Odd-quadratic Lie superalgebras with a weak filiform module as an odd part
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| Format: | Preprint |
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2024
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| author | Barreiro, Elisabete Benayadi, Saïd Navarro, Rosa M. Sánchez, J. M. |
| author_facet | Barreiro, Elisabete Benayadi, Saïd Navarro, Rosa M. Sánchez, J. M. |
| contents | The aim of this work is to study a very special family of odd-quadratic Lie superalgebras ${\mathfrak g}={\mathfrak g}_{\bar 0}\oplus {\mathfrak g}_{\bar 1}$ such that ${\mathfrak g}_{\bar 1}$ is a weak filiform ${\mathfrak g}_{\bar 0}$-module (weak filiform type). We introduce this concept after having proved that the unique non-zero odd-quadratic Lie superalgebra $({\mathfrak g},B)$ with ${\mathfrak g}_{\bar 1}$ a filiform ${\mathfrak g}_{\bar 0}$-module is the abelian $2$-dimensional Lie superalgebra ${\mathfrak g}={\mathfrak g}_{\bar 0} \oplus {\mathfrak g}_{\bar 1}$ such that $\mbox{\rm dim }{\mathfrak g}_{\bar 0}=\mbox{\rm dim }{\mathfrak g}_{\bar 1}=1$. Let us note that in this context the role of the center of ${\mathfrak g}$ is crucial. Thus, we obtain an inductive description of odd-quadratic Lie superalgebras of weak filiform type via generalized odd double extensions. Moreover, we obtain the classification, up to isomorphism, for the smallest possible dimensions, that is, six and eight. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_13017 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Odd-quadratic Lie superalgebras with a weak filiform module as an odd part Barreiro, Elisabete Benayadi, Saïd Navarro, Rosa M. Sánchez, J. M. Representation Theory The aim of this work is to study a very special family of odd-quadratic Lie superalgebras ${\mathfrak g}={\mathfrak g}_{\bar 0}\oplus {\mathfrak g}_{\bar 1}$ such that ${\mathfrak g}_{\bar 1}$ is a weak filiform ${\mathfrak g}_{\bar 0}$-module (weak filiform type). We introduce this concept after having proved that the unique non-zero odd-quadratic Lie superalgebra $({\mathfrak g},B)$ with ${\mathfrak g}_{\bar 1}$ a filiform ${\mathfrak g}_{\bar 0}$-module is the abelian $2$-dimensional Lie superalgebra ${\mathfrak g}={\mathfrak g}_{\bar 0} \oplus {\mathfrak g}_{\bar 1}$ such that $\mbox{\rm dim }{\mathfrak g}_{\bar 0}=\mbox{\rm dim }{\mathfrak g}_{\bar 1}=1$. Let us note that in this context the role of the center of ${\mathfrak g}$ is crucial. Thus, we obtain an inductive description of odd-quadratic Lie superalgebras of weak filiform type via generalized odd double extensions. Moreover, we obtain the classification, up to isomorphism, for the smallest possible dimensions, that is, six and eight. |
| title | Odd-quadratic Lie superalgebras with a weak filiform module as an odd part |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2401.13017 |