A Milnor-Moore Type Theorem for Primitively Generated Braided Bialgebras

TL;DR

This paper extends the Milnor-Moore theorem to primitively generated braided bialgebras, establishing their isomorphism to universal enveloping algebras of braided Lie algebras, and introduces a new braided Lie algebra concept.

Contribution

It proves a Milnor-Moore type theorem for braided bialgebras and introduces braided Lie algebras for arbitrary braided vector spaces.

Findings

Primitively generated braided bialgebras are isomorphic to universal enveloping algebras of their braided Lie algebras.

Introduces the concept of braided Lie algebra for arbitrary braided vector spaces.

Establishes a foundational result connecting braided bialgebras and braided Lie algebras.

Abstract

A braided bialgebra is called primitively generated if it is generated as an algebra by its space of primitive elements. We prove that any primitively generated braided bialgebra is isomorphic to the universal enveloping algebra of its infinitesimal braided Lie algebra, notions hereby introduced. This result can be regarded as a Milnor-Moore type theorem for primitively generated braided bialgebras and leads to the introduction of a concept of braided Lie algebra for an arbitrary braided vector space.