Geometric realizations of the Lie superalgebra D(2,1;a)
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918083265822720 |
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| author | Escofet, Anna Kruglikov, Boris The, Dennis |
| author_facet | Escofet, Anna Kruglikov, Boris The, Dennis |
| contents | For every parabolic subgroup $P$ of a Lie supergroup $G$, the homogeneous superspace $G/P$ carries a $G$-invariant supergeometry. We address the question whether $\mathfrak{g}=\text{Lie}(G)$ is the maximal supersymmetry of this supergeometry in the case of the exceptional Lie superalgebra $D(2,1;a)$. For each choice of parabolic $\mathfrak{p}\subset\mathfrak{g}$, we consider the corresponding negatively graded Lie subalgebra $\mathfrak{m}\subset\mathfrak{g}$, and compute its Tanaka--Weisfeiler prolongations, with reduction of the structure group when required, thus realizing $D(2,1;a)$ via symmetries of supergeometries.
This gives 6 inequivalent supergeometries: one of these is a vector superdistribution, two are given by cone fields of supervarieties, and the remaining three are higher order structure reductions (a novel feature). We describe those supergeometries and realize $D(2,1;a)$ supersymmetry explicitly in each case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_03418 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric realizations of the Lie superalgebra D(2,1;a) Escofet, Anna Kruglikov, Boris The, Dennis Differential Geometry Mathematical Physics Representation Theory For every parabolic subgroup $P$ of a Lie supergroup $G$, the homogeneous superspace $G/P$ carries a $G$-invariant supergeometry. We address the question whether $\mathfrak{g}=\text{Lie}(G)$ is the maximal supersymmetry of this supergeometry in the case of the exceptional Lie superalgebra $D(2,1;a)$. For each choice of parabolic $\mathfrak{p}\subset\mathfrak{g}$, we consider the corresponding negatively graded Lie subalgebra $\mathfrak{m}\subset\mathfrak{g}$, and compute its Tanaka--Weisfeiler prolongations, with reduction of the structure group when required, thus realizing $D(2,1;a)$ via symmetries of supergeometries. This gives 6 inequivalent supergeometries: one of these is a vector superdistribution, two are given by cone fields of supervarieties, and the remaining three are higher order structure reductions (a novel feature). We describe those supergeometries and realize $D(2,1;a)$ supersymmetry explicitly in each case. |
| title | Geometric realizations of the Lie superalgebra D(2,1;a) |
| topic | Differential Geometry Mathematical Physics Representation Theory |
| url | https://arxiv.org/abs/2507.03418 |