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 SIGMA, 2019, Volume 15, 101, 23 pp. (Mi sigma1537)

Commuting Ordinary Differential Operators and the Dixmier Test

Emma Previatoa, Sonia L. Ruedab, Maria-Angeles Zurroc

a Boston University, USA

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.

Keywords: Weyl algebra, Ore domain, spectral curve, higher-rank vector bundle.

 Funding Agency Grant Number Universidad Complutense de Madrid 910444 M.A. Zurro is partially supported by Grupo UCM 910444.

DOI: https://doi.org/10.3842/SIGMA.2019.101

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ArXiv: 1902.01361
MSC: 13P15, 14H70
Received: February 4, 2019; in final form December 23, 2019; Published online December 30, 2019
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Citation: Emma Previato, Sonia L. Rueda, Maria-Angeles Zurro, “Commuting Ordinary Differential Operators and the Dixmier Test”, SIGMA, 15 (2019), 101, 23 pp.

Citation in format AMSBIB
\Bibitem{PreRueZur19} \by Emma~Previato, Sonia~L.~Rueda, Maria-Angeles~Zurro \paper Commuting Ordinary Differential Operators and the Dixmier Test \jour SIGMA \yr 2019 \vol 15 \papernumber 101 \totalpages 23 \mathnet{http://mi.mathnet.ru/sigma1537} \crossref{https://doi.org/10.3842/SIGMA.2019.101} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000505603700001} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85077722124} 

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This publication is cited in the following articles:
1. Gulnara S. Mauleshova, Andrey E. Mironov, “Discretization of Commuting Ordinary Differential Operators of Rank 2 in the Case of Elliptic Spectral Curves”, Proc. Steklov Inst. Math., 310 (2020), 202–213
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