Leibniz Algebras, Lie Racks, and Digroups

Michael K. Kinyon¹

¹Dept. of Mathematics, University of Denver, Denver, CO 80208, U.S.A.

Journal of Lie Theory, Tome 17 (2007) no. 1, pp. 99-114

Résumé

The "coquecigrue" problem for Leibniz algebras is that of finding an appropriate generalization of Lie's third theorem, that is, of finding a generalization of the notion of group such that Leibniz algebras are the corresponding tangent algebra structures. The difficulty is determining exactly what properties this generalization should have. Here we show that Lie racks, smooth left distributive structures, have Leibniz algebra structures on their tangent spaces at certain distinguished points. One way of producing racks is by conjugation in digroups, a generalization of group which is essentially due to Loday. Using semigroup theory, we show that every digroup is a product of a group and a trivial digroup.
We partially solve the coquecigrue problem by showing that to each Leibniz algebra that splits over an ideal containing its ideal generated by squares, there exists a special type of Lie digroup with tangent algebra isomorphic to the given Leibniz algebra. The general coquecigrue problem remains open, but Lie racks seem to be a promising direction.

Détail

Classification : 17A32, 20M99, 22A30
Mots-clés : Leibniz algebra, Lie rack, digroup