Darboux Families and the Classification of Real Four-Dimensional Indecomposable Coboundary Lie Bialgebras
This work introduces a new concept, the so-called <i>Darboux family</i>, which is employed to determine coboundary Lie bialgebras on real four-dimensional, indecomposable Lie algebras, as well as geometrically analysying, and classifying them up to Lie algebra automorphisms, in a relativ...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-03-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/3/465 |
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| author | Javier de Lucas Daniel Wysocki |
| author_facet | Javier de Lucas Daniel Wysocki |
| author_sort | Javier de Lucas |
| collection | DOAJ |
| description | This work introduces a new concept, the so-called <i>Darboux family</i>, which is employed to determine coboundary Lie bialgebras on real four-dimensional, indecomposable Lie algebras, as well as geometrically analysying, and classifying them up to Lie algebra automorphisms, in a relatively easy manner. The Darboux family notion can be considered as a generalisation of the Darboux polynomial for a vector field. The classification of <i>r</i>-matrices and solutions to classical Yang–Baxter equations for real four-dimensional indecomposable Lie algebras is also given in detail. Our methods can further be applied to general, even higher-dimensional, Lie algebras. As a byproduct, a method to obtain matrix representations of certain Lie algebras with a non-trivial center is developed. |
| first_indexed | 2024-03-10T13:17:53Z |
| format | Article |
| id | doaj.art-eab150c89612430faea92a0a66a110a1 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T13:17:53Z |
| publishDate | 2021-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-eab150c89612430faea92a0a66a110a12023-11-21T10:18:14ZengMDPI AGSymmetry2073-89942021-03-0113346510.3390/sym13030465Darboux Families and the Classification of Real Four-Dimensional Indecomposable Coboundary Lie BialgebrasJavier de Lucas0Daniel Wysocki1Department of Mathematical Methods in Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, PolandDepartment of Mathematical Methods in Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, PolandThis work introduces a new concept, the so-called <i>Darboux family</i>, which is employed to determine coboundary Lie bialgebras on real four-dimensional, indecomposable Lie algebras, as well as geometrically analysying, and classifying them up to Lie algebra automorphisms, in a relatively easy manner. The Darboux family notion can be considered as a generalisation of the Darboux polynomial for a vector field. The classification of <i>r</i>-matrices and solutions to classical Yang–Baxter equations for real four-dimensional indecomposable Lie algebras is also given in detail. Our methods can further be applied to general, even higher-dimensional, Lie algebras. As a byproduct, a method to obtain matrix representations of certain Lie algebras with a non-trivial center is developed.https://www.mdpi.com/2073-8994/13/3/465coboundary Lie bialgebracocommutatorDarboux polynomialDarboux familygeneralised distributionindecomposable Lie algebra |
| spellingShingle | Javier de Lucas Daniel Wysocki Darboux Families and the Classification of Real Four-Dimensional Indecomposable Coboundary Lie Bialgebras Symmetry coboundary Lie bialgebra cocommutator Darboux polynomial Darboux family generalised distribution indecomposable Lie algebra |
| title | Darboux Families and the Classification of Real Four-Dimensional Indecomposable Coboundary Lie Bialgebras |
| title_full | Darboux Families and the Classification of Real Four-Dimensional Indecomposable Coboundary Lie Bialgebras |
| title_fullStr | Darboux Families and the Classification of Real Four-Dimensional Indecomposable Coboundary Lie Bialgebras |
| title_full_unstemmed | Darboux Families and the Classification of Real Four-Dimensional Indecomposable Coboundary Lie Bialgebras |
| title_short | Darboux Families and the Classification of Real Four-Dimensional Indecomposable Coboundary Lie Bialgebras |
| title_sort | darboux families and the classification of real four dimensional indecomposable coboundary lie bialgebras |
| topic | coboundary Lie bialgebra cocommutator Darboux polynomial Darboux family generalised distribution indecomposable Lie algebra |
| url | https://www.mdpi.com/2073-8994/13/3/465 |
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