Biderivations of Lie algebras
Qiufan Chen, Yufeng Yao, Kaiming Zhao
TL;DR
Introduces the symmetric biderivation radical $\mathrm{Rad}(L)$ and characteristic subalgebras as tools to study symmetric biderivations on Lie algebras. It determines all symmetric biderivations for two main classes: finite-dimensional classical simple Lie algebras over fields with $\operatorname{char}\mathbb{F} \neq 2,3$ and Witt algebras $\mathcal{W}^+_n$ over $\operatorname{char}=0$, showing these biderivations are trivial. Consequently, commutative post-Lie algebra structures on these algebras are also trivial. The approach leverages root-space decompositions, automorphism actions, and radical/characteristic subalgebras to obtain a uniform, computation-free treatment, with potential extension to other Lie algebras.
Abstract
In this paper, we first introduce the concept of symmetric biderivation radicals and characteristic subalgebras of Lie algebras, and study their properties. Based on these results, we precisely determine biderivations of some Lie algebras including finite-dimensional simple Lie algebras over arbitrary fields of characteristic not $2$ or $3$, and the Witt algebras $\mathcal{W}^+_n$ over fields of characteristic $0$. As an application, commutative post-Lie algebra structure on aforementioned Lie algebras is shown to be trivial.