{"id":{"repo_id":"dialnet","oai_identifier":"oai:dialnet.unirioja.es:TES0000023095"},"canonical_url":"https://search.dev.ndltd.org/etd/dialnet/oai:dialnet.unirioja.es:TES0000023095","repository":{"repo_id":"dialnet","name":"Dialnet","base_url":"https://dialnet.unirioja.es/oaites/OAIHandler"},"display":{"title":"Quadratic Lie algebras. Algorithms and (de)constructions","abstract":"In this dissertation we are going to study quadratic Lie algebras, with special interest in the ones which are 2-step nilpotent, and to give algorithmic procedures to build a wide range of examples. After an introduction and a overview of the known results in this matter, we start with a deconstruction process to reduce the study of general quadratic Lie algebras to the nilpotent ones. This is obtained undoing successive double extensions on quotients given from the location of some important ideals. The variety of nilpotent quadratic Lie algebras can be established from free nilpotent Lie algebras and their invariant bilinear forms. But this is a tough problem, so we focus ourselves in the 2-step case. We start by introducing a new method to obtain them using multilinear algebra. Later we prove this new method is equivalent to the two main classical techniques: double and T∗ -extensions. In combination with trivectors, we end up giving a classification of these algebras up to dimension 17. Once covered the 2-step nilpotent case, we start building larger and more general quadratic Lie algebras. This is achieved via double extensions using their skew-derivations, which can be described through the Universal Mapping Property of free nilpotent Lie algebras. After, we study the family of quadratic Lie algebras with only one maximal ideal: the local ones. These algebras have strong structural properties and include the well-known family of real oscillator algebras, which are the quadratic algebras attached to metric Lorentzian forms. The next part is devoted to the ideal structure of quadratic Lie algebras, specially those whose ideals form a chain by their inclusions. Finally, we introduce and explain how to use a computational package we have developed. This software is supported on the thesis results and includes many tools used along this work.","abstract_html":"In this dissertation we are going to study quadratic Lie algebras, with special interest in the ones which are 2-step nilpotent, and to give algorithmic procedures to build a wide range of examples. After an introduction and a overview of the known results in this matter, we start with a deconstruction process to reduce the study of general quadratic Lie algebras to the nilpotent ones. This is obtained undoing successive double extensions on quotients given from the location of some important ideals. The variety of nilpotent quadratic Lie algebras can be established from free nilpotent Lie algebras and their invariant bilinear forms. But this is a tough problem, so we focus ourselves in the 2-step case. We start by introducing a new method to obtain them using multilinear algebra. Later we prove this new method is equivalent to the two main classical techniques: double and T∗ -extensions. In combination with trivectors, we end up giving a classification of these algebras up to dimension 17. Once covered the 2-step nilpotent case, we start building larger and more general quadratic Lie algebras. This is achieved via double extensions using their skew-derivations, which can be described through the Universal Mapping Property of free nilpotent Lie algebras. After, we study the family of quadratic Lie algebras with only one maximal ideal: the local ones. These algebras have strong structural properties and include the well-known family of real oscillator algebras, which are the quadratic algebras attached to metric Lorentzian forms. The next part is devoted to the ideal structure of quadratic Lie algebras, specially those whose ideals form a chain by their inclusions. Finally, we introduce and explain how to use a computational package we have developed. This software is supported on the thesis results and includes many tools used along this work.","abstract_has_math":false,"creators":["Roldán López, Jorge"],"institution":"Universidad de La Rioja (España)","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["Benito Clavijo, María del Pilar (Universidad de La Rioja)"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2023,"date_issued":"2023","date_published":"2023","updated_at":"2026-07-24T06:27:07Z","subjects":[],"languages":["eng"],"rights":["LICENCIA DE USO: Los documentos a texto completo incluidos en Dialnet son de acceso libre y propiedad de sus autores y/o editores. Por tanto, cualquier acto de reproducción, distribución, comunicación pública y/o transformación total o parcial requiere el consentimiento expreso y escrito de aquéllos. Cualquier enlace al texto completo de estos documentos deberá hacerse a través de la URL oficial de éstos en Dialnet. Más información: https://dialnet.unirioja.es/info/derechosOAI | INTELLECTUAL PROPERTY RIGHTS STATEMENT: Full text documents hosted by Dialnet are protected by copyright and/or related rights. This digital object is accessible without charge, but its use is subject to the licensing conditions set by its authors or editors. Unless expressly stated otherwise in the licensing conditions, you are free to linking, browsing, printing and making a copy for your own personal purposes. All other acts of reproduction and communication to the public are subject to the licensing conditions expressed by editors and authors and require consent from them. Any link to this document should be made using its official URL in Dialnet. More info: https://dialnet.unirioja.es/info/derechosOAI"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://dialnet.unirioja.es/servlet/oaites?codigo=316881","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Benito Clavijo, María del Pilar (Universidad de La Rioja)"]},{"key":"dc:creator","label":"Author","values":["Roldán López, Jorge"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2023"]},{"key":"dc:publisher","label":"Institution","values":["Universidad de La Rioja (España)"]},{"key":"dc:type","label":"Dc Type","values":["text (thesis)"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["LICENCIA DE USO: Los documentos a texto completo incluidos en Dialnet son de acceso libre y propiedad de sus autores y/o editores. Por tanto, cualquier acto de reproducción, distribución, comunicación pública y/o transformación total o parcial requiere el consentimiento expreso y escrito de aquéllos. Cualquier enlace al texto completo de estos documentos deberá hacerse a través de la URL oficial de éstos en Dialnet. Más información: https://dialnet.unirioja.es/info/derechosOAI | INTELLECTUAL PROPERTY RIGHTS STATEMENT: Full text documents hosted by Dialnet are protected by copyright and/or related rights. This digital object is accessible without charge, but its use is subject to the licensing conditions set by its authors or editors. Unless expressly stated otherwise in the licensing conditions, you are free to linking, browsing, printing and making a copy for your own personal purposes. All other acts of reproduction and communication to the public are subject to the licensing conditions expressed by editors and authors and require consent from them. Any link to this document should be made using its official URL in Dialnet. More info: https://dialnet.unirioja.es/info/derechosOAI"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://dialnet.unirioja.es/servlet/oaites?codigo=316881"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In this dissertation we are going to study quadratic Lie algebras, with special interest in the ones which are 2-step nilpotent, and to give algorithmic procedures to build a wide range of examples. After an introduction and a overview of the known results in this matter, we start with a deconstruction process to reduce the study of general quadratic Lie algebras to the nilpotent ones. This is obtained undoing successive double extensions on quotients given from the location of some important ideals. The variety of nilpotent quadratic Lie algebras can be established from free nilpotent Lie algebras and their invariant bilinear forms. But this is a tough problem, so we focus ourselves in the 2-step case. We start by introducing a new method to obtain them using multilinear algebra. Later we prove this new method is equivalent to the two main classical techniques: double and T∗ -extensions. In combination with trivectors, we end up giving a classification of these algebras up to dimension 17. Once covered the 2-step nilpotent case, we start building larger and more general quadratic Lie algebras. This is achieved via double extensions using their skew-derivations, which can be described through the Universal Mapping Property of free nilpotent Lie algebras. After, we study the family of quadratic Lie algebras with only one maximal ideal: the local ones. These algebras have strong structural properties and include the well-known family of real oscillator algebras, which are the quadratic algebras attached to metric Lorentzian forms. The next part is devoted to the ideal structure of quadratic Lie algebras, specially those whose ideals form a chain by their inclusions. Finally, we introduce and explain how to use a computational package we have developed. This software is supported on the thesis results and includes many tools used along this work.","En esta tesis estudiamos las álgebras de Lie cuadráticas, con especial interés en aquellas que son nilpotentes de índice 2, dando métodos algorítmicos para construir una amplia gama de ejemplos. Después de una introducción y una visión general de los resultados conocidos sobre este tema, comenzamos con un proceso de deconstrucción que nos permite reducir el estudio de las álgebras de Lie cuadráticas generales a tan solo aquellas nilpotentes. Esta reducción se obtiene deshaciendo sucesivas doble extensiones sobre cocientes, una vez es conocida la ubicación de algunos ideales importantes. La variedad de álgebras de Lie cuadráticas nilpotentes se puede establecer a partir de las álgebras de Lie nilpotentes libres y sus formas bilineales invariantes. Pero hacerlo es difícil, así que nos enfocamos en el caso donde el índice de nilpotencia es 2. Empezamos presentando un nuevo método para obtener dichas álgebras empleando técnicas de álgebra multilineal, para luego demostrar que este nuevo método es equivalente a las dos técnicas clásicas principales: dobles extensiones y T∗ -extensiones. En combinación con trivectores, terminamos dando una clasificación de estas álgebras hasta dimensión 17. Una vez cubierto el caso nilpotente de índice 2, comenzamos la construcción álgebras de Lie cuadráticas más grandes y generales. Esto se logra mediante dobles extensiones usando sus derivaciones antisimétricas, que se pueden describir a través de la propiedad universal para álgebras de Lie nilpotentes libres. Después, estudiamos la familia de álgebras de Lie cuadráticas con un único ideal maximal: las álgebras locales. Estas tienen propiedades estructurales sólidas e incluyen a la conocida familia de álgebras osciladoras reales, que son las álgebras cuadráticas asociadas a formas métricas Lorentzianas. La siguiente parte está dedicada a la estructura que presentan los ideales de álgebras de Lie cuadráticas, especialmente aquellas cuyos ideales forman una cadena por inclusión. Finalmente, explicamos cómo usar un paquete computacional que hemos desarrollado. Este software está respaldado por los resultados de la tesis e incluye muchas herramientas utilizadas a lo largo de esta memoria."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Quadratic Lie algebras. Algorithms and (de)constructions"]}]}],"canonical_facts":{"dc:contributor":["Benito Clavijo, María del Pilar (Universidad de La Rioja)"],"dc:creator":["Roldán López, Jorge"],"dc:date":["2023"],"dc:description":["In this dissertation we are going to study quadratic Lie algebras, with special interest in the ones which are 2-step nilpotent, and to give algorithmic procedures to build a wide range of examples. After an introduction and a overview of the known results in this matter, we start with a deconstruction process to reduce the study of general quadratic Lie algebras to the nilpotent ones. This is obtained undoing successive double extensions on quotients given from the location of some important ideals. The variety of nilpotent quadratic Lie algebras can be established from free nilpotent Lie algebras and their invariant bilinear forms. But this is a tough problem, so we focus ourselves in the 2-step case. We start by introducing a new method to obtain them using multilinear algebra. Later we prove this new method is equivalent to the two main classical techniques: double and T∗ -extensions. In combination with trivectors, we end up giving a classification of these algebras up to dimension 17. Once covered the 2-step nilpotent case, we start building larger and more general quadratic Lie algebras. This is achieved via double extensions using their skew-derivations, which can be described through the Universal Mapping Property of free nilpotent Lie algebras. After, we study the family of quadratic Lie algebras with only one maximal ideal: the local ones. These algebras have strong structural properties and include the well-known family of real oscillator algebras, which are the quadratic algebras attached to metric Lorentzian forms. The next part is devoted to the ideal structure of quadratic Lie algebras, specially those whose ideals form a chain by their inclusions. Finally, we introduce and explain how to use a computational package we have developed. This software is supported on the thesis results and includes many tools used along this work.","En esta tesis estudiamos las álgebras de Lie cuadráticas, con especial interés en aquellas que son nilpotentes de índice 2, dando métodos algorítmicos para construir una amplia gama de ejemplos. Después de una introducción y una visión general de los resultados conocidos sobre este tema, comenzamos con un proceso de deconstrucción que nos permite reducir el estudio de las álgebras de Lie cuadráticas generales a tan solo aquellas nilpotentes. Esta reducción se obtiene deshaciendo sucesivas doble extensiones sobre cocientes, una vez es conocida la ubicación de algunos ideales importantes. La variedad de álgebras de Lie cuadráticas nilpotentes se puede establecer a partir de las álgebras de Lie nilpotentes libres y sus formas bilineales invariantes. Pero hacerlo es difícil, así que nos enfocamos en el caso donde el índice de nilpotencia es 2. Empezamos presentando un nuevo método para obtener dichas álgebras empleando técnicas de álgebra multilineal, para luego demostrar que este nuevo método es equivalente a las dos técnicas clásicas principales: dobles extensiones y T∗ -extensiones. En combinación con trivectores, terminamos dando una clasificación de estas álgebras hasta dimensión 17. Una vez cubierto el caso nilpotente de índice 2, comenzamos la construcción álgebras de Lie cuadráticas más grandes y generales. Esto se logra mediante dobles extensiones usando sus derivaciones antisimétricas, que se pueden describir a través de la propiedad universal para álgebras de Lie nilpotentes libres. Después, estudiamos la familia de álgebras de Lie cuadráticas con un único ideal maximal: las álgebras locales. Estas tienen propiedades estructurales sólidas e incluyen a la conocida familia de álgebras osciladoras reales, que son las álgebras cuadráticas asociadas a formas métricas Lorentzianas. La siguiente parte está dedicada a la estructura que presentan los ideales de álgebras de Lie cuadráticas, especialmente aquellas cuyos ideales forman una cadena por inclusión. Finalmente, explicamos cómo usar un paquete computacional que hemos desarrollado. Este software está respaldado por los resultados de la tesis e incluye muchas herramientas utilizadas a lo largo de esta memoria."],"dc:format":["application/pdf"],"dc:identifier":["https://dialnet.unirioja.es/servlet/oaites?codigo=316881"],"dc:language":["eng"],"dc:publisher":["Universidad de La Rioja (España)"],"dc:rights":["LICENCIA DE USO: Los documentos a texto completo incluidos en Dialnet son de acceso libre y propiedad de sus autores y/o editores. Por tanto, cualquier acto de reproducción, distribución, comunicación pública y/o transformación total o parcial requiere el consentimiento expreso y escrito de aquéllos. Cualquier enlace al texto completo de estos documentos deberá hacerse a través de la URL oficial de éstos en Dialnet. Más información: https://dialnet.unirioja.es/info/derechosOAI | INTELLECTUAL PROPERTY RIGHTS STATEMENT: Full text documents hosted by Dialnet are protected by copyright and/or related rights. This digital object is accessible without charge, but its use is subject to the licensing conditions set by its authors or editors. Unless expressly stated otherwise in the licensing conditions, you are free to linking, browsing, printing and making a copy for your own personal purposes. All other acts of reproduction and communication to the public are subject to the licensing conditions expressed by editors and authors and require consent from them. Any link to this document should be made using its official URL in Dialnet. More info: https://dialnet.unirioja.es/info/derechosOAI"],"dc:title":["Quadratic Lie algebras. Algorithms and (de)constructions"],"dc:type":["text (thesis)"]},"updated_at":"2026-07-24T06:27:07Z"}