Cup product, Frölicher-Nijenhuis bracket and the derived bracket associated to Hom-Lie algebras
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914934764339200 |
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| author | Baishya, Anusuiya Das, Apurba |
| author_facet | Baishya, Anusuiya Das, Apurba |
| contents | In this paper, we introduce some new graded Lie algebras associated with a Hom-Lie algebra. At first, we define the cup product bracket and its application to the deformation theory of Hom-Lie algebra morphisms. We observe an action of the well-known Hom-analogue of the Nijenhuis-Richardson graded Lie algebra on the cup product graded Lie algebra. Using the corresponding semidirect product, we define the Frölicher-Nijenhuis bracket and study its application to Nijenhuis operators. We show that the Nijenhuis-Richardson graded Lie algebra and the Frölicher-Nijenhuis algebra constitute a matched pair of graded Lie algebras. Finally, we define another graded Lie bracket, called the derived bracket that is useful to study Rota-Baxter operators on Hom-Lie algebras. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_01865 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cup product, Frölicher-Nijenhuis bracket and the derived bracket associated to Hom-Lie algebras Baishya, Anusuiya Das, Apurba Rings and Algebras Representation Theory 17B61, 17B70, 17B56 In this paper, we introduce some new graded Lie algebras associated with a Hom-Lie algebra. At first, we define the cup product bracket and its application to the deformation theory of Hom-Lie algebra morphisms. We observe an action of the well-known Hom-analogue of the Nijenhuis-Richardson graded Lie algebra on the cup product graded Lie algebra. Using the corresponding semidirect product, we define the Frölicher-Nijenhuis bracket and study its application to Nijenhuis operators. We show that the Nijenhuis-Richardson graded Lie algebra and the Frölicher-Nijenhuis algebra constitute a matched pair of graded Lie algebras. Finally, we define another graded Lie bracket, called the derived bracket that is useful to study Rota-Baxter operators on Hom-Lie algebras. |
| title | Cup product, Frölicher-Nijenhuis bracket and the derived bracket associated to Hom-Lie algebras |
| topic | Rings and Algebras Representation Theory 17B61, 17B70, 17B56 |
| url | https://arxiv.org/abs/2409.01865 |