Hypercomplex structures on special linear groups
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915239174340608 |
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| author | Andrada, Adrián Garrone, Agustín Tolcachier, Alejandro |
| author_facet | Andrada, Adrián Garrone, Agustín Tolcachier, Alejandro |
| contents | The purpose of this article is twofold. First, we prove that the $8$-dimensional Lie group $\operatorname{SL}(3,\mathbb{R})$ does not admit a left-invariant hypercomplex structure. To accomplish this we revise the classification of left-invariant complex structures on $\operatorname{SL}(3,\mathbb{R})$ due to Sasaki. Second, we exhibit a left-invariant hypercomplex structure on $\operatorname{SL}(2n+1,\mathbb{C})$, which arises from a complex product structure on $\operatorname{SL}(2n+1,\mathbb{R})$, for all $n\in \mathbb{N}$. We then show that there are no HKT metrics compatible with this hypercomplex structure. Additionally, we determine the associated Obata connection and we compute explicitly its holonomy group, providing thus a new example of an Obata holonomy group properly contained in $\operatorname{GL}(m,\mathbb{H})$ and not contained in $\operatorname{SL}(m,\mathbb{H})$, where $4m=\dim_\mathbb{R} \operatorname{SL}(2n+1,\mathbb{C})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_14715 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hypercomplex structures on special linear groups Andrada, Adrián Garrone, Agustín Tolcachier, Alejandro Differential Geometry 53C15, 53C26, 53C29 The purpose of this article is twofold. First, we prove that the $8$-dimensional Lie group $\operatorname{SL}(3,\mathbb{R})$ does not admit a left-invariant hypercomplex structure. To accomplish this we revise the classification of left-invariant complex structures on $\operatorname{SL}(3,\mathbb{R})$ due to Sasaki. Second, we exhibit a left-invariant hypercomplex structure on $\operatorname{SL}(2n+1,\mathbb{C})$, which arises from a complex product structure on $\operatorname{SL}(2n+1,\mathbb{R})$, for all $n\in \mathbb{N}$. We then show that there are no HKT metrics compatible with this hypercomplex structure. Additionally, we determine the associated Obata connection and we compute explicitly its holonomy group, providing thus a new example of an Obata holonomy group properly contained in $\operatorname{GL}(m,\mathbb{H})$ and not contained in $\operatorname{SL}(m,\mathbb{H})$, where $4m=\dim_\mathbb{R} \operatorname{SL}(2n+1,\mathbb{C})$. |
| title | Hypercomplex structures on special linear groups |
| topic | Differential Geometry 53C15, 53C26, 53C29 |
| url | https://arxiv.org/abs/2408.14715 |