Hopf algebras of planar binary trees: an operated algebra approach

Parallel to operated algebras built on top of planar rooted trees via the grafting operator $B^+$, we introduce and study $\vee$-algebras and more generally $\vee_\Omega$-algebras based on planar binary trees. Involving an analogy of the Hochschild 1-cocycle condition, cocycle $\vee_\Omega$-bialgebras (resp.~$\vee_\Omega$-Hopf algebras) are also introduced and their free objects are constructed via decorated planar binary trees. As a special case, the well-known Loday-Ronco Hopf algebra $H_{LR}$ is a free cocycle $\vee$-Hopf algebra. By means of admissible cuts, a combinatorial description of the coproduct $\Delta_{LR(\Omega)}$ on decorated planar binary trees is given, as in the Connes-Kreimer Hopf algebra by admissible cuts.