Biderivations of finite-dimensional complex simple Lie algebras

Abstract

In this paper, we prove that a biderivation of a finite-dimensional complex simple Lie algebra without the restriction of being skewsymmetric is an inner biderivation. 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 obtain the forms of the linear commuting maps on the finite-dimensional complex simple Lie algebra or general linear Lie algebra.

Keywords:

• Biderivation
• simple Lie algebra
• general linear Lie algebra
• linear commuting map

AMS Subject Classifications:

• 17B05
• 17B40
• 17B65

Acknowledgements

I would like to thank the referee and editor for their invaluable comments and suggestions. I also thank Dr. Kejia Zhang for selfless assistance in writing this article.

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported in part by the National Natural Science Foundation of China [grant number 11171294], the Natural Science Foundation of Heilongjiang Province of China [grant number A2015007], and the fund of the Heilongjiang Education Committee [grant number 12531483].