Ad-invariant metrics on nonnice nilpotent Lie algebras

Abstract

We proved in previous work that all real nilpotent Lie algebras of dimension up to 10 carrying an ad-invariant metric are nice. In this paper we show by constructing explicit examples that nonnice irreducible nilpotent Lie algebras admitting an ad-invariant metric exist for every dimension greater than 10 and every nilpotency step greater than 2.

New Special Einstein Pseudo-Riemannian Metrics on Solvable Lie Algebras

“…We studied intensively the nilpotent nice Lie algebras, and we obtain a classification up to equivalence for dimension ≤ 9 ( [10]). More recently we develop helpful techniques to handle with them in [6]. Those nice Lie algebras were introduced and studied in [30,29], and are an useful tool in the study of nilsoliton and Ricci flow in the Riemannian and pseudo-Riemannian setting (see e.g.…”

“…Those nice Lie algebras were introduced and studied in [30,29], and are an useful tool in the study of nilsoliton and Ricci flow in the Riemannian and pseudo-Riemannian setting (see e.g. [34,37,26,25,38,41,15]), and lately they were used to address the problem of ad-invariant metrics ( [8,6]). We also used nice Lie algebra to construct explicit left-invariant Einstein pseudo-Riemannian metrics on nilpotent Lie group with s = 0 ( [13]) or s = 0 ( [12,7]); even if those examples are difficult to construct, we know that for each n ≥ 8, there exist ndimensional nice nilpotent Lie algebras with an Einstein diagonal metric with s = 0 [13,Theorem 3.7].…”