paper

Leibniz algebras, Lie racks, and digroups

arXiv:math/0403509

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 \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.

15 pages, AMS-LaTeX, to appear in J. Lie Theory. v5: reformatted for JLoT's house style, other minor changes suggested by referees. v4: added references, added passage describing Lie rack associated to any Leibniz algebra, fixed various typos and miswordings. v3: (i) Improved notion of split Leibniz algebra and fixed corresponding results, (ii) Fixed left group/right group mix-up, (iii) Deleted auxiliary Thm 2.9; it will appear later in a paper by J.D. Phillips, (iv) Fixed statement of Thm 3.4 to something nontrivial. v2: turned off showkeys

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