Infinitely many left-symmetric structures on nilpotent Lie algebras
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909849865945088 |
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| author | Kato, Naoki |
| author_facet | Kato, Naoki |
| contents | Dekimpe and Ongenae constructed infinitely many pairwise non-isomorphic complete left-symmetric structures on $\mathbb{R}^n$ for $n\geq 6$. In this paper, we construct a family of complete left-symmetric structures on the cotangent Lie algebra $T^*\mathfrak{g}$ of a certain $n$-dimensional almost abelian nilpotent Lie algebra $\mathfrak{g}$ and give a condition under which two left-symmetric structures in this family are isomorphic. As a consequence of this result, we obtain infinitely many pairwise non-isomorphic left-symmetric structures on $T^{*}\mathfrak{g}$. As an application of this construction, we also obtain infinitely many symplectic structures on $T^{*}\mathfrak{g}$ which are pairwise non-symplectomorphic up to homothety. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_14610 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Infinitely many left-symmetric structures on nilpotent Lie algebras Kato, Naoki Rings and Algebras Symplectic Geometry 17B05, 17B30, 53D05 Dekimpe and Ongenae constructed infinitely many pairwise non-isomorphic complete left-symmetric structures on $\mathbb{R}^n$ for $n\geq 6$. In this paper, we construct a family of complete left-symmetric structures on the cotangent Lie algebra $T^*\mathfrak{g}$ of a certain $n$-dimensional almost abelian nilpotent Lie algebra $\mathfrak{g}$ and give a condition under which two left-symmetric structures in this family are isomorphic. As a consequence of this result, we obtain infinitely many pairwise non-isomorphic left-symmetric structures on $T^{*}\mathfrak{g}$. As an application of this construction, we also obtain infinitely many symplectic structures on $T^{*}\mathfrak{g}$ which are pairwise non-symplectomorphic up to homothety. |
| title | Infinitely many left-symmetric structures on nilpotent Lie algebras |
| topic | Rings and Algebras Symplectic Geometry 17B05, 17B30, 53D05 |
| url | https://arxiv.org/abs/2510.14610 |