Inductive description of quadratic Hom-Lie algebras with twist maps in the centroid
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909450818813952 |
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| author | García-Delgado, R. |
| author_facet | García-Delgado, R. |
| contents | In this work we give an inductive way to construct quadratic Hom-Lie algebras with twist maps in the centroid. We focus on those Hom-Lie algebras which are not Lie algebras. We prove that a Hom-Lie algebra of this type has trivial center and its twist map is nilpotent. We show that there exists a maximal ideal containing the kernel and the image of the twist map. Then we state an inductive way to construct this type of Hom-Lie algebras, similar to the double extension procedure for Lie algebras, and prove that any indecomposable quadratic Hom-Lie algebra with nilpotent twist map in the centroid, which is not a Lie algebra, can be constructed using this type of double extension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_04546 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Inductive description of quadratic Hom-Lie algebras with twist maps in the centroid García-Delgado, R. Rings and Algebras 17B61, 17A30 (Primary) 17B60, 17D30 (Secondary) In this work we give an inductive way to construct quadratic Hom-Lie algebras with twist maps in the centroid. We focus on those Hom-Lie algebras which are not Lie algebras. We prove that a Hom-Lie algebra of this type has trivial center and its twist map is nilpotent. We show that there exists a maximal ideal containing the kernel and the image of the twist map. Then we state an inductive way to construct this type of Hom-Lie algebras, similar to the double extension procedure for Lie algebras, and prove that any indecomposable quadratic Hom-Lie algebra with nilpotent twist map in the centroid, which is not a Lie algebra, can be constructed using this type of double extension. |
| title | Inductive description of quadratic Hom-Lie algebras with twist maps in the centroid |
| topic | Rings and Algebras 17B61, 17A30 (Primary) 17B60, 17D30 (Secondary) |
| url | https://arxiv.org/abs/2409.04546 |