Related papers: Algebraic structures on double and plane posets
We introduce a graded Hopf algebra based on the set of parking functions (hence of dimension (n+1)^{n-1} in degree n). This algebra can be embedded into a noncommutative polynomial algebra in infinitely many variables. We determine its…
This paper is an initial inquiry into the structure of the Hopf algebra of matroids with restriction-contraction coproduct. Using a family of matroids introduced by Crapo in 1965, we show that the subalgebra generated by a single point and…
Multiplier Hopf algebroids are algebraic versions of quantum groupoids that generalize Hopf algebroids to the non-unital case and weak (multiplier) Hopf algebras to non-separable base algebras. The main structure maps of a multiplier Hopf…
We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for…
We study weight posets of weight multiplicity free (=wmf) representations $R$ of reductive Lie algebras. Specifically, we are interested in relations between $\dim R$ and the number of edges in the Hasse diagram of the corresponding weight…
Let $\mathbb{K}$ be a field. We study the free bialgebra $\mathcal{T}$ generated by the coalgebra $C=\mathbb{K} g \oplus \mathbb{K} h$ and its quotient bialgebras (or Hopf algebras) over $\mathbb{K}$. We show that the free noncommutative…
This paper studies the duals of some finite dimensional pointed Hopf algebras, with abelian group of grouplikes, over an algebraically closed field of characteristic 0, which are either Radford biproducts or else nontrivial liftings of a…
We show that some associative algebras whose product splits up into the sum of several operations and are free, in a certain sense, with respect to these operations, admit a Hopf algebra structure. We show that the operad of dendriform…
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…
This paper defines a pairing of two finite Hopf C*-algebras $A$ and $B$, and investigates the interactions between them. If the pairing is non-degenerate, then the quantum double construction is given. This construction yields a new finite…
We give a new construction of a Hopf subalgebra of the Hopf algebra of Free quasi-symmetric functions whose bases are indexed by objects belonging to the Baxter combinatorial family (i.e. Baxter permutations, pairs of twin binary trees,…
A non-connected neither of finite type Hopf algebra $\mathcal{F}_{0}$ is defined over $\mathbb{Z}/ 2\mathbb{Z}$ and its hom dual turns out to be a tensor product of polynomial algebras. Certain quotient Hopf algebras include the Steenrod…
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 give a direct and self-contained proof that if $H$ is a Hopf algebra and $A\subset H$ is a right coideal subalgebra such $A$ is a direct summand in $H$ as an $A$-bimodule, then $H$ is faithfully flat as a left and right $A$-module.
We construct the analogue of Takeuchi's free Hopf algebra in the setting of Poisson Hopf algebras. More precisely, we prove that there exists a free Poisson Hopf algebra on any coalgebra or, equivalently that the forgetful functor from the…
A noncommutative and noncocommutative Hopf algebra on finite topologies H_T is introduced and studied (freeness, cofreeness, self-duality...). Generalizing Stanley's definition of P-partitions associated to a special poset, we define the…
To any poset or quasi-poset is attached a lattice polytope, whose Ehrhart polynomial we study from a Hopf-algebraic point of view. We use for this two interacting bialgebras on quasi-posets. The Ehrhart polynomial defines a Hopf algebra…
We study generating functions of strict and non-strict order polynomials of series-parallel posets, called order series. These order series are closely related to Ehrhart series and h*-polynomials of the associated order polytopes. We…
A commutative but not cocommutative graded Hopf algebra $\Hn$, based on ordered rooted trees, is studied. This Hopf algebra generalizes the Hopf algebraic structure of unordered rooted trees $\Hc$, developed by Butcher in his study of…
Each labeled rooted tree is associated with a hyperplane arrangement, which is free with exponents given by the depths of the vertices of this tree. The intersection lattices of these arrangements are described through posets of forests.…