Structure Theory of Rack-BialgebrasAlexandre C1, Bordemann M2, Rivière S3 and Wagemann F3*
- *Corresponding Author:
- Wagemann F
Department of Mathematics
Université de Nantes
1 Quai de Tourville, 44035
Nantes Cedex 1, France
E-mail: [email protected]
Received Date: July 17, 2016; Accepted Date: November 19, 2016; Published Date: November 30, 2016
Citation: Alexandre C, Bordemann M, Rivière S, Wagemann F (2016) Structure Theory of Rack-Bialgebras. J Generalized Lie Theory Appl 10:244. doi:10.4172/1736-4337.1000244
Copyright: © 2016 Alexandre C, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
In this paper we focus on a certain self-distributive multiplication on coalgebras, which leads to so-called rack bialgebra. Inspired by semi-group theory (adapting the Suschkewitsch theorem), we do some structure theory for rack bialgebras and cocommutative Hopf dialgebras. We also construct canonical rack bialgebras (some kind of enveloping algebras) for any Leibniz algebra and compare to the existing constructions. We are motivated by a differential geometric procedure which we call the Serre functor: To a pointed differentible manifold with multiplication is associated its distribution space supported in the chosen point. For Lie groups, it is wellknown that this leads to the universal enveloping algebra of the Lie algebra. For Lie racks, we get rack-bialgebras, for Lie digroups, we obtain cocommutative Hopf dialgebras.