Symmetries of modules of differential operators

TL;DR

This paper characterizes the symmetries of modules of differential operators acting on tensor densities on the circle and the line, revealing their algebraic structure and differences between compact and non-compact cases.

Contribution

It explicitly determines the algebra of symmetries for modules of differential operators on $S^1$ and $\,\mathbb{R}$, a novel analysis of their structure and differences.

Findings

Symmetry algebras are fully characterized for modules on $S^1$ and $\,\mathbb{R}$.

Differences between compact and non-compact cases are identified.

Results provide insight into the structure of differential operator modules under diffeomorphisms.

Abstract

Let ${\cal F}\_\lambda(S^1)$ be the space of tensor densities of degree (or weight) $\lambda$ on the circle $S^1$. The space ${\cal D}^k\_{\lambda,\mu}(S^1)$ of $k$-th order linear differential operators from ${\cal F}\_\lambda(S^1)$ to ${\cal F}\_\mu(S^1)$ is a natural module over $\mathrm{Diff}(S^1)$, the diffeomorphism group of $S^1$. We determine the algebra of symmetries of the modules ${\cal D}^k\_{\lambda,\mu}(S^1)$, i.e., the linear maps on ${\cal D}^k\_{\lambda,\mu}(S^1)$ commuting with the $\mathrm{Diff}(S^1)$-action. We also solve the same problem in the case of straight line $\mathbb{R}$ (instead of $S^1$) and compare the results in the compact and non-compact cases.