On Gradings Modulo 2 of Simple Lie Algebras in Characteristic 2
The ground field in the text is of characteristic 2. The classification of modulo 2 gradings of simple Lie algebras is vital for the classification of simple finite-dimensional Lie superalgebras: with each grading, a simple Lie superalgebra is associated (see arXiv:1407.1695). No classification of g...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2018 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2018
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/209874 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On Gradings Modulo 2 of Simple Lie Algebras in Characteristic 2 / A. Krutov, A. Lebedev // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 29 назв. — англ. |
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Krutov, A. Lebedev, A. 2025-11-28T09:38:12Z 2018 On Gradings Modulo 2 of Simple Lie Algebras in Characteristic 2 / A. Krutov, A. Lebedev // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 29 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B50; 17B20; 17B70 arXiv: 1711.00638 https://nasplib.isofts.kiev.ua/handle/123456789/209874 https://doi.org/10.3842/SIGMA.2018.130 The ground field in the text is of characteristic 2. The classification of modulo 2 gradings of simple Lie algebras is vital for the classification of simple finite-dimensional Lie superalgebras: with each grading, a simple Lie superalgebra is associated (see arXiv:1407.1695). No classification of gradings was known for any type of simple Lie algebras, bar restricted Jacobson-Witt algebras (i.e., the first derived of the Lie algebras of vector fields with truncated polynomials as coefficients) on not less than 3 indeterminates. Here we completely describe gradings modulo 2 for several series of Lie algebras and their simple relatives: the special linear series, its projectivizations, and projectivizations of the derived Lie algebras of two inequivalent orthogonal series (except for oΠ(8)). The classification of gradings is new, but all the corresponding superizations are known. For the simple derived Zassenhaus algebras of height n > 1, there is an (n − 2)-parametric family of modulo 2 gradings; all but one of the corresponding simple Lie superalgebras are new. Our classification also proves the non-triviality of a deformation of a simple 3|2-dimensional Lie superalgebra (new result). We are thankful to D. Leites, who raised the problem, I. Shchepochkina, and S. Skryabin for help, and to P. Grozman, whose code SuperLie, see [15], we used in our computer experiments. Thanks are due to S. Bouarroudj and V. Grandjean for discussions and useful comments. The first author thanks the Organising Committee of the symposium "Groningen Deformation Day" (October 7, 2016, Groningen, The Netherlands), where the results of this note were delivered, for hospitality and financial support; his research was partly supported by the WCMCS post-doctoral fellowship and the grant AD 065 NYUAD during his visits to NYUAD. For the possibility to perform the difficult computations of this research, we are grateful to M. Al Barwani, Director of the High Performance Computing resources at New York University Abu Dhabi. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On Gradings Modulo 2 of Simple Lie Algebras in Characteristic 2 Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
On Gradings Modulo 2 of Simple Lie Algebras in Characteristic 2 |
| spellingShingle |
On Gradings Modulo 2 of Simple Lie Algebras in Characteristic 2 Krutov, A. Lebedev, A. |
| title_short |
On Gradings Modulo 2 of Simple Lie Algebras in Characteristic 2 |
| title_full |
On Gradings Modulo 2 of Simple Lie Algebras in Characteristic 2 |
| title_fullStr |
On Gradings Modulo 2 of Simple Lie Algebras in Characteristic 2 |
| title_full_unstemmed |
On Gradings Modulo 2 of Simple Lie Algebras in Characteristic 2 |
| title_sort |
on gradings modulo 2 of simple lie algebras in characteristic 2 |
| author |
Krutov, A. Lebedev, A. |
| author_facet |
Krutov, A. Lebedev, A. |
| publishDate |
2018 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
The ground field in the text is of characteristic 2. The classification of modulo 2 gradings of simple Lie algebras is vital for the classification of simple finite-dimensional Lie superalgebras: with each grading, a simple Lie superalgebra is associated (see arXiv:1407.1695). No classification of gradings was known for any type of simple Lie algebras, bar restricted Jacobson-Witt algebras (i.e., the first derived of the Lie algebras of vector fields with truncated polynomials as coefficients) on not less than 3 indeterminates. Here we completely describe gradings modulo 2 for several series of Lie algebras and their simple relatives: the special linear series, its projectivizations, and projectivizations of the derived Lie algebras of two inequivalent orthogonal series (except for oΠ(8)). The classification of gradings is new, but all the corresponding superizations are known. For the simple derived Zassenhaus algebras of height n > 1, there is an (n − 2)-parametric family of modulo 2 gradings; all but one of the corresponding simple Lie superalgebras are new. Our classification also proves the non-triviality of a deformation of a simple 3|2-dimensional Lie superalgebra (new result).
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/209874 |
| citation_txt |
On Gradings Modulo 2 of Simple Lie Algebras in Characteristic 2 / A. Krutov, A. Lebedev // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 29 назв. — англ. |
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AT krutova ongradingsmodulo2ofsimpleliealgebrasincharacteristic2 AT lebedeva ongradingsmodulo2ofsimpleliealgebrasincharacteristic2 |
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2025-12-07T19:52:54Z |
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2025-12-07T19:52:54Z |
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1850886173621223424 |