Leibniz Algebras, Lie Racks, and Digroups
By Michael K. Kinyon. Journal of Lie theory, 2004
105 citations recorded by the source corpus.
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.
Subfields: Algebra and Number Theory, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/74faa17a25ec622e449a0ff8d5cacf5b2fd63a70
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In Algebra and Number Theory
In Mathematical Physics
Citations in the corpus
- n-ary algebras: a review with applications (2010, background): “However, the problem of integrating Leibniz algebras in general remains open, although progress has been made with the introduction of Lie racks [147].”