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


17B30 Solvable, nilpotent (super)algebras