On donne une majoration de l'indice de certaines algèbres de Lie introduites par V. Dergachev, A. Kirillov et D. Panyushev. On en déduit la preuve d'une conjecture de D. Panyushev. Nous formulons aussi une conjecture concernant l'indice de ces algèbres, et la prouvons dans des cas particuliers. Enfin, nous donnons un résultat concernant l'indice des sous-algèbres paraboliques d'une algèbre de Lie semi-simple.
Let be a Laurent polynomial algebra over a field of characteristic zero, the Lie algebra of -derivations of the algebra , the so-called Witt Lie algebra, and let be the Virasoro Lie algebra which is a -dimensional central extension of the Witt Lie algebra. The Lie algebras and are infinite dimensional Lie algebras. We prove that the following isomorphisms of the groups of Lie algebra automorphisms hold: , and give a short proof that .
The paper deals with the real classical Lie algebras and their finite dimensional irreducible representations. Signature formulae for Hermitian forms invariant relative to these representations are considered. It is possible to associate with the irreducible representation a Hurwitz matrix of special kind. So the calculation of the signatures is reduced to the calculation of Hurwitz determinants. Hence it is possible to use the Routh algorithm for the calculation.