A Milnor-Moore theorem for dendriform Hopf algebras (Q2708182)

Let \(k\) be a field. A dendriform algebra is a \(k\)-vector space \(D\) provided with two binary operations, \(\succ,\prec\colon D\otimes D\to D\) satisfying certain axioms; these axioms imply that \(x*y:=x\succ y+x\prec y\) is associative. See [\textit{J.-L Loday}, Lect. Notes Math 1763, 7-66 (2001; Zbl 0999.17002)]. A dendriform Hopf algebra is a dendriform algebra \(H\) provided with an associative coproduct \(\Delta\) satisfying certain compatibility conditions; as a consequence, in particular, the ``associated unitary'' algebra \(H_+=H\oplus k\) is actually a Hopf algebra. A brace algebra is a \(k\)-vector space \(P\) provided with a family of linear operations \(\langle\dots\rangle\colon P^{\otimes n}\to P\) for all \(n\geq 2\) satisfying certain compatibility axioms, regarding the natural actions of the symmetric groups \(S^n\) on \(P^{\otimes n}\). See [\textit{E. Getzler}, Isr. Math. Conf. Proc. 7, 65-78 (1993; Zbl 0844.18007)]. In [Contemp. Math. 267, 245-263 (2000; Zbl 0974.16035)] the author showed: any dendriform algebra has a natural structure of a brace algebra; the space of primitive elements of a dendriform Hopf algebra is a sub-brace algebra. In this note, to any brace algebra \(P\) a dendriform Hopf algebra \(U_{\text{Dend}}(P)\) is attached. The main result says: (a) the natural map \(P\to\text{Prim}(U_{\text{Dend}}(P))\) is an isomorphism for any brace algebra \(P\); (b) any graded connected dendriform Hopf algebra \(H\) is isomorphic to \(U_{\text{Dend}}(\text{Prim}(H))\). Statements (a) and (b) are versions of the famous Milnor-Moore theorem.