A Lie Bracket on the Space of (Right) Biderivations of a Lie Algebra
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| author | Di Bartolo, Alfonso La Rosa, Gianmarco |
| author_facet | Di Bartolo, Alfonso La Rosa, Gianmarco |
| contents | Derivations extend the concept of differentiation from functions to algebraic structures as linear operators satisfying the Leibniz rule. In Lie algebras, derivations form a Lie algebra via the commutator bracket of linear endomorphisms. Motivated by this, we study biderivations-bilinear maps capturing higher-order infinitesimal symmetries. This work focuses on right biderivations of Lie algebras, introducing Lie brackets on spaces of biderivations to explore their algebraic and geometric properties. We analyse the interplay between left and right biderivation brackets through symmetric and skew-symmetric cases, providing a coherent Lie algebra framework. Moreover, we initiate the construction of Lie groups corresponding to the Lie algebra of biderivations, linking infinitesimal and global structures. Our results offer new perspectives on higher-order derivations with potential applications in deformation theory and generalised symmetry studies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_12089 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Lie Bracket on the Space of (Right) Biderivations of a Lie Algebra Di Bartolo, Alfonso La Rosa, Gianmarco Rings and Algebras Differential Geometry 17B05, 17B40, 22E15, 22E60 Derivations extend the concept of differentiation from functions to algebraic structures as linear operators satisfying the Leibniz rule. In Lie algebras, derivations form a Lie algebra via the commutator bracket of linear endomorphisms. Motivated by this, we study biderivations-bilinear maps capturing higher-order infinitesimal symmetries. This work focuses on right biderivations of Lie algebras, introducing Lie brackets on spaces of biderivations to explore their algebraic and geometric properties. We analyse the interplay between left and right biderivation brackets through symmetric and skew-symmetric cases, providing a coherent Lie algebra framework. Moreover, we initiate the construction of Lie groups corresponding to the Lie algebra of biderivations, linking infinitesimal and global structures. Our results offer new perspectives on higher-order derivations with potential applications in deformation theory and generalised symmetry studies. |
| title | A Lie Bracket on the Space of (Right) Biderivations of a Lie Algebra |
| topic | Rings and Algebras Differential Geometry 17B05, 17B40, 22E15, 22E60 |
| url | https://arxiv.org/abs/2507.12089 |