Fine Gradings of Low-Rank Complex Lie Algebras and of Their Real Forms
In this review paper, we treat the topic of fine gradings of Lie algebras. This concept is important not only for investigating the structural properties of the algebras, but, on top of that, the fine gradings are often used as the starting point for studying graded contractions or deformations of t...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2008 |
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Інститут математики НАН України
2008
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/149045 |
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| Cite this: | Fine Gradings of Low-Rank Complex Lie Algebras and of Their Real Forms / M. Svobodová // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 26 назв. — англ. |
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Svobodová, M. 2019-02-19T13:16:57Z 2019-02-19T13:16:57Z 2008 Fine Gradings of Low-Rank Complex Lie Algebras and of Their Real Forms / M. Svobodová // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 26 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 17B45; 22E60 https://nasplib.isofts.kiev.ua/handle/123456789/149045 In this review paper, we treat the topic of fine gradings of Lie algebras. This concept is important not only for investigating the structural properties of the algebras, but, on top of that, the fine gradings are often used as the starting point for studying graded contractions or deformations of the algebras. One basic question tackled in the work is the relation between the terms 'grading' and 'group grading'. Although these terms have originally been claimed to coincide for simple Lie algebras, it was revealed later that the proof of this assertion was incorrect. Therefore, the crucial statements about one-to-one correspondence between fine gradings and MAD-groups had to be revised and re-formulated for fine group gradings instead. However, there is still a hypothesis that the terms 'grading' and 'group grading' coincide for simple complex Lie algebras. We use the MAD-groups as the main tool for finding fine group gradings of the complex Lie algebras A₃ = D₃, B₂ = C₂, and D₂. Besides, we develop also other methods for finding the fine (group) gradings. They are useful especially for the real forms of the complex algebras, on which they deliver richer results than the MAD-groups. Systematic use is made of the faithful representations of the three Lie algebras by 4 × 4 matrices: A₃ = sl(4,C), C₂ = sp(4,C), D₂ = o(4,C). The inclusions sl(4,C) É sp(4,C) and sl(4,C) É o(4,C) are important in our presentation, since they allow to employ one of the methods which considerably simplifies the calculations when finding the fine group gradings of the subalgebras sp(4,C) and o(4,C). This paper is a contribution to the Proceedings of the Seventh International Conference “Symmetry in Nonlinear Mathematical Physics” (June 24–30, 2007, Kyiv, Ukraine). The author acknowledges financial support by NCLF (Nadace ˇ Cesk´y liter´arn´ı fond) and by the grants LC06002 and MSM6840770039 of the Ministry of Education, Youth, and Sports of the Czech Republic en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Fine Gradings of Low-Rank Complex Lie Algebras and of Their Real Forms Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Fine Gradings of Low-Rank Complex Lie Algebras and of Their Real Forms |
| spellingShingle |
Fine Gradings of Low-Rank Complex Lie Algebras and of Their Real Forms Svobodová, M. |
| title_short |
Fine Gradings of Low-Rank Complex Lie Algebras and of Their Real Forms |
| title_full |
Fine Gradings of Low-Rank Complex Lie Algebras and of Their Real Forms |
| title_fullStr |
Fine Gradings of Low-Rank Complex Lie Algebras and of Their Real Forms |
| title_full_unstemmed |
Fine Gradings of Low-Rank Complex Lie Algebras and of Their Real Forms |
| title_sort |
fine gradings of low-rank complex lie algebras and of their real forms |
| author |
Svobodová, M. |
| author_facet |
Svobodová, M. |
| publishDate |
2008 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
In this review paper, we treat the topic of fine gradings of Lie algebras. This concept is important not only for investigating the structural properties of the algebras, but, on top of that, the fine gradings are often used as the starting point for studying graded contractions or deformations of the algebras. One basic question tackled in the work is the relation between the terms 'grading' and 'group grading'. Although these terms have originally been claimed to coincide for simple Lie algebras, it was revealed later that the proof of this assertion was incorrect. Therefore, the crucial statements about one-to-one correspondence between fine gradings and MAD-groups had to be revised and re-formulated for fine group gradings instead. However, there is still a hypothesis that the terms 'grading' and 'group grading' coincide for simple complex Lie algebras. We use the MAD-groups as the main tool for finding fine group gradings of the complex Lie algebras A₃ = D₃, B₂ = C₂, and D₂. Besides, we develop also other methods for finding the fine (group) gradings. They are useful especially for the real forms of the complex algebras, on which they deliver richer results than the MAD-groups. Systematic use is made of the faithful representations of the three Lie algebras by 4 × 4 matrices: A₃ = sl(4,C), C₂ = sp(4,C), D₂ = o(4,C). The inclusions sl(4,C) É sp(4,C) and sl(4,C) É o(4,C) are important in our presentation, since they allow to employ one of the methods which considerably simplifies the calculations when finding the fine group gradings of the subalgebras sp(4,C) and o(4,C).
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/149045 |
| citation_txt |
Fine Gradings of Low-Rank Complex Lie Algebras and of Their Real Forms / M. Svobodová // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 26 назв. — англ. |
| work_keys_str_mv |
AT svobodovam finegradingsoflowrankcomplexliealgebrasandoftheirrealforms |
| first_indexed |
2025-12-07T13:33:06Z |
| last_indexed |
2025-12-07T13:33:06Z |
| _version_ |
1850856583465009153 |