Structure theory of Rack-Bialgebras

TL;DR

This paper develops the structure theory of rack bialgebras, exploring their properties, connections to Leibniz algebras, and their geometric origins via the Serre functor, extending classical Lie theory concepts.

Contribution

It introduces a comprehensive structure theory for rack bialgebras, constructs canonical examples for Leibniz algebras, and relates these to geometric procedures like the Serre functor.

Findings

Rack bialgebras have a semi-group inspired structure.

Canonical rack bialgebras can be constructed for Leibniz algebras.

Connections established between rack bialgebras, Lie racks, and geometric distribution spaces.

Abstract

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 dialgebras.