Realizations of 3-Lie algebras

Abstract

3-Lie algebras have close relationships with many important fields in mathematics and mathematical physics. In this paper, we provide a construction of 3-Lie algebras in terms of Lie algebras and certain linear functions. Moreover, with the construction from γ-matrices and two-dimensional extensions of metric Lie algebras, all the complex 3-Lie algebras in dimension ≤5 are obtained along this approach. As a special case, we study the structure of the 3-Lie algebras constructed from the general linear Lie algebras with trace forms and prove that they are semisimple and local.