Derivation double Lie algebras

TL;DR

This paper investigates derivation double Lie algebras, focusing on classical R-matrices that are also derivations, and proves the nonexistence of nontrivial simple examples, contributing to the understanding of Lie algebra identities.

Contribution

It introduces derivation double Lie algebras and proves the nonexistence of nontrivial simple cases, expanding the theoretical framework of Lie algebra structures.

Findings

No nontrivial simple derivation double Lie algebras exist

Identifies Lie algebra identities related to derivation double structures

Connects derivation double Lie algebras to post-Lie algebra structures

Abstract

We study classical R-matrices D for Lie algebras L such that D is also a derivation of L. This yields derivation double Lie algebras (L,D). The motivation comes from recent work on post-Lie algebra structures on pairs of Lie algebras arising in the study of nil-affine actions of Lie groups. We prove that there are no nontrivial simple derivation double Lie algebras, and study related Lie algebra identities for arbitrary Lie algebras.