Related papers: Combinatorial Hopf algebra structure on packed squ…
The modular group algebra of an elementary abelian p-group is isomorphic to the restricted enveloping algebra of commutative restricted Lie algebra. The different ways of regarding this algebra result in different Hopf algebra structures…
In formulating a generalized framework to study certain noncommutative algebras naturally arising in representation theory, K. A. Brown asked if every finitely generated Hopf algebra satisfying a polynomial identity was finite over a normal…
Let $A$ be a non-degenerate algebra over the complex numbers and $\Delta$ a homomorphism from $A$ to the multiplier algebra $M(A\otimes A)$. Consider the linear maps $T_1$ and $T_2$ from $A\otimes A$ to $M(A\otimes A)$ defined by…
We provide examples of non-surjective epimorphisms $H\to K$ in the category of Hopf algebras over a field, even with the additional requirement that $K$ have bijective antipode, by showing that the universal map from a Hopf algebra to its…
If H is a connected, graded Hopf algebra, then Takeuchi's formula can be used to compute its antipode. However, there is usually massive cancellation in the result. We show how sign-reversing involutions can sometimes be used to obtain…
Let A and B be two connected graded commutative k-algebras of finite type, where k is a perfect field of positive characteristic p. We prove that the quasi--shuffle algebras generated by A and B are isomorphic as Hopf algebras if and only…
We study a Hopf algebra $H$, which is finitely generated and projective over a commutative ring $k$, as a $P$-Frobenius algebra. We define modular functions in this setting, and provide a complete proof of Radford's formula for the fourth…
Combinatorial Hopf algebras give a linear algebraic structure to infinite families of combinatorial objects, a technique further enriched by the categorification of these structure via the representation theory of families of algebras. This…
There is a long tradition of categorifying combinatorial Hopf algebras by the modules of a tower of algebras (or even better via the representation theory of a tower of groups). From the point of view of combinatorics, such a…
In this work, we develop systematically the ``Dirichlet Hopf algebra of arithmetics'' by dualizing addition and multiplication maps. We study the additive and multiplicative antipodal convolutions which fail to give rise to Hopf algebra…
We derive necessary and sufficient conditions for an ambiskew polynomial ring to have a Hopf algebra structure of a certain type. This construction generalizes many known Hopf algebras, for example U(sl2), U_q(sl2) and the enveloping…
In this article we discuss the Hopf algebras spanned by the adjacency matrices in detail. We show that there two Hopf algebraic structures concerning the adjacency matrices, one is the copy of Connes-Kreimer Hopf algebra, another one is the…
Descent algebras of graded bialgebras were introduced by F. Patras as a generalization of Solomon's descent algebras for Coxeter groups of type $A$, i.e. symmetric groups. The universal enveloping algebra of the free Lie algebra on a…
We construct Hopf algebra isomorphisms of discrete multiplier Hopf C*-algebras, and Hopf AF C*-algebras (generalized quantum UHF algebras), from K-theoretical data. Some of the intermediate results are of independent interest, such as a…
In Section 1 we introduce Frobenius coordinates in the general setting that includes Hopf subalgebras. In Sections 2 and 3 we review briefly the theories of Frobenius algebras and augmented Frobenius algebras with some new material in…
Using the standard filtration associated with a generalized lifting method, we determine all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose coradical generates a Hopf subalgebra isomorphic…
We propose a combinatorial formula for the coproduct in a Hopf algebra of decorated multi-indices that recently appeared in the literature, which can be briefly described as the graded dual of the enveloping algebra of the free Novikov…
We study the Hopf algebra of double posets and two of its Hopf subalgebras, the Hopf algebras of plane posets and of posets "without N". We prove that they are free, cofree, self-dual, and we give an explicit Hopf pairing on these Hopf…
We endow the set of isomorphic classes of matroids with a new Hopf algebra structure, in which the coproduct is implemented via the combinatorial operations of restriction and deletion. We also initiate the investigation of dendriform…
We build, from the collection of all groups of unitriangular matrices, Hopf monoids in Joyal's category of species. Such structure is carried by the collection of class function spaces on those groups, and also by the collection of…