Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 101, 23 pages      arXiv:1902.01361      https://doi.org/10.3842/SIGMA.2019.101
Contribution to the Special Issue on Algebraic Methods in Dynamical Systems

Commuting Ordinary Differential Operators and the Dixmier Test

Emma Previato a, Sonia L. Rueda b and Maria-Angeles Zurro c
a) Boston University, USA
b) Universidad Politécnica de Madrid, Spain
c) Universidad Autónoma de Madrid, Spain

Received February 04, 2019, in final form December 23, 2019; Published online December 30, 2019

Abstract
The Burchnall-Chaundy problem is classical in differential algebra, seeking to describe all commutative subalgebras of a ring of ordinary differential operators whose coefficients are functions in a given class. It received less attention when posed in the (first) Weyl algebra, namely for polynomial coefficients, while the classification of commutative subalgebras of the Weyl algebra is in itself an important open problem. Centralizers are maximal-commutative subalgebras, and we review the properties of a basis of the centralizer of an operator $L$ in normal form, following the approach of K.R. Goodearl, with the ultimate goal of obtaining such bases by computational routines. Our first step is to establish the Dixmier test, based on a lemma by J. Dixmier and the choice of a suitable filtration, to give necessary conditions for an operator $M$ to be in the centralizer of $L$. Whenever the centralizer equals the algebra generated by $L$ and $M$, we call $L$, $M$ a Burchnall-Chaundy (BC) pair. A construction of BC pairs is presented for operators of order $4$ in the first Weyl algebra. Moreover, for true rank $r$ pairs, by means of differential subresultants, we effectively compute the fiber of the rank $r$ spectral sheaf over their spectral curve.

Key words: Weyl algebra; Ore domain; spectral curve; higher-rank vector bundle.

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