Regularity of twisted spectral triples and pseudodifferential calculi

Abstract

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