Space of linear differential operators on the real line as a module over the Lie algebra of vector fields

TL;DR

This paper investigates the module structures of spaces of linear differential operators on the real line under the action of the Lie algebra of vector fields, identifying when these structures are isomorphic for different tensor density weights.

Contribution

It classifies the isomorphism classes of differential operator modules over the Lie algebra of vector fields on the real line, revealing specific conditions for equivalence based on the order and tensor density degree.

Findings

For third-order operators, modules are isomorphic for all but specific density weights.

For order four and higher, modules are isomorphic if the sum of their density weights equals one.

The paper provides explicit criteria for module isomorphisms depending on operator order and density parameters.

Abstract

Let ${\cal D}^k$ be the space of $k$-th order linear differential operators on ${\bf R}$: $A=a_k(x)\frac{d^k}{dx^k}+\cdots+a_0(x)$. We study a natural 1-parameter family of $\Diff(\bf R)$- (and $\Vect(\bf R)$)-modules on ${\cal D}^k$. (To define this family, one considers arguments of differential operators as tensor-densities of degree $\lambda$.) In this paper we solve the problem of isomorphism between $\Diff(\bf R)$-module structures on ${\cal D}^k$ corresponding to different values of $\lambda$. The result is as follows: for $k=3$ $\Diff(\bf R)$-module structures on ${\cal D}^3$ are isomorphic to each other for every values of $\lambda\not=0,\;1,\;{1\over 2},\;{1\over 2}\pm \frac{\sqrt 21}{6}$, in this case there exists a unique (up to a constant) intertwining operator $T:{\cal D}^3\to{\cal D}^3$. In the higher order case $(k\geq 4)$ $\Diff(\bf R)$-module structures on ${\cal D}^k$ corresponding to two different values of the degree: $\lambda$ and $\lambda^{\prime}$, are isomorphic if and only if $\lambda+\lambda^{\prime}=1$.