Let $R$ be an arbitrary commutative ring with identity, $gl(n,R)$ the general linear Lie algebra over $R$, $d(n,R)$ the diagonal subalgebra of $gl(n,R)$. In case 2 is a unit of $R$, all subalgebras of $gl(n,R)$ containing $d(n,R)$ are determined and their derivations are given. In case 2 is not a unit partial results are given.

La première partie de cet article est une adaptation au cadre des super groupes d’un théorème dû à Cartier qui assure que les groupes formels sont lisses en caractéristique zéro. La seconde partie donne une description des super groupes de Lie dits “vraiment classiques” comme groupes d’automorphismes de super algèbres semi-simples associatives à involution, selon une méthode de Weil.

The main goal of this paper is to show an application of Graph Theory to classifying Lie algebras over finite fields. It is rooted in the representation of each Lie algebra by a certain pseudo-graph. As partial results, it is deduced that there exist, up to isomorphism, four, six, fourteen and thirty-four $2$-, $3$-, $4$-, and $5$-dimensional algebras of the studied family, respectively, over the field $\mathbb{Z}/2\mathbb{Z}$. Over $\mathbb{Z}/3\mathbb{Z}$, eight and twenty-two $2$- and $3$-dimensional Lie algebras, respectively, are also found. Finally,...