Milnor–Moore theorem for nonassociative Hopf algebras

Determine whether a Milnor–Moore theorem holds for nonassociative Hopf algebras, in particular for the nonassociative bialgebra structures arising from the dual coproduct on rooted trees.

Background

The authors distinguish primitive rooted trees from the full space of primitive elements for the dual coproduct, because linear combinations of nonprimitive trees could themselves be primitive. They state that determining all primitive elements is less tractable and note that the relevant structural theory is uncertain because it is not clear whether a Milnor–Moore theorem is available for nonassociative Hopf algebras.

References

Finding all primitive elements is much harder but less interesting for our purpose, since we are in the dual and since it is not clear that we have a Milnor-Moore theorem (, see also and ) for non associative Hopf algebras.

— Coalgebras, bialgebras and Rota-Baxter algebras from shuffles of rooted forests  (2501.02557 - Clavier et al., 5 Jan 2025) in Remark following Definition of primitive trees, Section 2.4, "Dual primitive trees"