Characterization of Frobenius Lie algebras through deformation theory

Complex Frobenius Lie algebras are studied and characterized by means of deformations. We prove that every complex Frobenius Lie algebra of dimension 2n+2 is isomorphic to a linear or a quadratic deformation of a Lie algebra H=⟨D⟩⋉h2n+1 where h2n+1 is the (2n+1)-dimensional Heisenberg Lie algebra and D∈Der(h2n+1) is a semisimple derivation. As an application of this result, we revisit the classification of complex Frobenius Lie algebras of dimension 4 to obtain each one of these Lie algebras as an extension of the 3-dimensional Heisenberg Lie algebra h3 and a semisimple derivation D∈Der(h3).