Ad-invariant metrics on nonnice nilpotent Lie algebras

TL;DR

This paper demonstrates the existence of nonnice irreducible nilpotent Lie algebras with ad-invariant metrics in dimensions greater than 10, introducing a new construction method called single extension.

Contribution

It introduces the single extension method for constructing Lie algebras with ad-invariant metrics and provides explicit examples beyond dimension 10.

Findings

Nonnice irreducible nilpotent Lie algebras with ad-invariant metrics exist for dimensions >10.

The single extension method enables construction of such Lie algebras.

Explicit examples are constructed for all dimensions greater than 10 and nilpotency step greater than 2.

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 $2$. In the way of doing so, we introduce a method to construct Lie algebras with ad-invariant metrics called the single extension, as a parallel to the well-known double extension procedure.