Curvatures of metric Jordan algebras
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866913355310039040 |
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| author | Zhang, Hui Yan, Zaili Chen, Zhiqi |
| author_facet | Zhang, Hui Yan, Zaili Chen, Zhiqi |
| contents | In this paper, we investigate metric Jordan algebras, and follow the lines of the paper (J. Milnor: Curvatures of left invariant metrics on Lie groups. Adv. Math. (1976)). Firstly, we define the Jordan-Levi-Civita connection, then we show that every metric Jordan algebra admits a unique Jordan-Levi-Civita connection. Secondly, using the Jordan-Levi-Civita connection, we introduce three natural curvature tensors on metric Jordan algebras, and obtain the corresponding curvature formulas. Thirdly, based on these curvature formulas, we prove that every formally real Jordan algebra admits both a metric of non-positive Jordan curvature, and a Jordan-Einstein metric of negative Jordan scalar curvature. Besides, for nilpotent Jordan algebras, we prove that they admit no Jordan-Einstein metrics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_02682 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Curvatures of metric Jordan algebras Zhang, Hui Yan, Zaili Chen, Zhiqi Differential Geometry Commutative Algebra Rings and Algebras 17C37, 17C50, 53C25, 53C99, 22E60 In this paper, we investigate metric Jordan algebras, and follow the lines of the paper (J. Milnor: Curvatures of left invariant metrics on Lie groups. Adv. Math. (1976)). Firstly, we define the Jordan-Levi-Civita connection, then we show that every metric Jordan algebra admits a unique Jordan-Levi-Civita connection. Secondly, using the Jordan-Levi-Civita connection, we introduce three natural curvature tensors on metric Jordan algebras, and obtain the corresponding curvature formulas. Thirdly, based on these curvature formulas, we prove that every formally real Jordan algebra admits both a metric of non-positive Jordan curvature, and a Jordan-Einstein metric of negative Jordan scalar curvature. Besides, for nilpotent Jordan algebras, we prove that they admit no Jordan-Einstein metrics. |
| title | Curvatures of metric Jordan algebras |
| topic | Differential Geometry Commutative Algebra Rings and Algebras 17C37, 17C50, 53C25, 53C99, 22E60 |
| url | https://arxiv.org/abs/2309.02682 |