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UNIVERSIDADE ESTADUAL DE CAMPINAS Representations Of UNIVERSIDADE ESTADUAL DE CAMPINAS Instituto de Matemática, Estatística e Computação Científica YURY POPOV Representations of noncommutative Jordan superalgebras Representações de superálgebras não comutativas de Jordan Campinas 2020 Yury Popov Representations of noncommutative Jordan superalgebras Representações de superálgebras não comutativas de Jordan Tese apresentada ao Instituto de Matemática, Estatística e Computação Científica da Uni- versidade Estadual de Campinas como parte dos requisitos exigidos para a obtenção do título de Doutor em Matemática. Thesis presented to the Institute of Mathe- matics, Statistics and Scientific Computing of the University of Campinas in partial ful- fillment of the requirements for the degree of Doctor in Mathematics. Supervisor: Artem Lopatin Este exemplar corresponde à versão final da Tese defendida pelo aluno Yury Popov e orientada pelo Prof. Dr. Artem Lopatin. Campinas 2020 Ficha catalográfica Universidade Estadual de Campinas Biblioteca do Instituto de Matemática, Estatística e Computação Científica Ana Regina Machado - CRB 8/5467 Popov, Yury, 1993- P814r PopRepresentations of noncommutative Jordan superalgebras / Yury Popov. – Campinas, SP : [s.n.], 2020. PopOrientador: Artem Lopatin. PopTese (doutorado) – Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica. Pop1. Álgebras de Jordan. 2. Representações de álgebras. 3. Álgebras não- associativas. I. Lopatin, Artem, 1980-. II. Universidade Estadual de Campinas. Instituto de Matemática, Estatística e Computação Científica. III. Título. Informações para Biblioteca Digital Título em outro idioma: Representações de superálgebras não comutativas de Jordan Palavras-chave em inglês: Jordan algebras Representations of algebras Nonassociative algebras Área de concentração: Matemática Titulação: Doutor em Matemática Banca examinadora: Artem Lopatin [Orientador] Ivan Chestakov Henrique Guzzo Júnior Plamen Emilov Kochloukov Victor Mikhaylovich Petrogradsky Data de defesa: 30-09-2020 Programa de Pós-Graduação: Matemática Identificação e informações acadêmicas do(a) aluno(a) - ORCID do autor: https://orcid.org/0000-0002-3489-8838 - Currículo Lattes do autor: http://lattes.cnpq.br/3084813425763169 Powered by TCPDF (www.tcpdf.org) Tese de Doutorado defendida em 30 de setembro de 2020 e aprovada pela banca examinadora composta pelos Profs. Drs. Prof(a). Dr(a). ARTEM LOPATIN Prof(a). Dr(a). HENRIQUE GUZZO JÚNIOR Prof(a). Dr(a). PLAMEN EMILOV KOCHLOUKOV Prof(a). Dr(a). IVAN CHESTAKOV Prof(a). Dr(a). VICTOR MIKHAYLOVICH PETROGRADSKY A Ata da Defesa, assinada pelos membros da Comissão Examinadora, consta no SIGA/Sistema de Fluxo de Dissertação/Tese e na Secretaria de Pós-Graduação do Instituto de Matemática, Estatística e Computação Científica. Acknowledgements O presente trabalho foi realizado com apoio da Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Código de Financiamento 001. Agradeço a Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) pelo apoio financeiro (processo 2016/16445-0). Agradeço os membros da banca por terem participado na defesa e pelas su- gestões. Resumo Álgebras não comutativas de Jordan foram introduzidas por Albert. Ele observou que as teorias estruturais das álgebras alternativas e de Jordan compartilham tantas propriedades boas que é natural supor que essas álgebras são membros de uma classe mais geral com uma teoria estrutural semelhante. Então, ele introduziu a variedade de álgebras não comutativas de Jordan definidas pela identidade de Jordan e pela identidade da flexibilidade. A classe de (super)álgebras não-comutativas de Jordan se tornou vasta: por exemplo, além das (super)álgebras alternativas e de Jordan, ela contém álgebras quasiassociativas, (super)álgebras quadráticas flexíveis e (super)álgebras anticomutativas. No entanto, a teoria da estrutura dessa classe está longe de ser boa. No entanto, um certo progresso foi feito no estudo da teoria estrutural de álgebras (e, mais geralmente, superálgebras) não comutativas de Jordan. Particularmente, álgebras simples dessa classe foram estudadas por muitos autores. Superalgebras centrais não-comutativas simples de Jordan de dimensão finita foram descritas por Pozhidaev e Shestakov. Representações de superalgebras alternativas e de Jordan são um tópico popular atualmente, estudado por muitos autores. Neste trabalho, estudamos representações de álgebras não- comutativas de Jordan. Em particular, classificamos as representações irredutíveis de dimensões finitas de superalgebras simples não-comutativas de Jordan de dimensão finita sobre um corpo algebricamente fechado da característica 0 e mostramos o teorema de fatoração de Kronecker para algumas àlgebras. Palavras-chave: superálgebra não comutativa de Jordan, teoria de representações de superalgebras não-associativas, representações das superalgebras de Jordan, Teorema da fatoração de Kronecker. Abstract Noncommutative Jordan algebras were introduced by Albert. He noted that the structure theories of alternative and Jordan algebras share so many nice properties that it is natural to conjecture that these algebras are members of a more general class with a similar theory. So he introduced the variety of noncommutative Jordan algebras defined by the Jordan identity and the flexibility identity. The class of noncommutative Jordan (super)algebras turned out to be vast: for example, apart from alternative and Jordan (super)algebras it contains quasiassociative (super)algebras, quadratic flexible (super)algebras and (su- per)anticommutative (super)algebras. However, the structure theory of this class is far from being nice. Nevertheless, a certain progress was made in the study of structure theory of noncommuta- tive Jordan algebras (and, more generally, superalgebras). Particularly, simple algebras of this class were studied by many authors. Simple finite-dimensional central noncommutative Jordan superalgebras were described by Pozhidaev and Shestakov. Representations of alternative and Jordan superalgebras is a popular topic nowadays which was studied by many authors. In this work we study representations of noncommutative Jordan algebras. In particular, we classify the irreducible finite-dimensional representations of simple finite-dimensional noncommutative Jordan superalgebras over an algebraically closed field of characteristic 0 and prove Kronecker factorization theorems for some superalgebras. Keywords: noncommutative Jordan superalgebra, representation theory of nonassociative superalgebras, representations of Jordan superalgebras, Kronecker factorization theorem. Contents Introduction ............................... 10 1 PRELIMINARIES ............................ 13 1.1 Notations and defining identities ..................... 13 1.2 Peirce decomposition ........................... 15 1.3 Algebras with connected idempotents .................. 18 1.4 Mutations .................................. 20 1.5 Poisson brackets .............................. 22 1.6 Examples and classification in degree ¤ 2 ................ 23 1.7 The superalgebra Dtpα, β, γq ....................... 24 1.8 The superalgebra K3pα, β, γq ....................... 25 1.9 The superalgebra UpV, f, ‹q ........................ 26 1.10 The classification in degree 2 ....................... 27 1.11 Bimodules and representations ...................... 27 1.12 Noncommutative Jordan representations ................ 28 1.13 Discussion of main results ......................... 32 2 REPRESENTATIONS OF SIMPLE SUPERALGEBRAS OF DEGREE ¥ 3 AND JORDAN SUPERALGEBRAS................ 34 2.1 Superalgebras of degree ¥ 3 ....................... 34 2.2 Noncommutative Jordan representations of simple Jordan superal- gebras .................................... 36 2.2.1 Representations of the superalgebra JpV, fq. ................ 37 2.2.2 Representations of superalgebras Dt and K3. ................ 38 2.2.3 Representations of the superalgebra P p2q. .................. 40 2.2.4 Representations of the superalgebra Qp2qp`q ................. 41 2.2.5 Representations of superalgebras K10 and K9. ................ 42 2.2.6 Kronecker factorization theorem for K10 ................... 43 3 REPRESENTATIONS OF LOW-DIMENSIONAL ALGEBRAS . 46 3.1 Representations of Dtpα, β, γq and K3pα, β, γq .............. 46 3.1.1 Representations of Dtpλq, λ ‰ 1{2 ..................... 47 3.1.2 Representations of Dtp1{2, 1{2, 0q ...................... 55 3.1.3 Representations of K3pα, β, γq ........................ 63 3.2 Kronecker factorization theorem for Dtpα, β, γq ............ 64 3.2.1 Kronecker factorization theorem for Dtpλq .................. 64 3.2.2 Kronecker factorization theorem for Dtp1{2, 1{2, 0q ............. 69 3.3 Representations of Qp1q and Qp2q .................... 74 3.3.1 Representations of Qp1q ........................... 75 3.3.2 Representations of Qp2q ........................... 75 4 REPRESENTATIONS OF SUPERALGEBRAS UpV, fq AND KpΓn,Aq 77 4.1 Irreducibility of the symmetrized module ................ 77 4.2 Irreducible bimodules over UpV, f, ‹q ................... 81 4.2.1 The superalgebra UpV, f, ‹q ......................... 81 4.2.2 Classification of irreducible representations .................. 82 4.3 Irreducible bimodules over KpΓn,Aq ................... 88 4.3.1 The superalgebra KpΓn,Aq ......................... 88 4.3.2 Classification of irreducible representations .................. 89 BIBLIOGRAPHY ............................102 10 Introduction Noncommutative Jordan algebras were introduced by Albert in [Alb48]. He
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