Left-invariant ${\rm G}_2^*$-structures of type III
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913896527298560 |
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| author | del Barco, Viviana Ferreira, Ana Cristina Kath, Ines |
| author_facet | del Barco, Viviana Ferreira, Ana Cristina Kath, Ines |
| contents | We investigate left-invariant ${\rm G}_2^*$-structures on 7-dimensional Lie groups, focusing on those whose holonomy algebras are indecomposable and of type III, the latter meaning that the socle of the holonomy representation is maximal. Building on the classification of indecomposable holonomy algebras contained in $\mathfrak g_2^*$, we determine which ones arise as infinitesimal holonomy algebras of type III for left-invariant ${\rm G}_2^*$-structures. Our main result shows that only abelian subalgebras occur, and these are necessarily of dimension two or three. Moreover, we provide explicit Lie groups with left-invariant ${\rm G}_2^*$-structures realizing these abelian holonomies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_14031 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Left-invariant ${\rm G}_2^*$-structures of type III del Barco, Viviana Ferreira, Ana Cristina Kath, Ines Differential Geometry 53C29 (Primary), 53C50, 53C10, 53C30 (Secondary) We investigate left-invariant ${\rm G}_2^*$-structures on 7-dimensional Lie groups, focusing on those whose holonomy algebras are indecomposable and of type III, the latter meaning that the socle of the holonomy representation is maximal. Building on the classification of indecomposable holonomy algebras contained in $\mathfrak g_2^*$, we determine which ones arise as infinitesimal holonomy algebras of type III for left-invariant ${\rm G}_2^*$-structures. Our main result shows that only abelian subalgebras occur, and these are necessarily of dimension two or three. Moreover, we provide explicit Lie groups with left-invariant ${\rm G}_2^*$-structures realizing these abelian holonomies. |
| title | Left-invariant ${\rm G}_2^*$-structures of type III |
| topic | Differential Geometry 53C29 (Primary), 53C50, 53C10, 53C30 (Secondary) |
| url | https://arxiv.org/abs/2506.14031 |