
Journal of Lie Theory 17 (2007), No. 2, 399426 Copyright Heldermann Verlag 2007 Construction of Groups Associated to Lie and to LeibnizAlgebras Manon Didry Institut Elie Cartan, Université Henri Poincaré, BP 239, 54506 VandoeuvrelèsNancy, France Manon.Didry@iecn.unancy.fr [Abstractpdf] \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 CampbellHaussdorf formula. Keywords: Lie algebra, Leibniz algebra, polynomial group, formal group, exponential map, CampbellHaussdorf formula, dual numbers. MSC: 17B65, 14L05 [ Fulltextpdf (250 KB)] for subscribers only. 