Biderivations of complete Lie algebras

TL;DR

This paper investigates biderivations in complete Lie algebras, introducing a matricial approach, classifying biderivations in specific cases, and extending previous results to semisimple Lie algebras.

Contribution

It provides a new matricial method for studying biderivations and extends classification results to semisimple and complete Lie algebras.

Findings

Classified all biderivations of complete Lie algebras with zero center.

Extended biderivation classification to semisimple Lie algebras.

Presented results on symmetric and skew-symmetric biderivations.

Abstract

The authors of this article intend to present some results obtained in the study of biderivations of complete Lie algebras. Firstly they present a matricial approach to do this, which was a useful and explanatory tool not only in the study of biderivations but also in the synthesis of these results. Then they study all biderivations of a Lie algebra $L$ with $\operatorname{Z}(L)=0$ and $\operatorname{Der(L)}=\operatorname{ad(L)}$, called complete. Moreover, as an application of the previous result, they describe all biderivations of a semisimple Lie algebra (that are complete), extending a result obtained by X. Tang in ([20]) that describes all biderivations of a complex simple Lie algebra. And thirdly, results on symmetric and skew-symmetric biderivations are also presented.