Milnor-Moore theorems for bialgebras in characteristic zero

TL;DR

This paper establishes equivalences between categories of specific bialgebras and structured Lie algebras over fields of characteristic zero, extending classical results to broader classes of bialgebras.

Contribution

It generalizes Milnor-Moore theorems to include non-connected bialgebras without antipodes in characteristic zero.

Findings

Equivalence between certain bialgebra categories and structured Lie algebras.

Includes non-connected bialgebras that lack antipodes.

Extends classical Milnor-Moore theorems to broader classes.

Abstract

Over fields of characteristic zero, we construct equivalences between certain categories of bialgebras which are generated by grouplikes and generalized primitives, and certain categories of structured Lie algebras. The relevant families of bialgebras include many which are not connected, and which fail to admit antipodes.