Abstract

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.

Highlights

  • As is well known, Leibniz algebras are a noncommutative, or rather, non-anti-symmetric, generalization of Lie algebras

  • The best results to date for this case were given in the Ph.D. thesis of Covez, [2], who uses some cohomology groups associated with the Leibniz algebra to devise an integration procedure for arbitrary algebras

  • Since in a neighborhood of ξ, which can be regarded as a point in both D and E, both manifolds are diffeomorphic, and since by the computation done in Lemma 16 the product in E is expressed by formula (19), it is clear that their differentials at ξ coincide

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Summary

Introduction

As is well known, Leibniz algebras are a noncommutative, or rather, non-anti-symmetric, generalization of Lie algebras. The best results to date for this case were given in the Ph.D. thesis of Covez, [2], who uses some cohomology groups associated with the Leibniz algebra to devise an integration procedure for arbitrary algebras His solution is in terms of racks; by construction, his results are only local, as a condition of simple connexity of an underlying Lie group is required. No attempt is made to state the results in the most general possible context, and for the sake of definiteness we only consider finite dimensional real vector spaces

Split and Derived Leibniz Algebras
Integration for the Split Case
The Nonsplit Case
Towards a General Definition of a Coquecigrue
Some Concluding Remarks
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