Involutive and SAGBI bases for skew PBW extensions

ilustraciones, diagramas

Autores:: Suárez Gómez, Yésica Paola

Tipo de recurso:: Doctoral thesis

Fecha de publicación:: 2023

Institución:: Universidad Nacional de Colombia

Repositorio:: Universidad Nacional de Colombia

Idioma:: eng

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dc.title.eng.fl_str_mv	Involutive and SAGBI bases for skew PBW extensions
dc.title.translated.spa.fl_str_mv	bases involutivas y SAGBI para extensiones PBW torcidas
title	Involutive and SAGBI bases for skew PBW extensions
spellingShingle	Involutive and SAGBI bases for skew PBW extensions 510 - Matemáticas::512 - Álgebra Subalgebra Analogue to Gröbner Bases for Ideals SAGBI basis Quantum algebra Involutive basis Skew PBW extension Auslander-regular Artin-Schelter regular Skew Calabi-Yau Base SAGBI Base involutiva Extensión PBW torcida Álgebra cuántica Regularidad de Auslander Regularidad de Artin-Schelter Calabi-Yau torcida subálgebra anillo de polinomios álgebra no conmutativa subalgebra polynomial ring
title_short	Involutive and SAGBI bases for skew PBW extensions
title_full	Involutive and SAGBI bases for skew PBW extensions
title_fullStr	Involutive and SAGBI bases for skew PBW extensions
title_full_unstemmed	Involutive and SAGBI bases for skew PBW extensions
title_sort	Involutive and SAGBI bases for skew PBW extensions
dc.creator.fl_str_mv	Suárez Gómez, Yésica Paola
dc.contributor.advisor.spa.fl_str_mv	Reyes, Armando
dc.contributor.author.spa.fl_str_mv	Suárez Gómez, Yésica Paola
dc.subject.ddc.spa.fl_str_mv	510 - Matemáticas::512 - Álgebra
topic	510 - Matemáticas::512 - Álgebra Subalgebra Analogue to Gröbner Bases for Ideals SAGBI basis Quantum algebra Involutive basis Skew PBW extension Auslander-regular Artin-Schelter regular Skew Calabi-Yau Base SAGBI Base involutiva Extensión PBW torcida Álgebra cuántica Regularidad de Auslander Regularidad de Artin-Schelter Calabi-Yau torcida subálgebra anillo de polinomios álgebra no conmutativa subalgebra polynomial ring
dc.subject.other.none.fl_str_mv	Subalgebra Analogue to Gröbner Bases for Ideals
dc.subject.proposal.eng.fl_str_mv	SAGBI basis Quantum algebra Involutive basis Skew PBW extension Auslander-regular Artin-Schelter regular Skew Calabi-Yau
dc.subject.proposal.spa.fl_str_mv	Base SAGBI Base involutiva Extensión PBW torcida Álgebra cuántica Regularidad de Auslander Regularidad de Artin-Schelter Calabi-Yau torcida
dc.subject.wikidata.spa.fl_str_mv	subálgebra anillo de polinomios álgebra no conmutativa
dc.subject.wikidata.eng.fl_str_mv	subalgebra polynomial ring
description	ilustraciones, diagramas
publishDate	2023
dc.date.issued.none.fl_str_mv	2023
dc.date.accessioned.none.fl_str_mv	2024-07-03T21:23:06Z
dc.date.available.none.fl_str_mv	2024-07-03T21:23:06Z
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/86385
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/86385 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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dc.publisher.spa.fl_str_mv	Universidad Nacional de Colombia
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spelling	Atribución-NoComercial 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Reyes, Armando94408df5dafcee9e59989990d5187f68Suárez Gómez, Yésica Paolad2cce79de2d407fbf892ac290fd4db552024-07-03T21:23:06Z2024-07-03T21:23:06Z2023https://repositorio.unal.edu.co/handle/unal/86385Universidad Nacional de ColombiaRepositorio Institucional Universidad Nacional de Colombiahttps://repositorio.unal.edu.co/ilustraciones, diagramasIn this thesis, we study homological properties and SAGBI and Involutive bases of the noncommutative rings known as skew PBW extensions. First, we present some ring- theoretical notions of these extensions that are necessary throughout the thesis. With the aim of showing the generality of these objects in areas such as ring theory and noncommutative geometry, we include a non-exhaustive list of noncommutative algebras that are particular examples of these rings. Second, we characterize several homological properties of these ex- tensions. We provide a new and more general filtration to these extensions, and introduce the notion of σ-filtered skew PBW extension with the aim of studying its homological properties. We show that the homogenization of a σ-filtered skew PBW extension over a coefficient ring is a graded skew PBW extension over the homogenization of such a ring. By using this fact, we prove that if the homogenization of the coefficient ring is Auslander-regular, then the homogenization of the extension is a domain Noetherian, Artin-Schelter regular, Zariski and (ungraded) skew Calabi-Yau. Third, we present our proposal of SAGBI bases theory for skew PBW extensions over algebras. We consider the notion of reduction which is necessary in the characterization of these bases, and then establish an algorithm to find the normal form of an element. Then, we define what a SAGBI basis is, and formulate a criterion to determine when a subset of a skew PBW extension over a field is a SAGBI basis. In addition, we establish an algorithm to find a SAGBI basis from a subset contained in a subalgebra of a skew PBW extension. We illustrate our results with different examples of noncommutative algebras. We also investigate the problem of poly- nomial composition for SAGBI bases of subalgebras of skew PBW extensions. Finally, we present a theory of Involutive bases for skew PBW extensions over fields. We consider the notions of weak and strong Involutive bases, and then we define the involutive reduction process and involutive remainder that are necessary for the characterization of weak (strong) Involutive bases. Next, we introduce the notion of standard Involutive representation for elements of a subset of a skew PBW extension. Also, we give the definition of minimal Involutive basis and show the existence of a monic, involutively autoreduced, minimal Involutive basis. Finally, we establish different algorithms that compute involutive standard representations, principal involutive autoreduction, and an Involutive basis of a left ideal of a skew PBW extension. In this way, the existence of a finite Involutive basis for these ideals is proved by assuming that the involutive division is constructive Noetherian.En esta tesis, estudiamos propiedades homológicas y bases SAGBI e Involutivas de los anillos no conmutativos conocidos como extensiones PBW torcidas. Primero, presentamos algunas nociones teóricas de la teoría de anillos de estas extensiones que son necesarias a lo largo de la tesis. Con el propósito de mostrar la generalidad de estos objetos en áreas como la teoría de anillos y la geometría no conmutativa, incluimos una lista no exhaustiva de álgebras no conmutativas que son ejemplos particulares de estos anillos. Segundo, caracterizamos variadas propiedades homológicas de estas extensiones. Brindamos una nueva y más general filtración para estas extensiones, e introducimos la noción de extensión PBW torcida sigma-filtrada con el propósito de estudiar sus propiedades homológicas. Mostramos que la homogenización de una extensión PBW torcida sigma-filtrada sobre un anillo de coeficientes es una extensión PBW torcida graduada sobre la homogenización de dicho anillo. Utilizando este hecho, probamos que si la homogenización del anillo de coeficientes es Auslander-regular, entonces la homogenización de la extensión es un dominio noetheriano, Artin-Schelter regular, Zariski y Calabi-Yau torcida. Tercero, presentamos nuestra propuesta de teoría de bases SAGBI para extensiones PBW torcidas sobre álgebras. Consideramos la noción de reducción la cual es necesaria en la caracterización de estas bases, y luego establecemos un algoritmo para encontrar la forma normal de un elemento. Después, definimos lo que es una base SAGBI, y formulamos un criterio para determinar cuándo un subconjunto de una extensión PBW sobre un campo es una base SAGBI. De hecho, establecemos un algoritmo para encontrar una base SAGBI a partir de un subconjunto contenido en una subálgebra de una extensión PBW torcida. Ilustramos nuestros resultados con diferentes ejemplos de álgebras no conmutativas. También investigamos el problema de la composición polinomial para bases SAGBI de subálgebras de extensiones PBW torcidas. Finalmente, presentamos una teoría de bases Involutivas para extensiones PBW torcidas sobre campos. Consideramos las nociones de base Involutiva débil y fuerte, y luego definimos el proceso de reducción involutiva y el residuo involutivo que son necesarios para la caracterización de bases Involutivas débiles y fuertes. A continuación, presentamos la noción de representación involutiva estándar para elementos de un subconjunto de una extensión PBW torcida. Además, damos la definición de base Involutiva minimal y mostramos la existencia de una base Involutiva minimal, mónica, e involutivamente autorreducida. Finalmente, establecemos diferentes algoritmos que calculan representaciones estándar involutivas, autorreducción involutiva principal, y una base Involutiva de un ideal izquierdo de una extensión PBW torcida. De esta manera, la existencia de una base Involutiva finita para estos ideales se demuestra asumiendo que la división involutiva es noetheriana constructiva. (Texto tomado de la fuente).DoctoradoDoctor en Ciencias - Matemáticasiii, 111 páginasapplication/pdfengUniversidad Nacional de ColombiaBogotá - Ciencias - Doctorado en Ciencias - MatemáticasFacultad de CienciasBogotá, ColombiaUniversidad Nacional de Colombia - Sede Bogotá510 - Matemáticas::512 - ÁlgebraSubalgebra Analogue to Gröbner Bases for IdealsSAGBI basisQuantum algebraInvolutive basisSkew PBW extensionAuslander-regularArtin-Schelter regularSkew Calabi-YauBase SAGBIBase involutivaExtensión PBW torcidaÁlgebra cuánticaRegularidad de AuslanderRegularidad de Artin-SchelterCalabi-Yau torcidasubálgebraanillo de polinomiosálgebra no conmutativasubalgebrapolynomial ringInvolutive and SAGBI bases for skew PBW extensionsbases involutivas y SAGBI para extensiones PBW torcidasTrabajo de grado - Doctoradoinfo:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/acceptedVersionhttp://purl.org/coar/resource_type/c_db06Texthttp://purl.org/redcol/resource_type/TDN. Andruskiewitsch, F. 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Algebra, 322(2):373–409, 2009EstudiantesInvestigadoresMaestrosPúblico generalORIGINAL1049621918.2023.pdf1049621918.2023.pdfTesis de Doctorado en Ciencias - Matemáticasapplication/pdf1175394https://repositorio.unal.edu.co/bitstream/unal/86385/2/1049621918.2023.pdf74239c8ac35d8bc8edd6ec61a4184883MD52LICENSElicense.txtlicense.txttext/plain; charset=utf-85879https://repositorio.unal.edu.co/bitstream/unal/86385/1/license.txteb34b1cf90b7e1103fc9dfd26be24b4aMD51THUMBNAIL1049621918.2023.pdf.jpg1049621918.2023.pdf.jpgGenerated Thumbnailimage/jpeg4080https://repositorio.unal.edu.co/bitstream/unal/86385/3/1049621918.2023.pdf.jpgeab6ce7714cc7479a070bdccfb72506bMD53unal/86385oai:repositorio.unal.edu.co:unal/863852024-08-26 23:10:18.318Repositorio Institucional Universidad Nacional de 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Involutive and SAGBI bases for skew PBW extensions

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