# Almost inner derivations of Lie algebras

## TL;DR

This paper investigates almost inner derivations of Lie algebras, computing them for low-dimensional cases, and establishing conditions under which they are inner, with implications for various classes of nilpotent and solvable Lie algebras.

## Contribution

It introduces the concept of fixed basis vectors and characterizes almost inner derivations for multiple classes of Lie algebras, expanding understanding of their structure.

## Key findings

- All almost inner derivations are inner for certain 2-step nilpotent Lie algebras.
- Identified families of nilpotent Lie algebras with large spaces of non-inner almost inner derivations.
- Computed almost inner derivations for low-dimensional Lie algebras.

## Abstract

We study almost inner derivations of Lie algebras, which were introduced by Gordon and Wilson in their work on isospectral deformations of compact solvmanifolds. We compute all almost inner derivations for low-dimensional Lie algebras, and introduce the concept of fixed basis vectors for proving that all almost inner derivations are inner for $2$-step nilpotent Lie algebras determined by graphs, free $2$ and $3$-step nilpotent Lie algebras, free metabelian nilpotent Lie algebras on two generators, almost abelian Lie algebras and triangular Lie algebras. On the other hand we also exhibit families of nilpotent Lie algebras having an arbitrary large space of non-inner almost inner derivations.