On riemannian connections and semi-simplicity of a Lie algebra
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866916321053114368 |
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| author | Anona, Manelo |
| author_facet | Anona, Manelo |
| contents | Using a almost product structure defined by a spray, we give a necessary and sufficient condition, for a linear connection with vanishing torsion to be Riemannian and, for the semi-simplicity of Lie algebra of projectable vector fields which commute with a spray. We show the equivalence of the semi-simplicity of a finite dimensional Lie algebra to the coincidence of the derived ideal with its algebra, to the interiority of any derivation of the algebra and, to the semi-simplicity of its adjoint representation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_13475 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On riemannian connections and semi-simplicity of a Lie algebra Anona, Manelo Differential Geometry 53XX, 17B66, 53B05, 53C08 Using a almost product structure defined by a spray, we give a necessary and sufficient condition, for a linear connection with vanishing torsion to be Riemannian and, for the semi-simplicity of Lie algebra of projectable vector fields which commute with a spray. We show the equivalence of the semi-simplicity of a finite dimensional Lie algebra to the coincidence of the derived ideal with its algebra, to the interiority of any derivation of the algebra and, to the semi-simplicity of its adjoint representation. |
| title | On riemannian connections and semi-simplicity of a Lie algebra |
| topic | Differential Geometry 53XX, 17B66, 53B05, 53C08 |
| url | https://arxiv.org/abs/2211.13475 |