Pseudo-Kähler and hypersymplectic structures on semidirect products
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866910738273009664 |
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| author | Conti, Diego Gil-García, Alejandro |
| author_facet | Conti, Diego Gil-García, Alejandro |
| contents | We study left-invariant pseudo-Kähler and hypersymplectic structures on semidirect products $G\rtimes H$; we work at the level of the Lie algebra $\mathfrak{g}\rtimes\mathfrak{h}$. In particular we consider the structures induced on $\mathfrak{g}\rtimes\mathfrak{h}$ by existing pseudo-Kähler structures on $\mathfrak{g}$ and $\mathfrak{h}$; we classify all semidirect products of this type with $\mathfrak{g}$ of dimension $4$ and $\mathfrak{h}=\mathbb{R}^2$. In the hypersymplectic setting, we consider a more general construction on semidirect products. We construct a large class of hypersymplectic Lie algebras whose underlying complex structure is not abelian as well as non-flat hypersymplectic metrics on $k$-step nilpotent Lie algebras for every $k\geq3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_20660 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Pseudo-Kähler and hypersymplectic structures on semidirect products Conti, Diego Gil-García, Alejandro Differential Geometry 53C26, 53C50, 22E25, 53C15 We study left-invariant pseudo-Kähler and hypersymplectic structures on semidirect products $G\rtimes H$; we work at the level of the Lie algebra $\mathfrak{g}\rtimes\mathfrak{h}$. In particular we consider the structures induced on $\mathfrak{g}\rtimes\mathfrak{h}$ by existing pseudo-Kähler structures on $\mathfrak{g}$ and $\mathfrak{h}$; we classify all semidirect products of this type with $\mathfrak{g}$ of dimension $4$ and $\mathfrak{h}=\mathbb{R}^2$. In the hypersymplectic setting, we consider a more general construction on semidirect products. We construct a large class of hypersymplectic Lie algebras whose underlying complex structure is not abelian as well as non-flat hypersymplectic metrics on $k$-step nilpotent Lie algebras for every $k\geq3$. |
| title | Pseudo-Kähler and hypersymplectic structures on semidirect products |
| topic | Differential Geometry 53C26, 53C50, 22E25, 53C15 |
| url | https://arxiv.org/abs/2310.20660 |