Related papers: Hopf algebraic structures on hypergraphs and multi…
We describe the Hopf algebraic structure of Feynman graphs for non-abelian gauge theories, and prove compatibility of the so-called Slavnov-Taylor identities with the coproduct. When these identities are taken into account, the coproduct…
Quite recently, a ``coloured'' extension of the Yang-Baxter equation has appeared in the literature and various solutions of it have been proposed. In the present contribution, we introduce a generalization of Hopf algebras, to be referred…
A condition is identified which guarantees that the coinvariants of a coaction of a Hopf algebra on an algebra form a subalgebra, even though the coaction may fail to be an algebra homomorphism. A Hilbert Theorem (finite generation of the…
We study the symmetric function and polynomial combinatorial invariants of Hopf algebras of permutations, posets and graphs. We investigate their properties and the relations among them. In particular, we show that the chromatic symmetric…
We introduce new polynomial invariants of a finite-dimensional semisimple and cosemisimple Hopf algebra A over a field by using the braiding structures of A. We investigate basic properties of the polynomial invariants including stability…
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…
We generalize the notion of a coloring complex of a graph to linearized combinatorial Hopf monoids. We determine when a linearized combinatorial Hopf monoid has such a construction, and discover some inequalities that are satisfied by the…
We prove a cyclic cohomological analogue of Haefliger's van Est-type theorem for the groupoid of germs of diffeomorphisms of a manifold. The differentiable version of cyclic cohomology is associated to the algebra of transverse differential…
Let $H$ be a pointed Hopf algebra with abelian coradical. Let $A\supseteq B$ be left (or right) coideal subalgebras of $H$ that contain the coradical of $H$. We show that $A$ has a PBW basis over $B$, provided that $H$ satisfies certain…
We introduce the notions of Hopf quasigroup and Hopf coquasigroup $H$ generalising the classical notion of an inverse property quasigroup $G$ expressed respectively as a quasigroup algebra $k G$ and an algebraic quasigroup $k[G]$. We prove…
We consider a Hopf algebra of simplicial complexes and provide a cancellation-free formula for its antipode. We then obtain a family of combinatorial Hopf algebras by defining a family of characters on this Hopf algebra. The characters of…
We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in…
The abelian category of tetramodules over an associative bialgebra $A$ is related with the Gerstenhaber-Schack (GS) cohomology as $Ext_\Tetra(A,A)=H_\GS(A)$. We construct a 2-fold monoidal structure on the category of tetramodules of a…
We define a version of Hochschild homology and cohomology suitable for a class of algebras admitting compatible actions of bialgebras, called module algebras. We show this (co)homology, called Hopf--Hochschild (co)homology, can also be…
The Hopf algebra and the Rota-Baxter algebra are the two algebraic structures underlying the algebraic approach of Connes and Kreimer to renormalization of perturbative quantum field theory. In particular the Hopf algebra of rooted trees…
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,…
The zx-calculus and related theories are based on so-called interacting Frobenius algebras, where a pair of dagger-special commutative Frobenius algebras jointly form a pair of Hopf algebras. In this setting we introduce a generalisation of…
We show that the diagonal complex computing the Gerstenhaber-Schack cohomology of a bialgebra (that is, the cohomology theory governing bialgebra deformations) can be given the structure of an operad with multiplication if the bialgebra is…
We introduce the Hopf algebra of uniform block permutations and show that it is self-dual, free, and cofree. These results are closely related to the fact that uniform block permutations form a factorizable inverse monoid. This Hopf algebra…
We study several bialgebraic structures on boolean functions, that is to say maps defined on the set of subsets of a finite set $X$, taking the value $0$ on $\emptyset$. Examples of boolean functions are given by the indicator function of…