Rack-Bialgebras and Star-Products on duals of Leibniz algebras

Charles Alexandre , M. Bordemann (1) , Salim Rivière (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. The canonical rack bialgebras we have constructed for any Leibniz algebra lead then to a simple explicit formula of the rack-star-product on the dual of a Leibniz algebra recently constructed by Dherin and Wagemann in 2013. We clarify this framework doing some deformation theory.

Mots clés: racks; algèbres de Leibniz; star produits; quantification par déformations; rack bigèbres

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

Charles Alexandre, M. Bordemann, Salim Rivière, Friedrich Wagemann. Rack-Bialgebras and Star-Products on duals of Leibniz algebras. 2014. ⟨hal-01157529⟩