On differential Hopf algebras and $B_\infty$ algebras

TL;DR

This paper proves a structure theorem for differential graded Hopf algebras, showing they carry a $B_$ algebra structure that generalizes classical results and connects with $A_$ structures.

Contribution

It extends the classical Milnor-Moore theorem to differential graded Hopf algebras, introducing a $B_$ algebra framework and linking it with $A_$ structures.

Findings

Differential Hopf algebras have an underlying $B_$ structure.

The $B_$ structure originates from a twisting of a quasi-trivial structure.

The $B_$ and $A_$ structures are compatible within this framework.

Abstract

We establish a structure theorem, analogous to the classical result of Milnor and Moore, for differential graded Hopf algebras: any differential Hopf algebra $H$ that is free as a coalgebra carries an underlying $B_\infty$ algebra structure that restricts to the subspace of primitives, and conversely $H$ may be recovered via a universal enveloping differential-2-associative algebra. This extends the work of Loday and Ronco [12] where the ungraded non-differential case was treated, and only the multibrace part of the $B_\infty$ structure was found. We show that the multibrace structure of [12] originates from a twisting of a quasi-trivial structure, extending the work of Markl [14] on the $A_\infty$ structure underlying any algebra with a square-zero endomorphism. In this framework it is also clear that the multibrace and $A_\infty$ structures are compatible, and provide an appropriate $B_\infty$ structure for the structure theorem.