## Réductibilité de la variété des algèbres de Lie nilpotentes.(French)Zbl 0145.25902

Let $$L_n$$ be the set of $$n$$ dimensional Lie algebras over a field $$K$$ and let $$N_n$$ be the subset of nilpotent algebras. The main result is that $$N_n$$ is irreducible if $$n\geq 11$$. It is first shown that if $$n\geq 1$$ there exists an irreducible closed proper subset of $$N_n$$ having dimension $$\geq n^2 - 9$$. Next it is proved that the open subset of threadlike Lie algebras has dimension $$\leq 5n^2/4$$; by a threadlike Lie algebra is meant a nilpotent Lie algebra such that for each $$i$$ satisfying $$1 < i < n-2$$ the $$i$$-th term in the ascending central series has dimension $$i$$. Then it is shown that if $$n\geq 7$$ and $$G_n$$ is a threadlike Lie algebra such that $$\dim H^2(G_n, K) = a$$, every irreducible component containing $$G_n$$ has dimension $$\leq n^2 - 9+ (a - 3) (n - 6)$$. A threadlike Lie algebra $$G_n$$ such that $$\dim H^2(G_n, K) = 3$$ exists and this establishes the main theorem.
Reviewer: E. M. Patterson

### MSC:

 17B30 Solvable, nilpotent (super)algebras

### Keywords:

variety of nilpotent Lie algebras; reducibility