On Lie Algebras with Only Inner Derivations

TL;DR

This paper investigates non-semisimple Lie algebras with only inner derivations, introduces new families, and constructs examples with specific cohomological properties, advancing understanding of their structure.

Contribution

It generalizes existing methods to create new Lie algebra families, constructs complete non-perfect examples, and analyzes their cohomology, addressing open questions in the field.

Findings

Established vanishing of first adjoint cohomology for new Lie algebra families

Constructed a family of complete non-perfect Lie algebras

Reduced known examples to dimension 31 with specific properties

Abstract

This paper is devoted to the study of non-semisimple Lie algebras of the form $\mathcal{L} = \mathcal{S} \ltimes \mathcal{N}$ whose derivations are all inner. By generalizing the methods of Sato and Angelopoulos, we introduce new families of Lie algebras and establish the vanishing of their first adjoint cohomology. As an application, we construct a family of complete non-perfect Lie algebras, thereby providing examples that yield a positive answer to Carles' question on the existence of such algebras. In addition, we reduce the dimension of known examples of perfect Lie algebras with non-trivial center and only inner derivations to $31$. Furthermore, we employ the Hochschild--Serre factorization theorem to analyze the second adjoint cohomology groups, providing insights non-vanishing of the second adjoint cohomology groups for the algebras obtained through the paper.