Singular Degenerations of Lie Supergroups of Type D(2, 1; a)
The complex Lie superalgebras g of type D(2, 1; a) - also denoted by osp(4, 2; a) - are usually considered for "non-singular" values of the parameter a, for which they are simple. In this paper, we introduce five suitable integral forms of g that are well-defined at singular values too, gi...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2018 |
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Інститут математики НАН України
2018
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| Zitieren: | Singular Degenerations of Lie Supergroups of Type D(2, 1; a) / K. Iohara, F. Gavarini // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 18 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862650520887361536 |
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| author | Iohara, K. Gavarini, F. |
| author_facet | Iohara, K. Gavarini, F. |
| citation_txt | Singular Degenerations of Lie Supergroups of Type D(2, 1; a) / K. Iohara, F. Gavarini // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 18 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The complex Lie superalgebras g of type D(2, 1; a) - also denoted by osp(4, 2; a) - are usually considered for "non-singular" values of the parameter a, for which they are simple. In this paper, we introduce five suitable integral forms of g that are well-defined at singular values too, giving rise to "singular specializations" that are no longer simple: this extends the family of simple objects of type D(2, 1; a) in five different ways. The resulting five families coincide for general values of a but are different at "singular" ones: here they provide non-simple Lie superalgebras, whose structure we describe explicitly. We also perform the parallel construction for complex Lie supergroups and describe their singular specializations (or "degenerations") at singular values of a. Although one may work with a single complex parameter a, in order to stress the overall S3-symmetry of the whole situation, we shall work (following Kaplansky) with a two-dimensional parameter σ=(σ₁,σ₂,σ₃) ranging in the complex affine plane σ₁+σ₂+σ₃=0.
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| first_indexed | 2025-12-07T14:38:41Z |
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| id | nasplib_isofts_kiev_ua-123456789-209868 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T14:38:41Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
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| spelling | Iohara, K. Gavarini, F. 2025-11-28T09:33:51Z 2018 Singular Degenerations of Lie Supergroups of Type D(2, 1; a) / K. Iohara, F. Gavarini // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 18 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 14A22; 17B20; 13D10 arXiv: 1709.04717 https://nasplib.isofts.kiev.ua/handle/123456789/209868 https://doi.org/10.3842/SIGMA.2018.137 The complex Lie superalgebras g of type D(2, 1; a) - also denoted by osp(4, 2; a) - are usually considered for "non-singular" values of the parameter a, for which they are simple. In this paper, we introduce five suitable integral forms of g that are well-defined at singular values too, giving rise to "singular specializations" that are no longer simple: this extends the family of simple objects of type D(2, 1; a) in five different ways. The resulting five families coincide for general values of a but are different at "singular" ones: here they provide non-simple Lie superalgebras, whose structure we describe explicitly. We also perform the parallel construction for complex Lie supergroups and describe their singular specializations (or "degenerations") at singular values of a. Although one may work with a single complex parameter a, in order to stress the overall S3-symmetry of the whole situation, we shall work (following Kaplansky) with a two-dimensional parameter σ=(σ₁,σ₂,σ₃) ranging in the complex affine plane σ₁+σ₂+σ₃=0. The first author is partially supported by the French Agence Nationale de la Recherche (ANR GeoLie project ANR-15-CE40-0012). The second author acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome "Tor Vergata", CUP E83C18000100006. The authors would also like to thank the anonymous referees for their useful comments and suggestions to improve the presentation of this article. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Singular Degenerations of Lie Supergroups of Type D(2, 1; a) Article published earlier |
| spellingShingle | Singular Degenerations of Lie Supergroups of Type D(2, 1; a) Iohara, K. Gavarini, F. |
| title | Singular Degenerations of Lie Supergroups of Type D(2, 1; a) |
| title_full | Singular Degenerations of Lie Supergroups of Type D(2, 1; a) |
| title_fullStr | Singular Degenerations of Lie Supergroups of Type D(2, 1; a) |
| title_full_unstemmed | Singular Degenerations of Lie Supergroups of Type D(2, 1; a) |
| title_short | Singular Degenerations of Lie Supergroups of Type D(2, 1; a) |
| title_sort | singular degenerations of lie supergroups of type d(2, 1; a) |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209868 |
| work_keys_str_mv | AT ioharak singulardegenerationsofliesupergroupsoftyped21a AT gavarinif singulardegenerationsofliesupergroupsoftyped21a |