Hopf algebra structure on free Rota-Baxter algebras by angularly decorated rooted trees

Typed angularly decorated planar rooted trees and generalized Rota-Baxter algebras

We introduce a generalization of parametrized Rota-Baxter algebras, which includes family and matching Rota-Baxter algebras. We study the structure needed on the set $Ω$ of parameters in order to obtain that free Rota-Baxter algebras are described in terms of typed and angularly decorated planar rooted trees: we obtain the notion of $λ$-extended diassociative semigroup, which includes sets (for matching Rota-Baxter algebras) and semigroups (for family Rota-Baxter algebras), and many other examples. We also describe free commutative $Ω$-Rota-Baxter algebras generated by a commutative algebra A …

Reduced typed angularly decorated planar rooted trees and generalized tridendriform algebras

We introduce a generalization of tridendriform algebras, where each of the three products are replaced by a family of products indexed by a set $Ω$. We study the needed structure on $Ω$ for free $Ω$-tridendriform algebras to be built on Schr{ö}der trees (as it is the case in the classical case), with convenient decorations on their leaves. We obtain in this way extended triassociative semigroups. We describe commutative $Ω$-tridendriform algebras in terms of typed words. We also study links with generalizations of Rota-Baxter algebras and describe the Koszul duals of the corresponding operads.

Free Rota-Baxter family algebras and Free (tri)dendriform family algebras

In this paper, we first construct the free Rota-Baxter family algebra generated by some set $X$ in terms of typed angularly $X$-decorated planar rooted trees. As an application, we obtain a new construction of the free Rota-Baxter algebra only in terms of angularly decorated planar rooted trees (not forests), which is quite different from the known construction via angularly decorated planar rooted forests by K. Ebrahimi-Fard and L. Guo. We then embed the free dendriform (resp. tridendriform) family algebra into the free Rota-Baxter family algebra of weight zero (resp. one). Finally, we prove …

通过有根树、扭曲双代数与Fock函子构造的Rota-Baxter代数物种

Species of Rota-Baxter algebras by rooted trees, twisted bialgebras and Fock functors

本文引入简单角度装饰森林构造自由Rota-Baxter物种，建立其上的扭曲双代数结构，并通过Fock函子给出自由Rota-Baxter代数双代数结构的新证明。

On Differential Rota-Baxter Algebras

A Rota-Baxter operator of weight $λ$ is an abstraction of both the integral operator (when $λ=0$) and the summation operator (when $λ=1$). We similarly define a differential operator of weight $λ$ that includes both the differential operator (when $λ=0$) and the difference operator (when $λ=1$). We further consider an algebraic structure with both a differential operator of weight $λ$ and a Rota-Baxter operator of weight $λ$ that are related in the same way that the differential operator and the integral operator are related by the First Fundamental Theorem of Calculus. We construct free objec…

Hopf algebra of multi-decorated rooted forests, free matching Rota-Baxter algebras and Gröbner-Shirshov bases

Recent advances in stochastic PDEs, Hopf algebras of typed trees and integral equations have inspired the study of algebraic structures with replicating operations. To understand their algebraic and combinatorial nature, we first use rooted forests with multiple decoration sets to construct free Hopf algebras with multiple Hochschild 1-cocycle conditions. Applying the universal property of the underlying operated algebras and the method of Gröbner-Shirshov bases, we then construct free objects in the category of matching Rota-Baxter algebras which is a generalization of Rota-Baxter algebras to…