On Integral Manifolds for Leibniz Algebras

We discuss several partial solutions to the so-called “coquecigrue problem” of Loday; these solutions parallel, but also generalize in several directions, the classical Lie group-Lie algebra correspondence. Our study highlights some clear similarities between the split and nonsplit cases and leads us to a general unifying scheme that provides an answer to the problem of the algebraic structure of a coquecigrue. 1. Introduction As is now well known, Leibniz algebras are a noncommutative, or rather, non-anti-symmetric, generalization of Lie algebras. The so-called “coquecigrue problem” was proposed by J. L. Loday as an analogue for these algebras of Lie’s third fundamental theorem: given a finite dimensional Leibniz algebra, look for a manifold possessing an algebraic structure such that its linearization yields the original Leibniz algebra. Ideally, this algebraic structure would be a binary operation, but because there is no formal, precise, definition of what a coquecigrue is, this has proven to be a rather difficult question. To what extent this program can be fulfilled is, as of now, still not entirely clear, but we will describe here what we believe is the right approach. A first hint at a general answer to this problem first appeared in [1], where it was shown that any Leibniz algebra can in a sense be “integrated,” by the judicious choice of a so-called Lie rack, stemming from a construction due to Fenn and Rourke. But this was considered unsatisfactory, because for Lie algebras this does not, in general, give the corresponding Lie group(s), a condition that is normally regarded as a critical property of the solution. In the same paper it was also argued that, again via an appropriate rack, an answer for split Leibniz algebras can be given in terms of digroups. This is now generally regarded—including ourselves—as providing a reasonable solution to the coquecigrue problem for this kind of algebras, since the procedure used to obtain a Leibniz algebra from a digroup is in all respects identical to the one used to derive the Lie algebra of a Lie group; in particular, for Lie algebras in principle this reduces to the standard solution given by Lie's theorem. Nevertheless, because the splittings of a Leibniz algebra are not necessarily unique, this opens up questions of uniqueness of these integral manifolds that in a sense are even more delicate than the standard situation in the classical Lie theory, where groups that are locally but not globally diffeomorphic have the same algebra. One of our main objectives here (Section 3) is therefore to analyze