# Regularity of twisted spectral triples and pseudodifferential calculi

## TL;DR

This paper explores the regularity condition for twisted spectral triples, establishing its equivalence to a compatible pseudodifferential calculus, and provides examples from quantum groups and analysis of zeta functions.

## Contribution

It introduces a framework for constructing pseudodifferential calculi for twisted spectral triples using algebraic conditions on the twisting.

## Key findings

- Pseudodifferential calculus admits asymptotic expansion under certain conditions.
- Examples from quantum group theory illustrate the framework.
- Analysis of zeta functions and twisted residues on differential operators.

## Abstract

We investigate the regularity condition for twisted spectral triples. This condition is equivalent to the existence of an appropriate pseudodifferential calculus compatible with the spectral triple. A natural approach to obtain such a calculus is to start with a twisted algebra of abstract differential operators, in the spirit of Higson. Under an appropriate algebraic condition on the twisting, we obtain a pseudodifferential calculus which admits an asymptotic expansion, similarly to the untwisted case. We present some examples coming from the theory of quantum groups. Finally we discuss zeta functions and the residue (twisted) traces on differential operators.