On the operad of bigraft algebras

In this paper, we study the notion of a bigraft algebra, generalizing the notions of left and right graft algebras. We construct the free bigraft algebra on one generator in terms of certain planar rooted trees with decorated edges, and therefore describe explicitly the bigraft operad. We then compute its Koszul dual and show that the bigraft operad is Koszul. Moreover, we endow the free bigraft algebra on one generator with a universal Hopf algebra structure and a pairing. Finally, we prove an analogue of the Poincaré–Birkhoff–Witt and Cartier–Milnor–Moore theorems. For this, we define the notion of infinitesimal bigraft bialgebras and we prove the existence of a new good triple of operads.