Biderivations of finite dimensional complex simple Lie algebras

TL;DR

This paper proves that all biderivations of finite dimensional complex simple Lie algebras are inner, extending previous results by removing skew-symmetry restrictions, and explores their applications to linear maps.

Contribution

It establishes that biderivations of such Lie algebras are inner without skew-symmetry constraints and characterizes linear commuting maps.

Findings

All biderivations are inner for these Lie algebras.

Identified a class of non-inner, non-skewsymmetric biderivations.

Described forms of linear commuting maps on these algebras.

Abstract

In this paper, we prove that a biderivation of a finite dimensional complex simple Lie algebra without the restriction of skewsymmetric is inner. As an application, the biderivation of a general linear Lie algebra is presented. In particular, we find a class of a non-inner and non-skewsymmetric biderivations. Furthermore, we also get the forms of linear commuting maps on the finite dimensional complex simple Lie algebra or general linear Lie algebra.