The Algebra of differential operators for a matrix weight: An ultraspherical example

Autores
Zurrián, Ignacio Nahuel
Año de publicación
2017
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
In this article we study in detail algebraic properties of the algebra D(W) of differential operators associated to a matrix weight of Gegenbauer type. We prove that two secondorder operators generate the algebra, indeed D(W) is isomorphic to the free algebra generated by two elements subject to certain relations. Also, the center is isomorphic to the affine algebra of a singular rational curve. The algebra D(W) is a finitely generated torsion-free module over its center, but it is not flat and therefore it is not projective. This is the second detailed study of an algebra D(W) and the first one coming from spherical functions and group representations. We prove that the algebras for different Gegenbauer weights and the algebras studied previously, related to Hermite weights, are isomorphic to each other. We give some general results that allow us to regard the algebra D(W) as the centralizer of its center in the Weyl algebra. We do believe that this should hold for any irreducible weight and the case considered in this article represents a good step in this direction.
Fil: Zurrián, Ignacio Nahuel. Pontificia Universidad Católica de Chile; Chile. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina
Materia
matrix weights
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/58369

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spelling The Algebra of differential operators for a matrix weight: An ultraspherical exampleZurrián, Ignacio Nahuelmatrix weightshttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1In this article we study in detail algebraic properties of the algebra D(W) of differential operators associated to a matrix weight of Gegenbauer type. We prove that two secondorder operators generate the algebra, indeed D(W) is isomorphic to the free algebra generated by two elements subject to certain relations. Also, the center is isomorphic to the affine algebra of a singular rational curve. The algebra D(W) is a finitely generated torsion-free module over its center, but it is not flat and therefore it is not projective. This is the second detailed study of an algebra D(W) and the first one coming from spherical functions and group representations. We prove that the algebras for different Gegenbauer weights and the algebras studied previously, related to Hermite weights, are isomorphic to each other. We give some general results that allow us to regard the algebra D(W) as the centralizer of its center in the Weyl algebra. We do believe that this should hold for any irreducible weight and the case considered in this article represents a good step in this direction.Fil: Zurrián, Ignacio Nahuel. Pontificia Universidad Católica de Chile; Chile. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; ArgentinaOxford University Press2017-04info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/58369Zurrián, Ignacio Nahuel; The Algebra of differential operators for a matrix weight: An ultraspherical example; Oxford University Press; International Mathematics Research Notices; 2017; 8; 4-2017; 2402-24301073-7928CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1093/imrn/rnw104info:eu-repo/semantics/altIdentifier/url/https://academic.oup.com/imrn/article-abstract/2017/8/2402/3056814info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1505.03321info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:07:12Zoai:ri.conicet.gov.ar:11336/58369instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-29 10:07:12.742CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv The Algebra of differential operators for a matrix weight: An ultraspherical example
title The Algebra of differential operators for a matrix weight: An ultraspherical example
spellingShingle The Algebra of differential operators for a matrix weight: An ultraspherical example
Zurrián, Ignacio Nahuel
matrix weights
title_short The Algebra of differential operators for a matrix weight: An ultraspherical example
title_full The Algebra of differential operators for a matrix weight: An ultraspherical example
title_fullStr The Algebra of differential operators for a matrix weight: An ultraspherical example
title_full_unstemmed The Algebra of differential operators for a matrix weight: An ultraspherical example
title_sort The Algebra of differential operators for a matrix weight: An ultraspherical example
dc.creator.none.fl_str_mv Zurrián, Ignacio Nahuel
author Zurrián, Ignacio Nahuel
author_facet Zurrián, Ignacio Nahuel
author_role author
dc.subject.none.fl_str_mv matrix weights
topic matrix weights
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv In this article we study in detail algebraic properties of the algebra D(W) of differential operators associated to a matrix weight of Gegenbauer type. We prove that two secondorder operators generate the algebra, indeed D(W) is isomorphic to the free algebra generated by two elements subject to certain relations. Also, the center is isomorphic to the affine algebra of a singular rational curve. The algebra D(W) is a finitely generated torsion-free module over its center, but it is not flat and therefore it is not projective. This is the second detailed study of an algebra D(W) and the first one coming from spherical functions and group representations. We prove that the algebras for different Gegenbauer weights and the algebras studied previously, related to Hermite weights, are isomorphic to each other. We give some general results that allow us to regard the algebra D(W) as the centralizer of its center in the Weyl algebra. We do believe that this should hold for any irreducible weight and the case considered in this article represents a good step in this direction.
Fil: Zurrián, Ignacio Nahuel. Pontificia Universidad Católica de Chile; Chile. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina
description In this article we study in detail algebraic properties of the algebra D(W) of differential operators associated to a matrix weight of Gegenbauer type. We prove that two secondorder operators generate the algebra, indeed D(W) is isomorphic to the free algebra generated by two elements subject to certain relations. Also, the center is isomorphic to the affine algebra of a singular rational curve. The algebra D(W) is a finitely generated torsion-free module over its center, but it is not flat and therefore it is not projective. This is the second detailed study of an algebra D(W) and the first one coming from spherical functions and group representations. We prove that the algebras for different Gegenbauer weights and the algebras studied previously, related to Hermite weights, are isomorphic to each other. We give some general results that allow us to regard the algebra D(W) as the centralizer of its center in the Weyl algebra. We do believe that this should hold for any irreducible weight and the case considered in this article represents a good step in this direction.
publishDate 2017
dc.date.none.fl_str_mv 2017-04
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/58369
Zurrián, Ignacio Nahuel; The Algebra of differential operators for a matrix weight: An ultraspherical example; Oxford University Press; International Mathematics Research Notices; 2017; 8; 4-2017; 2402-2430
1073-7928
CONICET Digital
CONICET
url http://hdl.handle.net/11336/58369
identifier_str_mv Zurrián, Ignacio Nahuel; The Algebra of differential operators for a matrix weight: An ultraspherical example; Oxford University Press; International Mathematics Research Notices; 2017; 8; 4-2017; 2402-2430
1073-7928
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1093/imrn/rnw104
info:eu-repo/semantics/altIdentifier/url/https://academic.oup.com/imrn/article-abstract/2017/8/2402/3056814
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1505.03321
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
application/pdf
dc.publisher.none.fl_str_mv Oxford University Press
publisher.none.fl_str_mv Oxford University Press
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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