Local derivations on solvable Lie algebras of maximal rank

Abstract

The present paper is devoted to the description of local derivations on solvable Lie algebras of maximal rank. Namely, we consider a solvable Lie algebra of the form $\mathcal{R}=\mathcal{Q}\oplus \mathcal{N},$ where $\mathcal{Q}$ is the maximal torus subalgebra of $\mathcal{R},$ $\mathcal{N}$ is the nilradical of $\mathcal{R}$ and $\mathrm{dim\hspace{0.17em}}\mathcal{Q}=\mathrm{dim\hspace{0.17em}}\mathcal{N}/{\mathcal{N}}^2.$ We prove that any local derivation of such solvable Lie algebra $\mathcal{R}$ is a derivation.

Further, we present two examples of solvable Lie algebras which satisfy the condition $\mathrm{dim\hspace{0.17em}}\mathcal{Q}<\mathrm{dim\hspace{0.17em}}\mathcal{N}/{\mathcal{N}}^2$ and the first algebra admit a local derivation which is not a derivation, while for the second algebra we prove that any local derivation is a derivation. We also apply the main result of the paper to the description of local derivations on so-called standard Borel subalgebras of complex simple Lie algebras.

Keywords:

• Lie algebra
• solvable algebra
• nilradical
• complete algebra
• derivation
• local derivation

2020 Mathematics Subject Classification:

• Primary: 17B30
• Secondary: 17B40

Acknowledgments

The first author was partially supported by Russian Ministry of Education and Science, agreement no. 075-02-2022-896. The second author was partially supported by Agencia Estatal de Investigacion (Spain), grant PID2020-115155GB-I00 (European FEDER support included, UE).