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Isomorphism Problem for Universal Enveloping Algebras of Solvable Lie Algebras

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Isomorphism Problem for Universal Enveloping Algebras of Solvable Lie Algebras UNIVERSIDADE FEDERAL DE MINAS GERAIS - UFMG INSTITUTO DE CIENCIAS^ EXATAS DEPARTAMENTO DE POS-GRADUAC¸´ AO~ EM MATEMATICA´ Isomorphism Problem for Universal Enveloping Algebras of Solvable Lie Algebras Jos´eLuis Vilca Rodr´ıguez Belo Horizonte - MG 2019 UNIVERSIDADE FEDERAL DE MINAS GERAIS - UFMG INSTITUTO DE CIENCIAS^ EXATAS DEPARTAMENTO DE POS-GRADUAC¸´ AO~ EM MATEMATICA´ Jos´eLuis Vilca Rodr´ıguez Advisor: Csaba Schneider Isomorphism Problem for Universal Enveloping Algebras of Solvable Lie Algebras Thesis presented to the Federal Uni- versity of Minas Gerais in fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Belo Horizonte - MG 2019 Siempre que biene el tiempo fresco, o sea al medio del otonio, a m´ıme da la loca de pensar ideas de tipo es´entrico y es´otico, como ser por egenplo que me gustar´ıavenirme golondrina para agarrar y volar a los pa´ıxadonde haiga calor, o de ser hormiga para meterme bien adentro de una cueva y comer los productos guardados en el verano o de ser una b´ıvora como las del sol´ojico, que las tienen bien guardadas en una jaula de vidrio con calefaci´onpara que no se queden duras de fr´ıo,que es lo que les pasa a los pobres seres humanos que no pueden comprarse ropa con lo cara quest´a,ni pueden calentarse por la falta del queros´en,la falta del carb´on,la falta de lenia, la falta de petrolio y tami´enla falta de plata, porque cuando uno anda con biyuya ensima puede entrar a cualquier boliche y mandarse una buena grapa que hay que ver lo que calienta, aunque no conbiene abusar, porque del abuso entra el visio y del visio la dejenerad´estanto del cuerpo como de las taras moral de cada cual, y cuando se viene abajo por la pendiente fatal de la falta de buena condupta en todo sentido, ya nadie ni nadies lo salva de acabar en el m´asespantoso tacho de basura del desprasti- jio humano, y nunca le van a dar una mano para sacarlo de adentro del fango enmundo entre el cual se rebuelca, ni m´asni meno que si fuera un c´ondorque cuando joven supo correr y volar por la punta de las altas montanias, pero que al ser viejo cay´oparabajo como bombardero en picada que le falia el motor moral. <Y ojal´aque lo que estoy escribiendol´e sirbalguno para que mire bien su comportamiento y que no searrepienta cuando es tarde y ya todo se haiga ido al corno por culpa suya! CESAR´ BRUTO, Lo que me gustar´ıaser a m´ısi no fuera lo que soy. S´ı, ya s´e,hay m´asproblem´ologos que solu- cion´ologos, pero >qu´e vamos a hacerle? QUINO, Mafalda. 1 Agradecimientos El presente trabajo es el fruto de significativos esfuerzos de muchas personas. Juntos pudimos concretar la fabricaci´onde esta peque~napieza, en la gran maquina que es la matem´atica. As´ıcomo una pared no se construye s´olocon ladrillos, este trabajo no se construyo s´olocon investigaci´on,hizo falta la mano de muchas personas que me brindaron apoyo y motivaci´ona lo largo de mi vida. Lleg´ola hora de agradecer a todos ellos. • A mis padres. Gracias por transmitirme lo importante que es la educaci´on y apoyarme siempre en todas mis decisiones. Gracias por lo mucho que se sacrificaron por m´ı, para que pueda tener las oportunidades que ustedes no tuvieron. La conclusi´onde este doctorado es tambi´enla conclusi´onde sus esfuerzos. • A mis hermanos. Por la camarader´ıa,apoyo, motivaci´ony confianza en todos estos a~nosfuera de casa, y por todas las vivencias en todo el tiempo que estuve f´ısicamente m´ascerca de ustedes. Ustedes son mis mejores amigos. Para ustedes mis agradecimientos. • A mis abuelos, Leoncio y Valeriana. Gracias por todo el tiempo que me dedi- caron y por el apoyo incondicional que me brindaron. • A mi orientador, el profesor Csaba Schneider. Gracias por la dedicaci´on,tiempo, esfuerzo y paciencia para orientarme en mi vida profesional. Creo que debe ser muy dif´ıcilorientarme. Aprend´ımucho con usted. • A los profesores que examinaron este trabajo: Ana Cristina, Andr´eContiero, Lucas Calixto, V´ıctorPetrogradsky y Wagner Cortes. Gracias por el tiempo que dedicaron a leer este trabajo, y por las muchas sugerencias y contribuciones que sirvieron para enriquecerlo. • Al profesor Walter Torres, uno de mis grandes mentores. El ´unicoprofesor que sin usar una pizarra me hizo entender la matem´atica.Gracias por siempre motivarme a estudiar matem´atica. 2 • A la CNPq y la CAPES por el apoyo financiero. • A los funcionarios de la secretaria. En especial a Andrea y Kelli. Gracias por ayudarme en todos los momentos de correr´ıa. Siempre me atendieron con amabilidad y eficiencia, y sab´ıanm´asde mi vida acad´emicaque yo mismo. • A los amigos que conoc´ıdentro y fuera de mi vida acad´emica.A Carlos (papa frita), Eduardo (manito), Leo (el psic´ologo), Jhon (uai fai), Katy (la tata), Simeona (ya se ver´a...), gracias a ustedes por m´asde una d´ecadade amistad. A Ricardo (Homer), Joel (Desmond), Carlos Salazar (Andi), Alfonso (Matute), Matute (Alfonso), Tauan (baiano), Mario, Rafael, Vinicius, Ayane (Hahihani), Myrla (aTESTAda), Diogo (t´amaluco?), Moacir (Moacicinho), Aislan (pisado), Eduardo (Gordon), Aline (vamos pro forr´o?), Rensso, Guillermo, V´ıctor,An- drea, Elena. Gracias a todos ustedes por mostrarme que lejos de casa amigo es sin´onimode familia. 3 Abstract We dedicate this thesis to study the isomorphism problem for solvable Lie algebras of dimension at most 4. The principal contribution is to show that the isomorphism problem has a positive solution for 4-dimensional solvable Lie algebras over a field of characteristic zero. That is, we show that given two 4-dimensional Lie algebras L and H over a field F of characteristic 0, then UpLq – UpHq imply L – H. To reach our goal we use three tools. The first is the Frobenius semiradical and its relationship with the structure of a universal enveloping algebra of a Lie algebra. Using the Frobenius semiradical we show that for certain Lie algebras L the dimen- sion of ZpLq is determined by the isomorphism type of UpLq. The second tool is a construction. Given a Lie algebra L and an abelian ideal M of L, we produce a Lie algebra LM which is invariant under a group of automorphisms of UpLq. We show that in several cases the Lie algebras L and LM are isomorphic. Finally, the third tool is the study of structure of the universal enveloping algebras for Lie algebras with an ideal of codimension one. Analysing this type of Lie algebras we show that the isomorphism type of L{L2 is determined by the isomorphism type of UpLq, and as a consequence of this fact the isomorphisms problem has a positive solution for metabelian Lie algebras in which the codimension of the derived algebra is one. Although our main result is given in dimension four, in several of our results we require only that the Lie algebra be finite-dimensional. Further, many of our results hold for an arbitrary base field independently of the characteristic. Key words: Solvable Lie algebras, universal enveloping algebras, isomorphism prob- lem. 4 Resumo Dedicamos esta tese, basicamente, a estudar o problema do isomorfismo para ´algebrasde Lie sol´uveis de dimens~aono m´aximo4. A principal contribui¸c~ao ´emostrar que o problema do isomorfismo tem uma solu¸c~ao positiva para ´algebrasde Lie sol´uveis de dimens~ao4 sobre um corpo de caracter´ısticazero. Isto ´e,mostramos que, dadas duas ´algebras de Lie 4-dimensionais L e H sobre um corpo F de caracter´ıstica0, ent~ao UpLq – UpHq implica L – H. Para alcan¸carnosso objetivo, usamos basicamente tr^esferramentas. A primeira ´e o semirradical de Frobenius e sua rela¸c~aocom a estrutura de uma ´algebra envolvente universal de uma ´algebrade Lie. Usando o semirradical de Frobenius, mostramos que, para certas ´algebrade Lie L, a dimens~aode ZpLq ´edeterminada pelo tipo de isomorfismo de UpLq. A segunda ferramenta ´euma constru¸c~ao. Dada uma ´algebrade Lie L e um ideal abeliano M de L, produzimos uma ´algebrade Lie LM que ´einvariante por um grupo de automorfismos de UpLq. Mostramos que, em v´arioscasos, as ´algebras de Lie L e LM s~aoisomorfas. Finalmente, a terceira ferramenta ´eo estudo da estrutura das ´algebras envolventes universais para ´algebrasde Lie com um ideal de codimens~ao um. Analisando este tipo de ´algebrasde Lie, mostramos que o tipo de isomorfismo L{L2 ´edeterminado pelo tipo de isomorfismo de UpLq, e como consequ^enciadeste fato temos que o problema dos isomorfismos tem uma solu¸c~aopositiva para ´algebras de Lie metabelianas para as quais a codimens~aoda ´algebraderivada ´eum. Embora nosso principal resultado seja dado em dimens~ao4, em v´ariosdos nossos resultados, exigimos apenas que a ´algebrade Lie seja de dimens~aofinita. Al´emdisso, muitos dos nossos resultados valem para um corpo arbitr´ario,independentemente da caracter´ıstica. Palavras Chave: Algebras´ de Lie sol´uveis, ´algebrasenvolventes universais, problema do isomorfismo. 5 Contents Resumo Estendido 8 1 Introduction 17 1.1 Historical background . 17 1.2 Center, semicenter and Frobenius semiradical . 21 1.3 Structure of UpLq for certain Lie algebras . 23 2 Background 25 2.1 Lie algebras and their universal enveloping algebras . 25 2.2 Ring of quotients of the universal enveloping algebra . 36 2.3 Isomorphism between universal enveloping algebras of Lie algebras . 39 2.4 Hopf algebras . 41 3 Solvable Lie algebras of dimension at most 4 47 3.1 The classification by de Graaf .
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