We establish a structure theorem analogous to the classical
result of Milnor and Moore: any differential graded (not necessarily co commutative) Hopf algebra H that is cofree as a coalgebra carries an
underlying B∞ algebra structure that restricts to the subspace of prim itives, and conversely H may be recovered via a universal enveloping
2-associative differential algebra. This extends the work of Loday and
Ronco (J. reine angew. Math. 592: 123–155, 2006) where the ungraded
non-differential case was treated, and only the multibrace part of the
B∞ structure was found. We show that the multibrace algebras of Loday
and Ronco (J. reine angew. Math. 592: 123–155, 2006) originate from
twistings of quasi-trivial structures, complementing the work of Markl
(J. Homotopy Relat. Struct. 10, 637–667 (2015)) on the A∞ structure
underlying any algebra with a square-zero endomorphism. In this frame work we can prove the multibrace and A∞ algebras are compatible and
provide the appropriate B∞ algebra for the structure theorem.
