Cocommutative Hopf algebras have a very well-developed structure theory via the theorem of Cartier-Kostant-Milnor-Moore: over algebraically closed fields of characteristic zero, they split as a semidirect product of a group ring with the enveloping algebra of the space of primitives, a Lie algebra. Further, the Poincaré-Birkhoff-Witt theorem gives very explicit description of the graded algebra associated to the filtration of the enveloping algebra of the primitives.
This structure theory is in some sense governed by operads, particularly the relations between the associative, Lie, and commutative operads. Further, these constructions lead to computational tools in homological algebra, particularly relating Harrison and Hochschild homology. More structured formulations relate André-Quillen homology to the partition poset and spectral Lie algebras.
Unfortunately, in the setting of braided Hopf algebras, most of this theory falls apart due to the fact that the primitives no longer form a Lie algebra. I’ll present ongoing work in which we investigate what sort of structure the primitives support, and how much it can be leveraged to construct analogues of these results in a genuinely braided setting.