A Two Dimensional Adler-Manin Trace and Bi-singular Operators

TL;DR

This paper introduces a two-dimensional Adler-Manin trace for bi-singular pseudodifferential operators, generalizing the concept with algebra automorphisms and twisted derivations to expand noncommutative residue theory.

Contribution

It develops a novel two-dimensional Adler-Manin trace incorporating twists via algebra automorphisms, broadening the framework for bi-singular pseudodifferential operators.

Findings

Constructed a general algebra of formal twisted pseudodifferential symbols.

Defined a noncommutative residue in the twisted setting.

Provided examples illustrating the new trace and residue concepts.

Abstract

Motivated by the theory of bi-singular pseudodifferential operators, we introduce a two dimensional version of the Adler-Manin trace. Our construction is rather general in the sense that it involves a twist afforded by an algebra automorphism. That is, starting from an algebra equipped with an automorphism, two twisted derivations, and a twisted invariant trace, we construct an algebra of formal twisted pseudodifferential symbols and define a noncommutative residue. Also, we provide related examples.