Equivalence classes of subquotients of pseudodifferential operator modules

TL;DR

This paper classifies the equivalence classes of subquotients of pseudodifferential operator modules over the line, revealing geometric conditions involving conics and cubics, and identifies cases with no non-trivial equivalences.

Contribution

It provides a detailed classification of subquotients of pseudodifferential operator modules, including geometric criteria for equivalence and new insights into length-specific cases.

Findings

2-parameter families of subquotients with given Jordan-Holder series

Geometric conditions involving pencils of conics and cubics for equivalence

No non-obvious equivalences for subquotients of length > 7

Abstract

Consider the spaces of pseudodifferential operators between tensor density modules over the line as modules of the Lie algebra of vector fields on the line. We compute the equivalence classes of various subquotients of these modules. There is a 2-parameter family of subquotients with any given Jordan-Holder composition series. In the critical case of subquotients of length 5, the equivalence classes within each non-resonant 2-parameter family are specified by the intersections of a pencil of conics with a pencil of cubics. In the case of resonant subquotients of length 4 with self-dual composition series, as well as of lacunary subquotients of lengths 3 and 4, equivalence is specified by a single pencil of conics. Non-resonant subquotients of length exceeding 7 admit no non-obvious equivalences. The cases of lengths 6 and 7 are unresolved.