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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2411.17031 |
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| _version_ | 1866909815671881728 |
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| author | Gutierrez, Gonzalo Farinati, Marco |
| author_facet | Gutierrez, Gonzalo Farinati, Marco |
| contents | In this paper, we generalize the Tits construction for Lie superalgebras such that $\mathfrak{sl}_2$ acts by even derivations and decompose, as $\mathfrak{sl}_2$-module, into a direct sum of copies of the adjoint, the natural and the trivial representations.
This construction generalizes the one provided by Elduque et al in \cite{EBCC23}, and it is possible to described the $\mathfrak{sl}_2$-Lie superstructure in terms of $\mathcal{J}$-ternary superalgebras as a super version of the defined by Allison. We extend the Tits-Kantor-Koecher construction and the Tits-Allison-Gao functor that define a short $\mathfrak{sl}_2$-Lie superalgebra from a $\mathcal{J}$-ternary superalgebra $(\mathcal{J},\mathcal{M})$. Our setting includes and generalizes both \cite{EBCC23} and Shang's \cite{S22}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17031 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Tits construction for short $\mathfrak{sl}_2$-super-structures Gutierrez, Gonzalo Farinati, Marco Representation Theory Quantum Algebra 17C50, 17C70, 17B05, 17B99 In this paper, we generalize the Tits construction for Lie superalgebras such that $\mathfrak{sl}_2$ acts by even derivations and decompose, as $\mathfrak{sl}_2$-module, into a direct sum of copies of the adjoint, the natural and the trivial representations. This construction generalizes the one provided by Elduque et al in \cite{EBCC23}, and it is possible to described the $\mathfrak{sl}_2$-Lie superstructure in terms of $\mathcal{J}$-ternary superalgebras as a super version of the defined by Allison. We extend the Tits-Kantor-Koecher construction and the Tits-Allison-Gao functor that define a short $\mathfrak{sl}_2$-Lie superalgebra from a $\mathcal{J}$-ternary superalgebra $(\mathcal{J},\mathcal{M})$. Our setting includes and generalizes both \cite{EBCC23} and Shang's \cite{S22}. |
| title | The Tits construction for short $\mathfrak{sl}_2$-super-structures |
| topic | Representation Theory Quantum Algebra 17C50, 17C70, 17B05, 17B99 |
| url | https://arxiv.org/abs/2411.17031 |