Leibniz algebras, Lie racks, and digroups

TL;DR

This paper explores the relationship between Leibniz algebras and geometric structures like Lie racks and digroups, proposing a partial solution to the coquecigrue problem by linking Leibniz algebras to Lie digroups.

Contribution

It demonstrates that Lie racks have Leibniz algebra structures on their tangent spaces and introduces Lie digroups as a means to realize certain Leibniz algebras, advancing the understanding of their geometric counterparts.

Findings

Lie racks have Leibniz algebra structures on tangent spaces.

Every digroup can be decomposed into a product of a group and a trivial digroup.

Partial solution to the coquecigrue problem for Leibniz algebras that split over their squares.

Abstract

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 \emph{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 \emph{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 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.