Construction of Groups Associated to Lie- and to Leibniz-Algebras

Manon Didry¹

¹Institut Elie Cartan, Université Henri Poincaré, BP 239, 54506 Vandoeuvre-lès-Nancy, France

Journal of Lie Theory, Tome 17 (2007) no. 2, pp. 399-426

Résumé

\def\g{{\frak g}} \def\N{{\Bbb N}} We describe a method for associating to a Lie algebra $\g$ over a ring $\Bbb K$ a sequence of groups $(G_{n}(\g))_{n\in\N}$, which are {\it polynomial groups} in the sense that will be explained in Definition 5.1. Using a description of these groups by generators and relations, we prove the existence of an action of the symmetric group $\Sigma_{n}$ by automorphisms. The subgroup of fixed points under this action, denoted by $J_{n}(\g)$, is still a polynomial group and we can form the projective limit $J_{\infty}(\g)$ of the sequence $(J_{n}(\g))_{n\in\N}$. The formal group $J_{\infty}(\g)$ associated in this way to the Lie algebra $\g$ may be seen as a generalisation of the formal group associated to a Lie algebra over a field of characteristic zero by the Campbell-Haussdorf formula.

Détail

Classification : 17B65, 14L05
Mots-clés : Lie algebra, Leibniz algebra, polynomial group, formal group, exponential map, Campbell-Haussdorf formula, dual numbers