Commuting elements in the first Weyl algebra over \(\mathbb{Q} \)
type: Item
instance of: scholarly article
title: Commuting elements in the first Weyl algebra over \(\mathbb{Q} \)
zbMATH Open document ID: 1499.14059
DOI: 10.1134/S003744662205007X
author: A. F. Gundareva
published in: Siberian Mathematical Journal
publication date: 2022-10-06T00:00:00Z
review text: The aim of this paper is to study the commuting differential operators whose coefficients are polynomials over \(\mathbb{Q}\) and \(\mathbb{Z}\), i.e., the commuting elements of the first Weyl algebra over \(\mathbb{Q}\) and \(\mathbb{Z}\), as well as the actions of the groups Aut(\(A_1(\mathbb{Q})\)) and Aut(\(A_1(\mathbb{Z})\)) on the solution sets to some equations, where \(A_1(K)=K[x][\partial_x]\) is the first Weyl algebra over \(K\), with \(K=\mathbb{C}\), \(\mathbb{Q}\), and \(\mathbb{Z}\). The author considers the commuting self adjoint differential operators of rank \(2\) and orders \(4\) and \(6\) with the following elliptic spectral curve \[ w^2=z^3+c_2z^2+c_1z+c_0, \] and the Krichever-Novikov formally self adjoint operators \[ L_4==(\partial^2_x+V(x))^2+W(x), \] \[ L_6=(\partial^2_x+V(x))^3+\frac{1}{2}(c_2+3W(x))(\partial^2_x+V(x))+\frac{3}{2}W_x(x)\partial_x+\frac{5}{4}W_{xx}(x), \] where \(V(x)\) and \(W(x)\) are rational functions. The automorphism group of the first Weyl algebra over \(\mathbb{Q}\) acts on the commuting differential operators with polynomial coefficients over \(\mathbb{Q}\). The author shows that the orbit set is infinite for a fixed elliptic spectral curve over \(\mathbb{Q}\) with at least one rational point. The main result obtained in this paper can be formulated as follows: suppose that the above elliptic spectral curve with rational coefficients \(c_i\in\mathbb{Q}\) contains at least one point \((z_0,w_0)\) with rational coordinates, then there is a countable family \(\{(L_{4,\alpha},L_{6,\alpha}\in\mathbb{Q}[x][\partial_x]\}\) of commuting differential operators parametrized by rationals \(\alpha\in\mathbb{Q}\) with this spectral curve such that the pairs of operators \((L_{4,\alpha},L_{6,\alpha})\) and \((L_{4,\alpha'},L_{6,\alpha'})\) for distinct \(\alpha,\alpha'\in\mathbb{Q}\) cannot be taken into each other by the elements of Aut(\(A_1(\mathbb{Q})\)). It is also deduced that the orbit set of the action of Aut(\(A_1(\mathb…
reviewed by: Ahmed Lesfari
Mathematics Subject Classification ID: 14H70
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 13N10
Mathematics Subject Classification ID: 16S32
Mathematics Subject Classification ID: 35Q53
zbMATH DE Number: 7596649
zbMATH Keywords: automorphism
zbMATH Keywords: first Weyl algebra
zbMATH Keywords: commuting differential operator
zbMATH Keywords: polynomial coefficients over \(\mathbb{Q} \)
zbMATH Keywords: elliptic spectral curve
MaRDI profile type: Publication
cites work: Factorization of KdV Schrödinger operators using differential subresultants
cites work: Commuting ordinary differential operators and the Dixmier test
cites work: Centralizers of rank one in the first Weyl algebra
cites work: Algebraic Geometric Properties of Spectral Surfaces of Quantum Integrable Systems and Their Isospectral Deformations
cites work: Commuting Ordinary Differential Operators with Polynomial Coefficients and Automorphisms of the First Weyl Algebra
cites work: An algorithm for construction of commuting ordinary differential operators by geometric data
cites work: HOLOMORPHIC BUNDLES OVER ALGEBRAIC CURVES AND NON-LINEAR EQUATIONS
cites work: Self-adjoint commuting differential operators of rank two
cites work: Sur les algèbres de Weyl
cites work: Commuting rank 2 differential operators corresponding to a curve of genus 2
cites work: On commutative subalgebras of the Weyl algebra related to commuting operators of arbitrary rank and genus
cites work: Commuting ordinary differential operators of arbitrary genus and arbitrary rank with polynomial coefficients
cites work: Commuting differential operators of rank 2 with polynomial coefficients
cites work: On self-adjoint commuting differential operators of rank two
cites work: A ring of commuting differential operators of rank 2 corresponding to a curve of genus 2
full work available at URL: s003744662205007x
OpenAlex ID: W4300594542
description: scientific article; zbMATH DE number 7596649