Sur la variété des lois d'algèbres de Lie nilpotentes complexes

Abstract

Let N i be the variety of laws of i -dimensional nilpotent complex Lie algebras, N ˜ i the quotient space of orbits under the canonical action of the full linear group and U i ⊂N i the open subset composed of filiform Lie algebras. M. Vergne determined U 7 and showed that N i is reducible for i=7 and i≥11 . In a previous paper the authors proved that U ˜ 8 and N ˜ 8 are unions of points and lines. In this note they study N 9 and choose in U 9 four continuous families with two parameters. One may ask whether each of these families generates a component of N 9 . However, it seems that the authors may give a positive answer to the problem of reducibility for N i , 8≤i≤10 .