Biderivations of Lie algebras

TL;DR

This paper introduces symmetric biderivation radicals and characteristic subalgebras of Lie algebras, characterizes biderivations for certain Lie algebras, and shows that commutative post-Lie algebra structures are trivial on these algebras.

Contribution

It precisely determines biderivations of finite-dimensional simple Lie algebras and Witt algebras, and explores their structural properties and applications.

Findings

Biderivations of finite-dimensional simple Lie algebras are characterized.

Biderivations of Witt algebras over fields of characteristic 0 are determined.

Commutative post-Lie algebra structures on these Lie algebras are shown to be trivial.

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.