Research ArticleNo Access

Multi-derivations of Poisson algebras

Department of Mathematics, Jilin University, Changchun 130012, Jilin, P. R. China

https://doi.org/10.1142/S0219498825500276Cited by:0 (Source: Crossref)

Abstract

The purpose of this paper is to study multi-derivations of Poisson algebras. First, we show that the space of multi-derivations of a Lie algebra $𝔤$ with the Nijenhuis–Richardson bracket is a differential graded Lie algebra. Then we introduce the notion of a multi-derivation of a Poisson algebra, and show that the space of multi-derivations of a Poisson algebra with the Nijenhuis–Richardson bracket is also a graded Lie algebra. Finally, we study multi-derivations of three concrete Poisson algebras coming from linear Poisson manifolds, symplectic manifolds and Poisson manifolds with vanishing first cohomology groups. We establish the relationship between multi-derivations of a Lie algebra $𝔤$ and multi-derivations of the linear Poisson manifold $𝔤^{*}$ , and show that a two-derivation of a connected symplectic manifold $(M, ω)$ is $λ π$ , where $π$ is the Poisson bivector field induced by the symplectic structure $ω$ and $λ$ is a real number. We also prove that a two-derivation of a connected Poisson manifold $(M, π)$ with trivial Poisson cohomology group $H^{1} (M, π)$ is $f π$ , where $f$ is a Casimir function.

Communicated by Kailash C. Misra

Keywords:

AMSC: 17B63, 17B40, 53D17

References

1. M. Brešar , On generalized biderivations and related maps, J. Algebra 172 (1995) 764–786. Crossref, Web of Science, Google Scholar
2. M. Brešar and K. Zhao , Biderivations and commuting linear maps on Lie algebras, J. Lie Theory 28(3) (2018) 885–900. Web of Science, Google Scholar
3. Y. Chang and L. Chen , Biderivations and linear commuting maps on the restricted Cartan-type Lie algebras $W (n; 1)$ and $S (n; 1)$ , Linear Multilinear Algebra 67(8) (2019) 1625–1636. Crossref, Google Scholar
4. J. Conn , Normal forms for smooth Poisson structures, Ann. of. Math. (2) 121(3) (1985) 565–593. Crossref, Web of Science, Google Scholar
5. D. Eremita , Biderivations and commuting linear maps on current Lie algebras, J. Lie Theory 31(1) (2021) 119–126. Web of Science, Google Scholar
6. G. Fang and X. Dai , Super-biderivations of Lie superalgebras, Linear Multilinear Algebra 65(1) (2017) 58–66. Crossref, Google Scholar
7. X. Han, D. Wang and C. Xia , Linear commuting maps and biderivations on the Lie algebras $W (a, b)$ , J. Lie Theory 26 (2016) 777–786. Web of Science, Google Scholar
8. J. Jiang and X. Tang , Biderivations of the deformative Schrödinger–Virasoro algebras, Comm. Algebra 48(2) (2020) 609–624. Crossref, Web of Science, Google Scholar
9. X. Liu, X. Guo and K. Zhao , Biderivations of the Block Lie algebras, Linear Algebra Appl. 538 (2018) 43–55. Crossref, Web of Science, Google Scholar
10. B. Sun, Y. Ma and L. Chen, Biderivations and commuting linear maps on Hom–Lie algebras, preprint (2020), arXiv:2005.11117v2. Google Scholar
11. X. Tang , Biderivations of finite-dimensional complex simple Lie algebras, Linear Multilinear Algebra 66(2) (2018) 250–259. Crossref, Web of Science, Google Scholar
12. L. Tang, L. Meng and L. Chen , Super-biderivations and linear super-commuting maps on the Lie superalgebras, Comm. Algebra 48(12) (2020) 5076–5085. Crossref, Web of Science, Google Scholar
13. D. Wang and X. Yu , Biderivations and linear commuting maps on Schrödinger–Virasoro Lie algebra, Comm. Algebra 41(6) (2013) 2166–2173. Crossref, Google Scholar
14. D. Wang, X. Yu and Z. Chen , Biderivations of the parabolic subalgebras of simple Lie algebras, Comm. Algebra 39 (2011) 4097–4104. Crossref, Web of Science, Google Scholar
15. C. Xia, D. Wang and X. Hou , Linear super-commuting maps and super-biderivations on the super-Virasoro algebras, Comm. Algebra 44(12) (2016) 5342–5350. Crossref, Web of Science, Google Scholar
16. Y. Zhao, Y. Chen and K. Zhao , $2$ -Local derivations on Witt algebras, J. Algebra Appl. 20(4) (2021) 2150068. Link, Web of Science, Google Scholar
17. Y. Zhong and X. Tang , $n$ -Derivations of the extended Schrödinger–Virasoro Lie algebra, Algebra Colloq. 28(1) (2021) 105–118. Link, Web of Science, Google Scholar