On differential Hopf algebras and $B_{\infty }$ algebras

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.

18pp