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Csaba Schneiderhttp://lattes.cnpq.br/0326577563802136Letterio GattoLucas Henrique CalixtoRenato Vidal da Silva MartinsTiago Rodrigues Macedohttp://lattes.cnpq.br/0816225004191018Vanderlei Lopes de Jesus2022-01-08T03:19:48Z2022-01-08T03:19:48Z2021-08-13http://hdl.handle.net/1843/39048Seja $\mathfrak{g}$ uma álgebra de Lie de dimensão finita sobre um corpo $\mathbb{F}$ de característica prima $p$ e seja $U(\mathfrak{g})$ a álgebra envolvente universal de $\mathfrak{g}$. Sabemos da literatura que o centro $Z(\mathfrak{g})$ da álgebra envolvente $U(\mathfrak{g})$ é um domínio integralmente fechado. Nesta tese, descrevemos $Z(\mathfrak{g})$ para todas as álgebras de Lie nilpotentes indecomponíveis de dimensão menor ou igual a $6$ sobre um corpo $\mathbb{F}$ de característica prima $p$ assumindo que a classe de nilpotência $\mbox{cl}(\mathfrak{g})\leq p$. Apresentamos exemplos em que $Z(\mathfrak{g})=Z_{p}(\mathfrak{g})$, onde $Z_{p}(\mathfrak{g})$ é o $p$-centro de $U(\mathfrak{g})$. No entanto, encontramos casos em que $Z(\mathfrak{g})\neq Z_{p}(\mathfrak{g})$. Nestes casos, vamos lidar com extensões integrais $Z_{p}(\mathfrak{g})[z_{1},\ldots z_{s}]\subseteq Z(\mathfrak{g})$ onde $z_{1},\ldots, z_{s}\in Z(\mathfrak{g})\setminus Z_{p}(\mathfrak{g})$. Nas condições em que $Z_{p}(\mathfrak{g})[z_{1},\ldots z_{s}]\subseteq Z(\mathfrak{g})$ é uma extensão integral cujos corpos de frações coincidem, para ocorrer a igualdade $Z_{p}(\mathfrak{g})[z_{1},\ldots z_{s}]=Z(\mathfrak{g})$, é suficiente que $Z_{p}(\mathfrak{g})[z_{1},\ldots z_{s}]$ seja um domínio integralmente fechado. Boa parte de nosso trabalho é mostrar essa propriedade. Nesse ponto, precisamos dos conceitos de anéis regulares e anéis Cohen-Macaulay. Denotamos por $S(\mathfrak{g})$ a álgebra simétrica de $\mathfrak{g}$. Em característica zero, os espaços $U(\mathfrak{g})$ e $S(\mathfrak{g})$ são $\mathfrak{g}$-módulos isomorfos com respeito a representação adjunta. O conjunto de invariantes do $\mathfrak{g}$-módulo $U(\mathfrak{g})$ é o seu centro $Z(\mathfrak{g})$ e o conjunto de invariantes do $\mathfrak{g}$-módulo $S(\mathfrak{g})$ é denotado por $S(\mathfrak{g})^{\mathfrak{g}}$. Nesta tese, também, explicitamos a álgebra de invariantes $S(\mathfrak{g})^{\mathfrak{g}}$ para todas as álgebras de Lie nilpotentes indecomponíveis de dimensão menor ou igual a $6$ sobre um corpo $\mathbb{F}$ de característica prima $p$ com $\mbox{cl}(\mathfrak{g})\leq p$. Bem como, mostramos a incidência de um isomorfismo de álgebras entre $Z(\mathfrak{g})$ e $S(\mathfrak{g})^{\mathfrak{g}}$. Particularmente, determinamos $Z(\mathfrak{g})$ e $S(\mathfrak{g})^{\mathfrak{g}}$ para as álgebras de Lie standard filiform de dimensão até $6$ em característica prima $p$. Para essas álgebras, em geral, não é uma tarefa fácil determinar geradores explicitos para $Z(\mathfrak{g})$ e $S(\mathfrak{g})^{\mathfrak{g}}$ por causa da complexidade de seus geradores e suas relações.Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over a field $\mathbb{F}$ of prime characteristic $p$ and let $U(\mathfrak{g})$ be the universal enveloping algebra of $\mathfrak{g}$. We know from the literature that the center $Z(\mathfrak{g})$ of the enveloping algebra $U(\mathfrak{g})$ is an integrally closed domain. In this thesis, we describe $Z(\mathfrak{g})$ for all indecomposable nilpotent Lie algebras of dimension less than or equal to $6$ over a field $\mathbb{F}$ of prime characteristic $p$ assuming that $\mbox{cl}(\mathfrak{g})\leq p$. We present examples where $Z(\mathfrak{g})=Z_{p}(\mathfrak{g})$, where $Z_{p}(\mathfrak{g})$ is the $p$-center of $U(\mathfrak{g})$. However, we found cases where $Z(\mathfrak{g})\neq Z_{p}(\mathfrak{g})$. In these cases, we will deal with integral extensions $Z_{p}(\mathfrak{g})[z_{1},\ldots z_{s}]\subseteq Z(\mathfrak{g})$ where $z_{1},\ldots, z_{s}\in Z(\mathfrak{g})\setminus Z_{p}(\mathfrak{g})$. When $Z_{p}(\mathfrak{g})[z_{1},\ldots z_{s}]\subseteq Z(\mathfrak{g})$ is an integral extension whose fraction fields coincide, for equality $Z_{p}(\mathfrak{g})[z_{1},\ldots z_{s}]=Z(\mathfrak{g})$ to occur, it is sufficient that $Z_{p}(\mathfrak{g})[z_{1},\ldots z_{s}]$ is an integrally closed domain. A good part of our job is to show this property. At this point, we need the concepts of regular rings and Cohen-Macaulay rings. We denote by $S(\mathfrak{g})$ the symmetric algebra of $\mathfrak{g}$. In characteristic zero, the spaces $U(\mathfrak{g})$ and $S(\mathfrak{g})$ are isomorphic $\mathfrak{g}$-modules with respect to the adjoint representation. The set of invariants of the $\mathfrak{g}$-module $U(\mathfrak{g})$ is its center $Z(\mathfrak{g})$ and the set of invariants of the $\mathfrak{g}$-module $S(\mathfrak{g})$ is denoted by $S(\mathfrak{g})^{\mathfrak{g}}$. In this thesis, we describe the algebra of invariants $S(\mathfrak{g})^{\mathfrak{g}}$ for all indecomposable nilpotent Lie algebras of a dimension less than or equal to $6$ over a field $\mathbb{F}$ of prime caracteristic $p$ with $\mbox{cl}(\mathfrak{g})\leq p$. Furthermore, we show the existence of an algebra isomorphism between $Z(\mathfrak{g})$ and $S(\mathfrak{g})^{\mathfrak{g}}$. We also describe $Z(\mathfrak{g})$ and $S(\mathfrak{g})^{\mathfrak{g}}$ for the standard filiform Lie algebras of dimension up to $6$ in prime characteristic $p$. For these algebras, in general, it is not an easy task to determine explicit generators for $Z(\mathfrak{g})$ and $S(\mathfrak{g})^{\mathfrak{g}}$ because of the complexity of their generators and their relations. Keywords: nilpotent Lie algebras, universal enveloping algebra, center, Poisson center, Cohen-Macaulay rings.CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível SuperiorporUniversidade Federal de Minas GeraisPrograma de Pós-Graduação em MatemáticaUFMGBrasilICX - DEPARTAMENTO DE MATEMÁTICAhttp://creativecommons.org/licenses/by/3.0/pt/info:eu-repo/semantics/openAccessMatemática - TesesLie, Álgebra de - TesesGrupos nilpotentes - TesesPoisson, Distribuição de - TesesAneis comutativos -TesesÁlgebras de Lie nilpotentesÁlgebra envolvente universalCentroCentro PoissonAnéis Cohen-MacaulayO centro das álgebras envolventes universais de álgebras de Lie nilpotentes em caraterística primaThe center of universal enveloping algebras of nilpotent Lie algebras in prime characteristicinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesisreponame:Repositório Institucional da UFMGinstname:Universidade Federal de Minas Gerais (UFMG)instacron:UFMGORIGINALO centro das álgebras envolventes universais de álgebras de Lie nilpotentes em característica prima.pdfO centro das álgebras envolventes universais de álgebras de Lie nilpotentes em característica prima.pdfapplication/pdf1179190https://repositorio.ufmg.br/bitstream/1843/39048/1/O%20centro%20das%20%c3%a1lgebras%20envolventes%20universais%20de%20%c3%a1lgebras%20de%20Lie%20nilpotentes%20em%20caracter%c3%adstica%20prima.pdf6cafc683e73ccfd3bc21fc10ed80877aMD51LICENSElicense.txtlicense.txttext/plain; charset=utf-82118https://repositorio.ufmg.br/bitstream/1843/39048/3/license.txtcda590c95a0b51b4d15f60c9642ca272MD53CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8914https://repositorio.ufmg.br/bitstream/1843/39048/2/license_rdff9944a358a0c32770bd9bed185bb5395MD521843/390482022-01-08 00:19:48.9oai:repositorio.ufmg.br:1843/39048TElDRU7Dh0EgREUgRElTVFJJQlVJw4fDg08gTsODTy1FWENMVVNJVkEgRE8gUkVQT1NJVMOTUklPIElOU1RJVFVDSU9OQUwgREEgVUZNRwoKQ29tIGEgYXByZXNlbnRhw6fDo28gZGVzdGEgbGljZW7Dp2EsIHZvY8OqIChvIGF1dG9yIChlcykgb3UgbyB0aXR1bGFyIGRvcyBkaXJlaXRvcyBkZSBhdXRvcikgY29uY2VkZSBhbyBSZXBvc2l0w7NyaW8gSW5zdGl0dWNpb25hbCBkYSBVRk1HIChSSS1VRk1HKSBvIGRpcmVpdG8gbsOjbyBleGNsdXNpdm8gZSBpcnJldm9nw6F2ZWwgZGUgcmVwcm9kdXppciBlL291IGRpc3RyaWJ1aXIgYSBzdWEgcHVibGljYcOnw6NvIChpbmNsdWluZG8gbyByZXN1bW8pIHBvciB0b2RvIG8gbXVuZG8gbm8gZm9ybWF0byBpbXByZXNzbyBlIGVsZXRyw7RuaWNvIGUgZW0gcXVhbHF1ZXIgbWVpbywgaW5jbHVpbmRvIG9zIGZvcm1hdG9zIMOhdWRpbyBvdSB2w61kZW8uCgpWb2PDqiBkZWNsYXJhIHF1ZSBjb25oZWNlIGEgcG9sw610aWNhIGRlIGNvcHlyaWdodCBkYSBlZGl0b3JhIGRvIHNldSBkb2N1bWVudG8gZSBxdWUgY29uaGVjZSBlIGFjZWl0YSBhcyBEaXJldHJpemVzIGRvIFJJLVVGTUcuCgpWb2PDqiBjb25jb3JkYSBxdWUgbyBSZXBvc2l0w7NyaW8gSW5zdGl0dWNpb25hbCBkYSBVRk1HIHBvZGUsIHNlbSBhbHRlcmFyIG8gY29udGXDumRvLCB0cmFuc3BvciBhIHN1YSBwdWJsaWNhw6fDo28gcGFyYSBxdWFscXVlciBtZWlvIG91IGZvcm1hdG8gcGFyYSBmaW5zIGRlIHByZXNlcnZhw6fDo28uCgpWb2PDqiB0YW1iw6ltIGNvbmNvcmRhIHF1ZSBvIFJlcG9zaXTDs3JpbyBJbnN0aXR1Y2lvbmFsIGRhIFVGTUcgcG9kZSBtYW50ZXIgbWFpcyBkZSB1bWEgY8OzcGlhIGRlIHN1YSBwdWJsaWNhw6fDo28gcGFyYSBmaW5zIGRlIHNlZ3VyYW7Dp2EsIGJhY2stdXAgZSBwcmVzZXJ2YcOnw6NvLgoKVm9jw6ogZGVjbGFyYSBxdWUgYSBzdWEgcHVibGljYcOnw6NvIMOpIG9yaWdpbmFsIGUgcXVlIHZvY8OqIHRlbSBvIHBvZGVyIGRlIGNvbmNlZGVyIG9zIGRpcmVpdG9zIGNvbnRpZG9zIG5lc3RhIGxpY2Vuw6dhLiBWb2PDqiB0YW1iw6ltIGRlY2xhcmEgcXVlIG8gZGVww7NzaXRvIGRlIHN1YSBwdWJsaWNhw6fDo28gbsOjbywgcXVlIHNlamEgZGUgc2V1IGNvbmhlY2ltZW50bywgaW5mcmluZ2UgZGlyZWl0b3MgYXV0b3JhaXMgZGUgbmluZ3XDqW0uCgpDYXNvIGEgc3VhIHB1YmxpY2HDp8OjbyBjb250ZW5oYSBtYXRlcmlhbCBxdWUgdm9jw6ogbsOjbyBwb3NzdWkgYSB0aXR1bGFyaWRhZGUgZG9zIGRpcmVpdG9zIGF1dG9yYWlzLCB2b2PDqiBkZWNsYXJhIHF1ZSBvYnRldmUgYSBwZXJtaXNzw6NvIGlycmVzdHJpdGEgZG8gZGV0ZW50b3IgZG9zIGRpcmVpdG9zIGF1dG9yYWlzIHBhcmEgY29uY2VkZXIgYW8gUmVwb3NpdMOzcmlvIEluc3RpdHVjaW9uYWwgZGEgVUZNRyBvcyBkaXJlaXRvcyBhcHJlc2VudGFkb3MgbmVzdGEgbGljZW7Dp2EsIGUgcXVlIGVzc2UgbWF0ZXJpYWwgZGUgcHJvcHJpZWRhZGUgZGUgdGVyY2Vpcm9zIGVzdMOhIGNsYXJhbWVudGUgaWRlbnRpZmljYWRvIGUgcmVjb25oZWNpZG8gbm8gdGV4dG8gb3Ugbm8gY29udGXDumRvIGRhIHB1YmxpY2HDp8OjbyBvcmEgZGVwb3NpdGFkYS4KCkNBU08gQSBQVUJMSUNBw4fDg08gT1JBIERFUE9TSVRBREEgVEVOSEEgU0lETyBSRVNVTFRBRE8gREUgVU0gUEFUUk9Dw41OSU8gT1UgQVBPSU8gREUgVU1BIEFHw4pOQ0lBIERFIEZPTUVOVE8gT1UgT1VUUk8gT1JHQU5JU01PLCBWT0PDiiBERUNMQVJBIFFVRSBSRVNQRUlUT1UgVE9ET1MgRSBRVUFJU1FVRVIgRElSRUlUT1MgREUgUkVWSVPDg08gQ09NTyBUQU1Cw4lNIEFTIERFTUFJUyBPQlJJR0HDh8OVRVMgRVhJR0lEQVMgUE9SIENPTlRSQVRPIE9VIEFDT1JETy4KCk8gUmVwb3NpdMOzcmlvIEluc3RpdHVjaW9uYWwgZGEgVUZNRyBzZSBjb21wcm9tZXRlIGEgaWRlbnRpZmljYXIgY2xhcmFtZW50ZSBvIHNldSBub21lKHMpIG91IG8ocykgbm9tZXMocykgZG8ocykgZGV0ZW50b3IoZXMpIGRvcyBkaXJlaXRvcyBhdXRvcmFpcyBkYSBwdWJsaWNhw6fDo28sIGUgbsOjbyBmYXLDoSBxdWFscXVlciBhbHRlcmHDp8OjbywgYWzDqW0gZGFxdWVsYXMgY29uY2VkaWRhcyBwb3IgZXN0YSBsaWNlbsOnYS4KRepositório InstitucionalPUBhttps://repositorio.ufmg.br/oaiopendoar:2022-01-08T03:19:48Repositório Institucional da UFMG - Universidade Federal de Minas Gerais (UFMG)false
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