Deformations of Symmetric Simple Modular Lie (Super)Algebras
We say that a Lie (super)algebra is ''symmetric'' if, with every root (with respect to the maximal torus) it has the opposite root of the same multiplicity. Over algebraically closed fields of positive characteristics (up to 7 or 11, enough to formulate a general conjecture), we...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2023 |
| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2023
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/211912 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Deformations of Symmetric Simple Modular Lie (Super)Algebras. Sofiane Bouarroudj, Pavel Grozman and Dimitry Leites. SIGMA 19 (2023), 031, 66 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862551262465097728 |
|---|---|
| author | Bouarroudj, Sofiane Grozman, Pavel Leites, Dimitry |
| author_facet | Bouarroudj, Sofiane Grozman, Pavel Leites, Dimitry |
| citation_txt | Deformations of Symmetric Simple Modular Lie (Super)Algebras. Sofiane Bouarroudj, Pavel Grozman and Dimitry Leites. SIGMA 19 (2023), 031, 66 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We say that a Lie (super)algebra is ''symmetric'' if, with every root (with respect to the maximal torus) it has the opposite root of the same multiplicity. Over algebraically closed fields of positive characteristics (up to 7 or 11, enough to formulate a general conjecture), we computed the cohomology corresponding to the infinitesimal deformations of all known simple finite-dimensional symmetric Lie (super)algebras of rank < 9, except for superizations of the Lie algebras with ADE root systems, and queerified Lie algebras, considered only partly. The moduli of deformations of any Lie superalgebra constitute a supervariety. Any infinitesimal deformation given by an odd cocycle is integrable. All deformations corresponding to odd cocycles are new. Among new results are classifications of the cocycles describing deformations (results of deformations) of the 29-dimensional Brown algebra in characteristic 3, of Weisfeiler-Kac algebras, and orthogonal Lie algebras without Cartan matrix in characteristic 2. Open problems: describe non-isomorphic deforms and equivalence classes of cohomology theories. Appendix: For several modular analogs of complex simple Lie algebras, and simple Lie algebras indigenous to characteristics 3 and 2, we describe the space of cohomology with trivial coefficients. We show that the natural multiplication in this space is very complicated.
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| first_indexed | 2026-03-13T04:30:39Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211912 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T04:30:39Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bouarroudj, Sofiane Grozman, Pavel Leites, Dimitry 2026-01-16T11:18:10Z 2023 Deformations of Symmetric Simple Modular Lie (Super)Algebras. Sofiane Bouarroudj, Pavel Grozman and Dimitry Leites. SIGMA 19 (2023), 031, 66 pages 1815-0659 2020 Mathematics Subject Classification: 17B50; 17B55; 17B56; 17B20 arXiv:0807.3054 https://nasplib.isofts.kiev.ua/handle/123456789/211912 https://doi.org/10.3842/SIGMA.2023.031 We say that a Lie (super)algebra is ''symmetric'' if, with every root (with respect to the maximal torus) it has the opposite root of the same multiplicity. Over algebraically closed fields of positive characteristics (up to 7 or 11, enough to formulate a general conjecture), we computed the cohomology corresponding to the infinitesimal deformations of all known simple finite-dimensional symmetric Lie (super)algebras of rank < 9, except for superizations of the Lie algebras with ADE root systems, and queerified Lie algebras, considered only partly. The moduli of deformations of any Lie superalgebra constitute a supervariety. Any infinitesimal deformation given by an odd cocycle is integrable. All deformations corresponding to odd cocycles are new. Among new results are classifications of the cocycles describing deformations (results of deformations) of the 29-dimensional Brown algebra in characteristic 3, of Weisfeiler-Kac algebras, and orthogonal Lie algebras without Cartan matrix in characteristic 2. Open problems: describe non-isomorphic deforms and equivalence classes of cohomology theories. Appendix: For several modular analogs of complex simple Lie algebras, and simple Lie algebras indigenous to characteristics 3 and 2, we describe the space of cohomology with trivial coefficients. We show that the natural multiplication in this space is very complicated. We are thankful to A. Krutov and A. Lebedev for their huge help. We are thankful to N. Chebochko and M. Kuznetsov for helpful discussions of their unpublished results pertaining to this paper. Wethank A. Dzhumadildaev for pointing out [67], thus correcting an error. We are very thankful to the referees, carefully selected by SIGMA, for their constructive criticism and extremely careful job. D.L. is thankful to MPIMiS, Leipzig, where he was Sophus-Lie-Professor (2004-07, when certain results of this paper were obtained), for financial support and the most creative environment. We are thankful to M. Al Barwani, Director of the High Performance Computing resources at New York University Abu Dhabi, for the possibility to perform the difficult computations of this research. S.B. and D.L. were supported by the grant AD 065 NYUAD. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Deformations of Symmetric Simple Modular Lie (Super)Algebras Article published earlier |
| spellingShingle | Deformations of Symmetric Simple Modular Lie (Super)Algebras Bouarroudj, Sofiane Grozman, Pavel Leites, Dimitry |
| title | Deformations of Symmetric Simple Modular Lie (Super)Algebras |
| title_full | Deformations of Symmetric Simple Modular Lie (Super)Algebras |
| title_fullStr | Deformations of Symmetric Simple Modular Lie (Super)Algebras |
| title_full_unstemmed | Deformations of Symmetric Simple Modular Lie (Super)Algebras |
| title_short | Deformations of Symmetric Simple Modular Lie (Super)Algebras |
| title_sort | deformations of symmetric simple modular lie (super)algebras |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211912 |
| work_keys_str_mv | AT bouarroudjsofiane deformationsofsymmetricsimplemodularliesuperalgebras AT grozmanpavel deformationsofsymmetricsimplemodularliesuperalgebras AT leitesdimitry deformationsofsymmetricsimplemodularliesuperalgebras |