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

1996/02/04 by Hichem Gargoubi, Gargoubi, H., Valentin Ovsienko +1
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Differential Geometry Research #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.dg-ga/9602004

openalex publication_date 1996/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \cal D^k be the space of k-th order linear differential operators on \bf R: A=a_k(x)(d^k)/(dx^k)+⋯+a₀(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 λ.) In this paper we solve the problem of isomorphism between \Diff(\bf R)-module structures on \cal D^k corresponding to different values of λ. The result is as follows: for k=3 \Diff(\bf R)-module structures on \cal D³ are isomorphic to each other for every values of λ\not=0, 1, 1\over 2, 1\over 2± (√ 21)/(6), in this case there exists a unique (up to a constant) intertwining operator T:\cal D³→\cal D³. In the higher order case (k≥ 4) \Diff(\bf R)-module structures on \cal D^k corresponding to two different values of the degree: λ and λ^′, are isomorphic if and only if λ+λ^′=1.

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