form an abelian group surjecting onto KK. The technical innovation which underpins these results is a novel multiplicative perturbation theory. By these means, we obtain Kasparov classes from the bounded transform with minimal side conditions.