---
abstract: 'In this work we find necessary and sufficient conditions for a free nilpotent or a free metabelian nilpotent Lie algebra to be endowed with an ad-invariant metric. For such nilpotent Lie algebras admitting an ad-invariant metric the corresponding automorphisms groups are studied.'
address: 'CONICET and ECEN-FCEIA, Universidad Nacional de Rosario, Pellegrini 250, 2000 Rosario, Argentina. '
author:
- 'Viviana J. del Barco'
- 'Gabriela P. Ovando'
title: 'Free nilpotent Lie algebras admitting ad-invariant metrics'
---

[^1] [^2]

Introduction
============

An ad-invariant metric on a Lie algebra ${\mathfrak g }$ is a nondegenerate symmetric bilinear form $\la \,,\, \ra$ which satisfies $$\label{adme}
\la [x, y],  z\ra + \la y, [x, z]\ra = 0 \qquad \mbox{ for all }x, y, z \in  {\mathfrak g }.$$

Lie algebras endowed with ad-invariant metrics (also called “metric” or “quadratic”) became relevant some years ago when they were useful in the formulation of some physical problems such as the known Adler-Kostant-Symes scheme. They also constitute the basis for the construction of bialgebras and they give rise to interesting pseudo-Riemannian geometry [@Co]. For instance in [@Ov1] a result originally due to Kostant [@Kos] was revalidated for pseudo-Riemannian metrics: it states that the Lie algebra of 