In this paper, we define a new coproduct on the space of decorated planar rooted forests to equip it with a weighted infinitesimal unitary bialgebraic structure. We introduce the concept of Ω-cocycle infinitesimal bialgebras of weight λ and then prove that the space of decorated planar rooted forests H RT (X, Ω), together with a set of grafting operations {B + ω | ω ∈ Ω}, is the free Ωcocycle infinitesimal unitary bialgebra of weight λ on a set X, involving a weighted version of a Hochschild 1-cocycle condition. As an application, we equip a free cocycle infinitesimal unitary bialgebraic structure on the undecorated planar rooted forests, which is the object studied in the well-known (noncommutative) Connes-Kreimer Hopf algebra.
We introduce the notion of a matching Rota-Baxter algebra motivated by the recent work on multiple pre-Lie algebras arising from the study of algebraic renormalization of regularity structures [10,18]. This notion is also related to iterated integrals with multiple kernels and solutions of the associative polarized Yang-Baxter equation. Generalizing the natural connection of Rota-Baxter algebras with dendriform algebras to matching Rota-Baxter algebras, we obtain the notion of matching dendriform algebras. As in the classical case of one operation, matching Rota-Baxter algebras and matching dendriform algebras are related to matching pre-Lie algebras which coincide with the aforementioned multiple pre-Lie algebras. More general notions and results on matching tridendriform algebras and matching PostLie algebras are also obtained.
Parallel to operated algebras built on top of planar rooted trees via the grafting operator B + , we introduce and study ∨-algebras and more generally ∨ Ω -algebras based on planar binary trees. Involving an analogy of the Hochschild 1-cocycle condition, cocycle ∨ Ω -bialgebras (resp. ∨ Ω -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 ∨-Hopf algebra. By means of admissible cuts, a combinatorial description of the coproduct ∆ LR(Ω) on decorated planar binary trees is given, as in the Connes-Kreimer Hopf algebra by admissible cuts.
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