The local integration of Leibniz algebras

Abstract

This article gives a local answer to the coquecigrue problem. Hereby we mean the problem, formulated by J-L. Loday in \cite{LodayEns}, is that of finding a generalization of the Lie's third theorem for Leibniz algebra. That is, we search a manifold provided with an algebraic structure which generalizes the structure of a (local) Lie group, and such that the tangent space at a distinguished point is a Leibniz algebra structure. Moreover, when the Leibniz algebra is a Lie algebra, we want that the integrating manifold is a Lie group. In his article \cite{Kinyon}, M.K. Kinyon solves the particular case of split Leibniz algebras. He shows, in particular, that the tangent space at the neutral element of a Lie rack is provided with a Leibniz algebra structure. Hence it seemed reasonable to think that Lie racks give a solution to the coquecigrue problem, but M.K. Kinyon also showed that a Lie algebra can be integrated into a Lie rack which is not a Lie group. Therefore, we have to specify inside the category of Lie racks, which objects are the coquecigrues. In this article we give a local solution to this problem. We show that every Leibniz algebra becomes integrated into a \textit{local augmented Lie rack}. The proof is inspired by E. Cartan's proof of Lie's third theorem, and, viewing a Leibniz algebra as a central extension by some center, proceeds by integrating explicitely the corresponding Leibniz 2-cocycle into a rack 2-cocycle. This proof gives us a way to construct local augmented Lie racks which integrate Leibniz algebras, and this article ends with examples of the integration of non split Leibniz algebras in dimension 4 and 5.