Deconstructing and constructing quadratic Lie Algebra

A quadratic Lie algebra (L, φ) is a Lie algebra L equipped with a symmetric, non-degenerate, invariant bilinear form φ, i.e. φ([x,y],z) + φ(x,[y,z]) = 0. While simple and semisimple Lie algebras are naturally quadratic via their Killing form, the classification of non-semisimple quadratic Lie algebras remains an open problem. In this talk, we explore the deconstruction of a general Lie algebra down to a nilpotent one. This process is achieved by reversing successive double extensions. However, determining the variety of nilpotent quadratic Lie algebras is a tough problem. Given this difficulty, we focus on the 2-step nilpotent case, where we provide a classification of these algebras up to dimension 17, utilizing the existing classification of trivectors of dimension less or equal than 8. Finally, we use these 2-step nilpotent quadratic Lie algebras as building blocks to construct larger, more general quadratic Lie algebras via double extensions using their skew-derivations.