Related papers: Order structure on the algebra of permutations and…
The paper describes the algebraic structure of the graded algebra of differentially homogeneous polynomials of fixed finite order. We show that it is a finitely generated algebra, and we exhibit a minimal set of generators. Along the way,…
The objects of study in this paper are Hopf algebras $H$ which are finitely generated algebras over an algebraically closed field and are extensions of a commutative Hopf algebra by a finite dimensional Hopf algebra. Basic structural and…
Weak (Hopf) bialgebras are described as (Hopf) bimonoids in appropriate duoidal (also known as 2-monoidal) categories. This interpretation is used to define a category wba of weak bialgebras over a given field. As an application, the "free…
We construct and study new generalisations to rooted trees and forests of some properties of shuffles of words. First, we build a coproduct on rooted trees which, together with their shuffle, endow them with bialgebra structure. We then…
In this paper, we construct a bialgebraic and further a Hopf algebraic structure on top of subgraphs of a given graph. Further, we give the dual structure of this Hopf algebraic structure. We study the algebra morphisms induced by graph…
We study partially and totally associative ternary algebras of first and second kind. Assuming the vector space underlying a ternary algebra to be a topological space and a triple product to be continuous mapping we consider the trivial…
We get new Hopf algebras (HA): 1. A wealth of quotient HA's of the Malvenuto-Reutenauer HA (the Loday-Ronco HA being a special case). They consist of the permutations avoiding an {\it arbitrary} set of permutations without global descents,…
We exhibit a monoidal structure on the category of finite sets indexed by P-trees for a finitary polynomial endofunctor P. This structure categorifies the monoid scheme (over Spec N) whose semiring of functions is (a P-version of) the…
We construct an addition and a multiplication on the set of planar binary trees, closely related to addition and multiplication on the integers. This gives rise to a new kind of (noncommutative) arithmetic theory. The price to pay for this…
We study the (so-called bilinear) factorization problem answered by a weak wreath product (of monads and, more specifically, of algebras over a commutative ring) in the works by Street and by Caenepeel and De Groot. A bilinear factorization…
A quantitative model of concurrent interaction is introduced. The basic objects are linear combinations of partial order relations, acted upon by a group of permutations that represents potential non-determinism in synchronisation. This…
We study ideals in Hall algebras of monoid representations on pointed sets corresponding to certain conditions on the representations. These conditions include the property that the monoid act via partial permutations, that the…
We give a geometric realization of the polyhedra governed by the structure of associative algebras with co-inner products, or more precisely, governed by directed planar trees. Our explicit realization of these polyhedra, which include the…
We recapture Kuperberg's numerical invariant of 3-manifolds associated to a semisimple and cosemisimple Hopf algebra through a `planar algebra construction'. A result of possibly independent interest, used during the proof, which relates…
We prove several new results on the structure of the subgroup generated by a small doubling subset of an ordered group, abelian or not. We obtain precise results generalizing Freiman's 3k-3 and 3k-2 theorems in the integers and several…
In theory of Coxeter groups, bigrassmannian elements are well known as elements which have precisely one left descent and precisely one right descent. In this article, we prove formulas on enumeration of bigrassmannian permutations weakly…
Building on Retakh's approach to Ext groups through categories of extensions, Schwede reobtained the well-known Gerstenhaber algebra structure on Ext groups over bimodules of associative algebras both from splicing extensions (leading to…
We introduce the notion of generalized bialgebra, which includes the classical notion of bialgebra (Hopf algebra) and many others. We prove that, under some mild conditions, a connected generalized bialgebra is completely determined by its…
We analyze the structure of the Malvenuto-Reutenauer Hopf algebra of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determine its primitive elements and its coradical filtration, and…
Trees are partial orders in which every element has a linearly ordered set of predecessors. Here we initiate the exploration of the structural theory of trees with the study of different notions of \emph{branching in trees} and of…