Algebro-geometric characterizations of commuting differential operators in semi-graded rings

In this thesis, we study algebro-geometric characterizations of commuting differential operators in families of semi-graded rings. First, we present some ring-theoretical notions of semi-graded rings that are necessary throughout the thesis. We include a non-exhaustive list of noncommutative rings t...

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Autores:
Niño Torres, Diego Arturo
Tipo de recurso:
Doctoral thesis
Fecha de publicación:
2023
Institución:
Universidad Nacional de Colombia
Repositorio:
Universidad Nacional de Colombia
Idioma:
eng
OAI Identifier:
oai:repositorio.unal.edu.co:unal/86565
Acceso en línea:
https://repositorio.unal.edu.co/handle/unal/86565
https://repositorio.unal.edu.co/
Palabra clave:
510 - Matemáticas::516 - Geometría
510 - Matemáticas::512 - Álgebra
Operadores diferenciales
Differential operators
Anillos (Álgebra)
Rings (Algebra)
Semi-graded ring
Quantum algebra
Ore extension
PBW basis
Valuation
Sylvester matrix
Resultant
Determinant polynomial
Centralizer
Gelfand-Kirillov dimension
Anillo semi-graduado
Álgebra cuántica
Extensión de Ore
Base PBW
Valuación
Matriz de Sylvester
Resultante
Polinomio determinante
Centralizador
Dimensión de Gelfand-Kirillov
Teoría de anillos
Ring theory
Matriz de Sylveste
Rights
openAccess
License
Reconocimiento 4.0 Internacional
id UNACIONAL2_0d3656e357ec16d735a82edb26162050
oai_identifier_str oai:repositorio.unal.edu.co:unal/86565
network_acronym_str UNACIONAL2
network_name_str Universidad Nacional de Colombia
repository_id_str
dc.title.eng.fl_str_mv Algebro-geometric characterizations of commuting differential operators in semi-graded rings
dc.title.translated.spa.fl_str_mv Caracterizaciones algebro-geométricas de operadores diferenciales conmutativos en anillos semi-graduados
title Algebro-geometric characterizations of commuting differential operators in semi-graded rings
spellingShingle Algebro-geometric characterizations of commuting differential operators in semi-graded rings
510 - Matemáticas::516 - Geometría
510 - Matemáticas::512 - Álgebra
Operadores diferenciales
Differential operators
Anillos (Álgebra)
Rings (Algebra)
Semi-graded ring
Quantum algebra
Ore extension
PBW basis
Valuation
Sylvester matrix
Resultant
Determinant polynomial
Centralizer
Gelfand-Kirillov dimension
Anillo semi-graduado
Álgebra cuántica
Extensión de Ore
Base PBW
Valuación
Matriz de Sylvester
Resultante
Polinomio determinante
Centralizador
Dimensión de Gelfand-Kirillov
Teoría de anillos
Ring theory
Matriz de Sylveste
title_short Algebro-geometric characterizations of commuting differential operators in semi-graded rings
title_full Algebro-geometric characterizations of commuting differential operators in semi-graded rings
title_fullStr Algebro-geometric characterizations of commuting differential operators in semi-graded rings
title_full_unstemmed Algebro-geometric characterizations of commuting differential operators in semi-graded rings
title_sort Algebro-geometric characterizations of commuting differential operators in semi-graded rings
dc.creator.fl_str_mv Niño Torres, Diego Arturo
dc.contributor.advisor.none.fl_str_mv Reyes Villamil, Milton Armando
dc.contributor.author.none.fl_str_mv Niño Torres, Diego Arturo
dc.subject.ddc.spa.fl_str_mv 510 - Matemáticas::516 - Geometría
510 - Matemáticas::512 - Álgebra
topic 510 - Matemáticas::516 - Geometría
510 - Matemáticas::512 - Álgebra
Operadores diferenciales
Differential operators
Anillos (Álgebra)
Rings (Algebra)
Semi-graded ring
Quantum algebra
Ore extension
PBW basis
Valuation
Sylvester matrix
Resultant
Determinant polynomial
Centralizer
Gelfand-Kirillov dimension
Anillo semi-graduado
Álgebra cuántica
Extensión de Ore
Base PBW
Valuación
Matriz de Sylvester
Resultante
Polinomio determinante
Centralizador
Dimensión de Gelfand-Kirillov
Teoría de anillos
Ring theory
Matriz de Sylveste
dc.subject.lemb.none.fl_str_mv Operadores diferenciales
Differential operators
Anillos (Álgebra)
Rings (Algebra)
dc.subject.proposal.eng.fl_str_mv Semi-graded ring
Quantum algebra
Ore extension
PBW basis
Valuation
Sylvester matrix
Resultant
Determinant polynomial
Centralizer
Gelfand-Kirillov dimension
dc.subject.proposal.spa.fl_str_mv Anillo semi-graduado
Álgebra cuántica
Extensión de Ore
Base PBW
Valuación
Matriz de Sylvester
Resultante
Polinomio determinante
Centralizador
Dimensión de Gelfand-Kirillov
dc.subject.wikidata.none.fl_str_mv Teoría de anillos
Ring theory
Matriz de Sylveste
description In this thesis, we study algebro-geometric characterizations of commuting differential operators in families of semi-graded rings. First, we present some ring-theoretical notions of semi-graded rings that are necessary throughout the thesis. We include a non-exhaustive list of noncommutative rings that are particular examples of these rings. Second, to motivate the study of commuting differential operators beloging to noncommutative algebras, and hence to develop a possible Burchnall-Chaundy (BC) theory for them, we review algebraic and matrix results appearing in the literature on the theory of these operators in some families of semi-graded rings. Third, we introduce the notion of pseudo-multidegree function as a generalization of pseudo-degree function, and hence we establish a criterion to determine whether the centralizer of an element has finite dimension over a noncommutative ring having PBW basis. In this way, we formulate a BC theorem for rings having pseudo-multidegree functions. We illustrate our results with families of algebras appearing in ring theory and noncommutative geometry. Fourth, we develop a first approach to the BC theory for quadratic algebras having PBW bases defined by Golovashkin and Maksimov. We prove combinatorial properties on products of elements in these algebras, and then consider the notions of Sylvester matrix and resultant for quadratic algebras with the purpose of exploring common right factors. Then, by using the concept of determinant polynomial, we formulate the version of BC theory for these algebras. We present illustrative examples of the assertions about these algebras. Finally, we establish some bridging ideas with the aim of extending results on centralizers for graded rings to the setting of semi-graded rings.
publishDate 2023
dc.date.issued.none.fl_str_mv 2023
dc.date.accessioned.none.fl_str_mv 2024-07-18T16:08:55Z
dc.date.available.none.fl_str_mv 2024-07-18T16:08:55Z
dc.type.spa.fl_str_mv Trabajo de grado - Doctorado
dc.type.driver.spa.fl_str_mv info:eu-repo/semantics/doctoralThesis
dc.type.version.spa.fl_str_mv info:eu-repo/semantics/acceptedVersion
dc.type.coar.spa.fl_str_mv http://purl.org/coar/resource_type/c_db06
dc.type.content.spa.fl_str_mv Text
dc.type.redcol.spa.fl_str_mv http://purl.org/redcol/resource_type/TD
format http://purl.org/coar/resource_type/c_db06
status_str acceptedVersion
dc.identifier.uri.none.fl_str_mv https://repositorio.unal.edu.co/handle/unal/86565
dc.identifier.instname.spa.fl_str_mv Universidad Nacional de Colombia
dc.identifier.reponame.spa.fl_str_mv Repositorio Institucional Universidad Nacional de Colombia
dc.identifier.repourl.spa.fl_str_mv https://repositorio.unal.edu.co/
url https://repositorio.unal.edu.co/handle/unal/86565
https://repositorio.unal.edu.co/
identifier_str_mv Universidad Nacional de Colombia
Repositorio Institucional Universidad Nacional de Colombia
dc.language.iso.spa.fl_str_mv eng
language eng
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spelling Reconocimiento 4.0 Internacionalhttp://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Reyes Villamil, Milton Armando767f31307c790697ee2bf5d2c4f57583600Niño Torres, Diego Arturod7596f25bf9c21fa97f66fa0103e5dbb2024-07-18T16:08:55Z2024-07-18T16:08:55Z2023https://repositorio.unal.edu.co/handle/unal/86565Universidad Nacional de ColombiaRepositorio Institucional Universidad Nacional de Colombiahttps://repositorio.unal.edu.co/In this thesis, we study algebro-geometric characterizations of commuting differential operators in families of semi-graded rings. First, we present some ring-theoretical notions of semi-graded rings that are necessary throughout the thesis. We include a non-exhaustive list of noncommutative rings that are particular examples of these rings. Second, to motivate the study of commuting differential operators beloging to noncommutative algebras, and hence to develop a possible Burchnall-Chaundy (BC) theory for them, we review algebraic and matrix results appearing in the literature on the theory of these operators in some families of semi-graded rings. Third, we introduce the notion of pseudo-multidegree function as a generalization of pseudo-degree function, and hence we establish a criterion to determine whether the centralizer of an element has finite dimension over a noncommutative ring having PBW basis. In this way, we formulate a BC theorem for rings having pseudo-multidegree functions. We illustrate our results with families of algebras appearing in ring theory and noncommutative geometry. Fourth, we develop a first approach to the BC theory for quadratic algebras having PBW bases defined by Golovashkin and Maksimov. We prove combinatorial properties on products of elements in these algebras, and then consider the notions of Sylvester matrix and resultant for quadratic algebras with the purpose of exploring common right factors. Then, by using the concept of determinant polynomial, we formulate the version of BC theory for these algebras. We present illustrative examples of the assertions about these algebras. Finally, we establish some bridging ideas with the aim of extending results on centralizers for graded rings to the setting of semi-graded rings.En esta tesis, estudiamos caracterizaciones algebro-geométricas de operadores diferenciales conmutativos en familias de anillos semi-graduados. Primero, presentamos algunas nociones de la teoría de anillos de anillos semi-graduados que son necesarias a lo largo de la tesis. Incluimos una lista no exhaustiva de anillos no conmutativos que son ejemplos particulares de estos anillos. Segundo, para motivar el estudio de operadores diferenciales conmutativos pertenecientes a álgebras no conmutativas, y así desarrollar una posible teoría Burchnall-Chaundy (BC) para ellos, consideramos resultados algebraicos y matriciales presentes en la literatura sobre la teoría de estos operadores en algunas familias de anillos semi-graduados. Tercero, introducimos la noción de función pseudo-multigrado como una generalización de función pseudo-grado, y así establecemos un criterio para determinar si el centralizador de un elemento tiene dimensión finita sobre un anillo no conmutativo con base PBW. De esta manera, formulamos un teorema BC para anillos que tienen funciones pseudo-multigrado. Ilustramos nuestros resultados con familias de álgebras presentes en la teoría de anillos y la geometría no conmutativa. Cuarto, desarrollamos un primer acercamiento a la teoría BC para las álgebras cuadráticas con base PBW definidas por Golovashkin y Maksimov. Demostramos propiedades combinatoriales sobre productos de elementos en estas álgebras, y luego consideramos las nociones de matriz de Sylvester y resultante para álgebras cuadráticas con el fin de explorar factores comunes a derecha. Después, utilizando el concepto de determinante polinomial, formulamos la versión de la teoría BC para estas álgebras. Presentamos ejemplos ilustrativos de las afirmaciones sobre estas álgebras. Finalmente, formulamos algunas ideas con el propósito de extender resultados sobre centralizadores para anillos graduados al contexto de los anillos semi-graduados. (Texto tomado de la fuente)DoctoradoDoctor en Ciencias - Matemáticasvii, 120 páginasapplication/pdfengUniversidad Nacional de ColombiaBogotá - Ciencias - Doctorado en Ciencias - MatemáticasFacultad de CienciasBogotá, ColombiaUniversidad Nacional de Colombia - Sede Bogotá510 - Matemáticas::516 - Geometría510 - Matemáticas::512 - ÁlgebraOperadores diferencialesDifferential operatorsAnillos (Álgebra)Rings (Algebra)Semi-graded ringQuantum algebraOre extensionPBW basisValuationSylvester matrixResultantDeterminant polynomialCentralizerGelfand-Kirillov dimensionAnillo semi-graduadoÁlgebra cuánticaExtensión de OreBase PBWValuaciónMatriz de SylvesterResultantePolinomio determinanteCentralizadorDimensión de Gelfand-KirillovTeoría de anillosRing theoryMatriz de SylvesteAlgebro-geometric characterizations of commuting differential operators in semi-graded ringsCaracterizaciones algebro-geométricas de operadores diferenciales conmutativos en anillos semi-graduadosTrabajo de grado - Doctoradoinfo:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/acceptedVersionhttp://purl.org/coar/resource_type/c_db06Texthttp://purl.org/redcol/resource_type/TDV. 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Springer Berlin, Heidelberg, 2014.BibliotecariosEstudiantesInvestigadoresPúblico generalORIGINAL1016069203.2024.pdf1016069203.2024.pdfTesis de Doctorado en Ciencias Matemáticasapplication/pdf1063455https://repositorio.unal.edu.co/bitstream/unal/86565/2/1016069203.2024.pdff641b57734c17e9a7e96a131773cf9a2MD52LICENSElicense.txtlicense.txttext/plain; charset=utf-85879https://repositorio.unal.edu.co/bitstream/unal/86565/3/license.txteb34b1cf90b7e1103fc9dfd26be24b4aMD53THUMBNAIL1016069203.2024.pdf.jpg1016069203.2024.pdf.jpgGenerated Thumbnailimage/jpeg4284https://repositorio.unal.edu.co/bitstream/unal/86565/4/1016069203.2024.pdf.jpg6a49e2c9aaf789014e06eb0d33422894MD54unal/86565oai:repositorio.unal.edu.co:unal/865652024-08-26 23:10:57.101Repositorio Institucional Universidad Nacional de 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