Two-step nilpotent Lie algebras with ad-invariant metrics and a special kind of skew-symmetric maps

Abstract

We prove that a 2-step nilpotent Lie algebras admitting an ad-invariant metric can be constructed from a vector space $\mathfrak v$ endowed with a inner product $<, >$ and an injective homomorphism $\rho: \mathfrak v \to \mathfrak{so}(\mathfrak v)$ satisfying $\rho(v)v=0$ for all $v\in \mathfrak v$. The corresponding simply connected pseudo-Riemannian Lie groups are flat and any isometry fixing the identity element does not depend on $\rho$. The description allows to construct examples starting with a compact semisimple Lie algebra and it is useful to show some applications.