A universal enveloping algebra for cocommutative rack bialgebras

TL;DR

This paper constructs a bialgebra in the category of linear maps from cocommutative rack bialgebras, extending to some non-cocommutative cases, and relates the Loday complex to rack bialgebra deformations.

Contribution

It introduces a universal enveloping algebra for cocommutative rack bialgebras and explores its extension to non-cocommutative cases, linking to Leibniz algebra deformations.

Findings

Constructed a bialgebra object in the category of linear maps from cocommutative rack bialgebras.

Extended the construction to certain non-cocommutative rack bialgebras.

Embedded the Loday complex into the rack bialgebra deformation complex.

Abstract

We construct a bialgebra object in the category of linear maps LM from a cocommutative rack bialgebra. The construction does extend to some non-cocommutative rack bialgebras, as is illustrated by a concrete example. As a separate result, we show that the Loday complex with adjoint coefficients embeds into the rack bialgebra deformation complex for the rack bialgebra defined by a Leibniz algebra.