A New Definition of Superbiderivations for Lie Superalgebras
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908428187729920 |
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| author | Di Bartolo, Alfonso Di Fatta, Francesco Paolo La Rosa, Gianmarco |
| author_facet | Di Bartolo, Alfonso Di Fatta, Francesco Paolo La Rosa, Gianmarco |
| contents | In this paper, we study superbiderivations on Lie superalgebras from structural and geometric perspectives. Motivated by the classical fact that the bracket of a Lie algebra is itself a biderivation, we propose a new definition of superbiderivation for Lie superalgebras, one that requires the bracket to be a superbiderivation, a condition not satisfied by existing definitions in the literature. Our focus is on complete Lie superalgebras, a natural generalization of semisimple Lie algebras that has emerged as a promising framework in the search for alternative structural notions. In this setting, we introduce and study linear supercommuting maps, comparing our definition with previous proposals. Finally, we present two applications: one involving the superalgebra of superderivations of the Heisenberg Lie superalgebra and another offering initial geometric insights into deformation theory via superbiderivations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_23710 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A New Definition of Superbiderivations for Lie Superalgebras Di Bartolo, Alfonso Di Fatta, Francesco Paolo La Rosa, Gianmarco Rings and Algebras 17B40, 17B65, 17B66 In this paper, we study superbiderivations on Lie superalgebras from structural and geometric perspectives. Motivated by the classical fact that the bracket of a Lie algebra is itself a biderivation, we propose a new definition of superbiderivation for Lie superalgebras, one that requires the bracket to be a superbiderivation, a condition not satisfied by existing definitions in the literature. Our focus is on complete Lie superalgebras, a natural generalization of semisimple Lie algebras that has emerged as a promising framework in the search for alternative structural notions. In this setting, we introduce and study linear supercommuting maps, comparing our definition with previous proposals. Finally, we present two applications: one involving the superalgebra of superderivations of the Heisenberg Lie superalgebra and another offering initial geometric insights into deformation theory via superbiderivations. |
| title | A New Definition of Superbiderivations for Lie Superalgebras |
| topic | Rings and Algebras 17B40, 17B65, 17B66 |
| url | https://arxiv.org/abs/2506.23710 |