Almost inner derivations of Lie superalgebras
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911131824553984 |
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| author | Serganova, Vera Vaintrob, Arkady |
| author_facet | Serganova, Vera Vaintrob, Arkady |
| contents | An almost inner derivation of a Lie algebra $L$ is a derivation that coincides with an inner derivation on each one-dimensional subspace of $L$. The almost inner derivations form a subalgebra ${aDer}(L)$ of the Lie algebra ${Der}(L)$ of all derivations of $L$, containing the inner derivations ${iDer}(L)$ as an ideal. If $L$ is a simple finite-dimensional Lie algebra, then ${aDer}(L)={iDer}(L)$, since all derivations of $L$ are inner.
In this paper, we introduce and study almost inner derivations derivations of Lie superalgebras. Since simple Lie superalgebras may admit non-inner outer derivations, the existence of non-inner almost inner derivations becomes a nontrivial question. Nevertheless, we show that all almost inner derivations of finite-dimensional simple Lie superalgebras over $\mathbb C$ are inner. We also give examples of naturally occurring non-inner almost inner derivations derivations of some pseudo-reductive Lie superalgebras related to the Sato-Kimura classification of prehomogeneous vector spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_00689 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Almost inner derivations of Lie superalgebras Serganova, Vera Vaintrob, Arkady Rings and Algebras 17 An almost inner derivation of a Lie algebra $L$ is a derivation that coincides with an inner derivation on each one-dimensional subspace of $L$. The almost inner derivations form a subalgebra ${aDer}(L)$ of the Lie algebra ${Der}(L)$ of all derivations of $L$, containing the inner derivations ${iDer}(L)$ as an ideal. If $L$ is a simple finite-dimensional Lie algebra, then ${aDer}(L)={iDer}(L)$, since all derivations of $L$ are inner. In this paper, we introduce and study almost inner derivations derivations of Lie superalgebras. Since simple Lie superalgebras may admit non-inner outer derivations, the existence of non-inner almost inner derivations becomes a nontrivial question. Nevertheless, we show that all almost inner derivations of finite-dimensional simple Lie superalgebras over $\mathbb C$ are inner. We also give examples of naturally occurring non-inner almost inner derivations derivations of some pseudo-reductive Lie superalgebras related to the Sato-Kimura classification of prehomogeneous vector spaces. |
| title | Almost inner derivations of Lie superalgebras |
| topic | Rings and Algebras 17 |
| url | https://arxiv.org/abs/2509.00689 |