Algèbres de Lie rigides

Abstract

Let Ln Kn3 be the affine variety of all n-dimensional Lie algebras over an algebraically closed field of characteristic zero. An element g 2 Ln can be identified with the bilinear mapping μ defining the multiplication in g. If f is an element of the general linear group GL(n,K) then f−1μ(f, f) defines a Lie algebra in Ln. In this way GL(n,K) operates on Ln. μ 2 Ln is called rigid iff the orbit O(μ) is Zariski-open in Ln. The vanishing of the second cohomology group H2(μ, μ) implies rigidity of the Lie algebra μ. Hence semisimple Lie algebras are rigid. The classification of rigid Lie algebras is somewhat more accessible than the general case. The authors classify all rigid solvable Lie algebras up to dimension eight. A nilpotent Lie algebra n of dimension n is called filamentous iff its characteristic sequence is (n − 1, 1). The rank of a filamentous Lie algebra is at most two. A rigid Lie algebra with filamentous nilradical is necessarily solvable. The authors answer the question as to when a filamentous Lie algebra occurs as the nilradical of a rigid Lie algebra: This is always the case if the rank of the filamentous Lie algebra n is two (Theorem 3), and in the case of rank one it depends on the existence of diagonalizable derivations of n with certain integral eigenvalues (Theorem 4).