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 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 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.
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