Deformations, Contractions and Classification of Lie Algebras of Order 3 TeX

Lie algebras of order $F$ (or $F-$Lie algebras) are possible generalisations of Lie algebras (F=1) and Lie superalgebras (F=2). These structures have been used to implement new non-trivial extensions of the Poincaré algebra. In this paper we set the basis of the theory of the deformations (in the Gerstenhaber sense) and contractions for Lie algebras of order 3. We then initiated a general classification for Lie algebras of order 3 and we give all Lie algebras of order 3 based on $\mathfraksl(2,\mathbb C)$ and $\mathfrakiso(1,3)$ the four-dimensional Poincaré algebra.