A $\delta$-first Whitehead Lemma

TL;DR

This paper extends the classical Whitehead Lemma by characterizing $oldsymbol{ ext{δ}}$-derivations of simple finite-dimensional Lie algebras over characteristic zero fields, revealing they are mostly inner or scalar multiplications, with special cases involving $oldsymbol{ ext{sl}(2)}$.

Contribution

It provides a comprehensive classification of $oldsymbol{ ext{δ}}$-derivations for simple Lie algebras, extending Whitehead's lemma to include new cases and clarifying their structure.

Findings

$oldsymbol{ ext{δ}}$-derivations are mostly inner derivations.

In the adjoint module, $oldsymbol{ ext{δ}}$-derivations are scalar multiplications.

Exceptional cases involve $oldsymbol{ ext{sl}(2)}$ related structures.

Abstract

We prove that $\delta$-derivations of a simple finite-dimensional Lie algebra over a field of characteristic zero, with values in a finite-dimensional module, are either inner derivations, or, in the case of adjoint module, multiplications by a scalar, or some exceptional cases related to $\mathfrak{sl}(2)$. This can be viewed as an extension of the classical first Whitehead Lemma.