Classification of filiform Lie algebras in dimension 8

Abstract

The classification of real and complex filiform Lie algebras is known in dimension less or equal than 7 (cf. [3] for dimension less or equal than 6 and [1] for dimension 7). The set of isomorphism classes has a finite number of points up to dimension 6. In dimension 7 we get a line 8real or complex on the case) and 9 points (resp. 8) for the real case (resp. complex). Note that dimension p=7 is the smallest for which it does not exit any rigid filiform law in the algebraic variety Np of nilpotent Li algebra laws in dimension p (cf. [1]). In this work we give the classification of complex filiform Lie algebras in dimension 8, and we obtain that the set of isomorphism classes is union of a finite number of lines (only two intersecting) and a finite number of points. In this case, we find a unique rigid filiform law in N8.