Structure theory of Rack-Bialgebras

Charles Alexandre , Martin Bordemann (1) , Salim Riviere (2) , Friedrich Wagemann (2)

1 LMIA - Laboratoire de Mathématiques Informatique et Applications [UHA]
2 LMJL - Laboratoire de Mathématiques Jean Leray

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 well-known 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 di-algebras.

Domaines: Topologie algébrique [math.AT]; Algèbres quantiques [math.QA]

Charles Alexandre, Martin Bordemann, Salim Riviere, Friedrich Wagemann. Structure theory of Rack-Bialgebras. Journal of Generalized Lie Theory and Applications, 2016, 10 (1). ⟨hal-01891841⟩