Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Lie Theory 17 (2007), No. 2, 399--426Copyright Heldermann Verlag 2007 Construction of Groups Associated to Lie- and to Leibniz-Algebras Manon Didry Institut Elie Cartan, Université Henri Poincaré, BP 239, 54506 Vandoeuvre-lès-Nancy, France Manon.Didry@iecn.u-nancy.fr [Abstract-pdf] \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. Keywords: Lie algebra, Leibniz algebra, polynomial group, formal group, exponential map, Campbell-Haussdorf formula, dual numbers. MSC: 17B65, 14L05 [ Fulltext-pdf  (250  KB)] for subscribers only.