Related papers: Cofree compositions of coalgebras (extended abstra…
In this paper, we define a Hopf algebra structure on the vector space spanned by packed words using a selection-quotient coproduct. We show that this algebra is free on its irreducible packed words. Finally, we give some brief explanations…
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…
We prove that both, the embedding of the category of Hopf algebras into that of bialgebras and the forgetful functor from the category of Hopf algebras to the category of algebras, have right adjoints; in other words: every bialgebra has a…
We endow the category of bialgebras over a pair of operads in distribution with a cofibrantly generated model category structure. We work in the category of chain complexes over a field of characteristic zero. We split our construction in…
Combinatorial Hopf algebras of trees exemplify the connections between operads and bialgebras. Painted trees were introduced recently as examples of how graded Hopf operads can bequeath Hopf structures upon compositions of coalgebras. We…
The Hopf envelope of a bialgebra is the free Hopf algebra generated by the given bialgebra. Its existence, as well as that of the cofree Hopf algebra, is a well-known fact in Hopf algebra theory, but their construction is not particularly…
Two coalgebra structures are used in quantum field theory. The first one is the coalgebra part of a Hopf algebra leading to deformation quantization. The second one is a co-module co-algebra over the first Hopf algebra and it is used to…
In this paper, we obtain a canonical factorization of basis elements in free Rota-Baxter algebras built on bracketed words. This canonical factorization is applied to give a coalgebra structure on the free Rota-Baxter algebras. Together…
This article introduces a method, which starting from simple and quite general mathematical data, allows to construct linear algebras of operators which are, each of them, endowed with a bialgebra structure (coproduct and counity). Moreover…
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…
We study the category of graded Hopf algebras that are free noncommutative, cocommutative, graded and connected from the perspective of the sequences of dimensions of the graded pieces. We show that a Hopf algebra exists with a given…
Let k be a field. Let also (F, G) be a matched pair of groups. We give necessary and sufficient conditions on a pair (\sigma, \tau) of 2-cocycles in order that the crossed product algebra and the crossed coproduct coalgebra…
Let $A$ and $B$ be algebras and coalgebras in a braided monoidal category $\Cc$, and suppose that we have a cross product algebra and a cross coproduct coalgebra structure on $A\ot B$. We present necessary and sufficient conditions for…
The notion of a modified Rota-Baxter algebra comes from the combination of those of a Rota-Baxter algebra and a modified Yang-Baxter equation. In this paper, we first construct free modified Rota-Baxter algebras. We then equip a free…
We construct a coherent Hopf 2-algebra in terms of Hopf coquasigroups, which relax the coassociativity condition and generalize the results in \cite{XH2023}. We also study quasi coassociative Hopf coquasigroups, and show that they give rise…
We give the explicit construction of the product of an arbitrary family of coalgebras, bialgebras and Hopf algebras: it turns out that the product of an arbitrary family of coalgebras (resp. bialgebras, Hopf algebras) is the sum of a family…
By means of a new notion of subforests of an angularly decorated rooted forest, we give a combinatorial construction of a coproduct on the free Rota-Baxter algebra on angularly decorated rooted forests. We show that this coproduct equips…
The operads of Poisson and Gerstenhaber algebras are generated by a single binary element if we consider them as Hopf operads (i.e. as operads in the category of cocommutative coalgebras). In this note we discuss in details the Hopf operads…
We study the Hopf structure of a class of dual operator algebras corresponding to certain semigroups. This class of algebras arises in dilation theory, and includes the noncommutative analytic Toeplitz algebra and the multiplier algebra of…
We study different algebraic structures associated to an operad and their relations: to any operad $\mathbf{P}$ is attached a bialgebra,the monoid of characters of this bialgebra, the underlying pre-Lie algebra and its enveloping algebra;…