Related papers: Preordered forests, packed words and contraction a…
We re-examine all the contractions related with the ${\cal U}_q(su(2))$ deformed algebra and study the consequences that the contraction process has for their structure. We also show using ${\cal U}_q(su(2))\times{\cal U}(u(1))$ as an…
The main properties of the crossed product in the category of Hopf algebras are investigated. Let $A$ and $H$ be two Hopf algebras connected by two morphism of coalgebras $\triangleright : H\ot A \to A$, $f:H\ot H\to A$. The crossed product…
Hopf algebroids are generalization of Hopf algebras over non-commutative base rings. It consists of a left- and a right-bialgebroid structure related by a map called the antipode. However, if the base ring of a Hopf algebroid is commutative…
We examine languages of unranked forests definable using the temporal operators EF and EX. We characterize the languages definable in this logic, and various fragments thereof, using the syntactic forest algebras introduced by Bojanczyk and…
We give a new construction of a Hopf algebra defined first by Reading whose bases are indexed by objects belonging to the Baxter combinatorial family (i.e., Baxter permutations, pairs of twin binary trees, etc.). Our construction relies on…
We prove that the structure algebra of a Bruhat moment graph of a finite real root system is a Hopf algebroid with respect to the Hecke and the Weyl actions. We introduce new techniques (reconstruction and push-forward formula of a product,…
A known fundamental Theorem for braided pointed Hopf algebras states that for each coideal subalgebra, that fulfils a few properties, there is an associated quotient coalgebra right module such that the braided Hopf algebra can be…
A non-commutative, planar, Hopf algebra of rooted trees was proposed in L. Foissy, Bull. Sci. Math. 126 (2002) 193-239. In this paper we propose such a non-commutative Hopf algebra for graphs. In order to define a non-commutative product we…
We prove a universal characterization of Hopf algebras among cocommutative bialgebras over a field: a cocommutative bialgebra is a Hopf algebra precisely when every split extension over it admits a join decomposition. We also explain why…
We gives examples of Com-PreLie bialgebras, that is to say bialgebras with a preLie product satisfying certain compatibilities. Three families are defined on shuffle algebras: one associated to linear endomorphisms, one associated to linear…
In this paper we construct "structural" pre-braidings characterizing different algebraic structures: a rack, an associative algebra, a Leibniz algebra and their representations. Some of these pre-braidings seem original. On the other hand,…
We continue the study of twisted automorphisms of Hopf algebras started in "Twisted automorphisms of Hopf algebras". In this paper we concentrate on the group algebra case. We describe the group of twisted automorphisms of the group algebra…
We introduce the combinatorial notion of posetted trees and we use it in order to write an explicit expression of the Baker-Campbell-Hausdorff formula.
Bergeron and Ceballos defined a Hopf algebra structure on equivalence classes of subword complexes. We introduce a category of subword complexes, endow it with a proto-exact-like structure, and show that the corresponding dual Hall Hopf…
A leaf path language is a Boolean combination of sets of the form $\mathsf{{}^mE}^k L$, with $k \ge 1$ and $L$ a regular word language, which consist of those forests where the node labels in at least $k$ leaf-to-root paths make up a word…
We apply a combinatorial formula of the first author and Rosso, for products in Hopf quiver algebras, to determine the structure of Nichols algebras. We illustrate this technique by explicitly constructing new examples of Nichols algebras…
Lifting attempts to speed up probabilistic inference by exploiting symmetries in the model. Exact lifted inference methods, like their propositional counterparts, work by recursively decomposing the model and the problem. In the…
We show that the # product of binary trees introduced by Aval and Viennot [arXiv:0912.0798] is in fact defined at the level of the free associative algebra, and can be extended to most of the classical combinatorial Hopf algebras.
We study the structure of the category of graded, connected, countable-dimensional, commutative and cocommutative Hopf algebras over a perfect field $k$ of characteristic $p$. Every $p$-torsion object in this category is uniquely a direct…
Hopf braces have been introduced as a Hopf-theoretic generalization of skew braces. Under the assumption of cocommutativity, these algebraic structures are equivalent to matched pairs of actions on Hopf algebras, that can be used to produce…