Invariant metrics on nilpotent Lie algebras

Abstract

In this work we state criteria for a nilpotent Lie algebra  g  to admit an invariant metric. We use that g possesses two  canonical abelian ideals  i(g) ⊂ J(g) to decompose the underlying vector space of g. By using this decomposition we state sufficiency conditions for g to admit an invariant metric. The properties of the ideal J(g) allow to prove that if a current Lie algebra g ⊗ S admits an invariant metric, then there must be an invariant and non-degenerate bilinear map from S × S into the space of centroids of g/J(g). We also prove that in any nilpotent Lie algebra g there exists a non-zero, symmetric and invariant bilinear form. This bilinear form allows to reconstruct g by means of an algebra with unit. We prove that this algebra is simple if and only if the bilinear form is an invariant metric on g.