Hopf Structures Research Articles

Hopf Structures in the Representation Theory of Direct Products

Combinatorial Hopf algebras give a linear algebraic structure to infinite families of combinatorial objects, a technique further enriched by the categorification of these structures via the representation theory of families of algebras. This paper examines a fundamental construction in group theory, the direct product, and how it can be used to build representation theoretic Hopf algebras out of towers of groups. A key special case gives us the noncommutative symmetric functions NSym, but there are many things that we can say for the general Hopf algebras, including the structure of their character groups and a formula for the antipode.

Primary and secondary instabilities of the Mixed mode solution in a reaction diffusion system near the codimension-two Turing-Hopf point

Abstract Space-time structures near the codimension-two Turing-Hopf point (CTHP) that combine a spatial pattern with temporal oscillations are studied for a particular reaction diffusion (RD) system, which has been useful to model many different biological processes. We show that in this sytem, these Mixed mode solutions arise in regions of the parameter space where the analysis of the linearized RD system predicts only Turing or Hopf structures. Therefore, a standard weakly non linear analysis is used to find the amplitude equations describing the existence and stability regions of these three primary solutions and to study how they are affected by the proximity to the CTHP, the relative strength of the nonlinear terms of the system and the diffusion coefficient. By means of exhaustive numerical calculations, it is shown that the stability analysis to homogeneous perturbations qualitatively predicts the region where Mixed solutions arise. We also show that if the size of the domain grows, non-homogeneous perturbations become important and secondary instabilities can arise in the system. Two types of secondary structures are numerically identified: (1) spatial patterns that change its effective wave number while they oscillate as an Eckhaus instability, or (2) a chaotic oscillations mixed with a relatively fixed spatial pattern in a Benjamin–Feir–Newell phase turbulence. Our study provides a characterization of the time oscillating patterns observed numerically in the RD system studied here and a panoramic view of the Mixed mode primary and secondary instabilities that arise near the CTHP.

Species substitution, graph suspension, and graded Hopf algebras of painted tree polytopes

Combinatorial Hopf algebras of trees exemplify the connections between operads and bialgebras. Painted trees were introduced recently as examples of how graded Hopf operads can bequeath Hopf structures upon compositions of coalgebras. We put these trees in context by exhibiting them as the minimal elements of face posets of certain convex polytopes. The full face posets themselves often possess the structure of graded Hopf algebras (with one-sided unit). We can enumerate faces using the fact that they are structure types of substitutions of combinatorial species. Species considered here include ordered and unordered binary trees and ordered lists (labeled corollas). Some of the polytopes that constitute our main results are well known in other contexts. First we see the classical permutohedra, and then certain generalized permutohedra: specifically the graph associahedra of suspensions of certain simple graphs. As an aside we show that the stellohedra also appear as liftings of generalized permutohedra: graph composihedra for complete graphs. Thus our results give examples of Hopf algebras of tubings and marked tubings of graphs. We also show an alternative associative algebra structure on the graph tubings of star graphs.

Generalized path algebra and its significance in representation theory

In 2000, Shaoxue Liu and Coelho introduced the concept of generalized path algebras in order tomake a more direct characterization of the structure and representations of algebras. Over the following decade,the method for generalized path algebras received some developments and applications. This article isa summary and demonstration in this respect. Firstly, we give an introduction to structures and representationsof generalized path algebras and some related notions, such as triangular matrix algebras, etc. We showthe characterization of generalized path coalgebras with Hopf structures and then suggest a programme for classificationof non-basic (resp. non-pointed) Hopf algebras by generalized path (co-)algebras. We also define the mutation ofgeneralized path algebras and then investigate the additive categorification of cluster algebras determined by a classof generalized path algebras.

Skew Monoidal Monoids

Skew monoidal categories are monoidal categories with non-invertible “coherence” morphisms. As shown in a previous article, bialgebroids over a ring R can be characterized as the closed skew monoidal structures on the category Mod-R in which the unit object is RR. This offers a new approach to bialgebroids and Hopf algebroids. Little is known about skew monoidal structures on general categories. In the present article, we study the one-object case: skew monoidal monoids (SMMs). We show that they possess a dual pair of bialgebroids describing the symmetries of the (co)module categories of the SMM. These bialgebroids are submonoids of their own base and are rank 1 free over the base on the source side. We give various equivalent definitions of SMM, study the structure of their (co)module categories, and discuss the possible closed and Hopf structures on a SMM.

Generalized Clifford Algebras as Algebras in Suitable Symmetric Linear Gr-Categories

By viewing Clifford algebras as algebras in some suitable symmetric Gr-categories, Albuquerque and Majid were able to give a new derivation of some well known results about Clifford algebras and to generalize them. Along the same line, Bulacu observed that Clifford algebras are weak Hopf algebras in the aforementioned categories and obtained other interesting properties. The aim of this paper is to study generalized Clifford algebras in a similar manner and extend the results of Albuquerque, Majid and Bulacu to the generalized setting. In particular, by taking full advantage of the gauge transformations in symmetric linear Gr-categories, we derive the decomposition theorem and provide categorical weak Hopf structures for generalized Clifford algebras in a conceptual and simpler manner.

The Green Rings of Minimal Hopf Quivers

AbstractLet be a field and let Q be a minimal Hopf quiver, i.e. a cyclic quiver or the infinite linear quiver, and let repln(Q) denote the category of locally nilpotent finite-dimensional -representations of Q. The category repln(Q) has natural tensor structures induced from graded Hopf structures on the path coalgebra . Tensor categories of the form repln(Q) are an interesting class of tame hereditary pointed tensor categories that are not finite. The aim of this paper is to compute the Clebsch–Gordan formulae and Green rings of such tensor categories.

INDUCED HOPF CORING STRUCTURES

Hopf corings are dened in this work as coring objects in the category of algebras over a commutative ring R. Using the Dieudonn<TEX>$\'{e}$</TEX> equivalences from [7] and [19], one can associate coring structures built from the Hopf algebra <TEX>$F_p[x_0,x_1,{\ldots}]$</TEX>, p a prime, with Hopf ring structures with same underlying connected Hopf algebra. We have that <TEX>$F_p[x_0,x_1,{\ldots}]$</TEX> coring structures classify thus Hopf ring structures for a given Hopf algebra. These methods are applied to dene new ring products in the Hopf algebras underlying known Hopf rings that come from connective Morava <TEX>${\kappa}$</TEX>-theory.

On a homeomorphism between orbit spaces of linear systems and matrix polynomials

The orbit space of controllable systems under system similarity and the orbit space of matrix polynomials with determinant degree equal to the order of the state matrix under right equivalence are proved to be homeomorphic when the quotient compact–open topology is considered in the latter. As a consequence, the variation of the finite and left Wiener–Hopf structures under small perturbations of matrix polynomials with fixed degree for their determinants is described.

Hopf structures on the Hopf quiverQ(⟨g⟩,g)

We investigate pointed Hopf algebras via quiver methods. We classify all the possible Hopf structures arising from the simplest Hopf quiver Q(〈g〉, g), which serves as a basic ingredient for the general ones. This provides the very local structure information for the general pointed Hopf algebras.

New Hopf Structures on Binary Trees

The multiplihedra $\mathcal{M}_{\bullet} = (\mathcal{M}_n)_{n \geq 1}$ form a family of polytopes originating in the study of higher categories and homotopy theory. While the multiplihedra may be unfamiliar to the algebraic combinatorics community, it is nestled between two families of polytopes that certainly are not: the permutahedra $\mathfrak{S}_{\bullet}$ and associahedra $\mathcal{Y}_{\bullet}$. The maps $\mathfrak{S}_{\bullet} \twoheadrightarrow \mathcal{M}_{\bullet} \twoheadrightarrow \mathcal{Y}_{\bullet}$ reveal several new Hopf structures on tree-like objects nestled between the Hopf algebras $\mathfrak{S}Sym$ and $\mathcal{Y}Sym$. We begin their study here, showing that $\mathcal{M}Sym$ is a module over $\mathfrak{S}Sym$ and a Hopf module over $\mathcal{Y}Sym$. An elegant description of the coinvariants for $\mathcal{M}Sym$ over $\mathcal{Y}Sym$ is uncovered via a change of basis-using Möbius inversion in posets built on the $1$-skeleta of $\mathcal{M}_{\bullet}$. Our analysis uses the notion of an $\textit{interval retract}$ that should be of independent interest in poset combinatorics. It also reveals new families of polytopes, and even a new factorization of a known projection from the associahedra to hypercubes. Les multiplièdres $\mathcal{M}_{\bullet} = (\mathcal{M}_n)_{n \geq 1}$ forment une famille de polytopes en provenant de l'étude des catégories supérieures et de la théorie de l'homotopie. Tandis que les multiplihèdres sont peu connus dans la communauté de la combinatoire algébrique, ils sont nichés entre deux familles des polytopes qui sont bien connus: les permutahèdres $\mathfrak{S}_{\bullet}$ et les associahèdres $\mathcal{Y}_{\bullet}$. Les morphismes $\mathfrak{S}_{\bullet} \twoheadrightarrow \mathcal{M}_{\bullet} \twoheadrightarrow \mathcal{Y}_{\bullet}$ dévoilent plusieurs nouvelles structures de Hopf sur les arbres binaires entre les algèbres de Hopf $\mathfrak{S}Sym$ et $\mathcal{Y}Sym$. Nous commençons son étude ici, en démontrant que $\mathcal{M}Sym$ est un module sur $\mathfrak{S}Sym$ et un module de Hopf sur $\mathcal{Y}Sym$. Une description élégante des coinvariants de $\mathcal{M}Sym$ sur $\mathcal{Y}Sym$ est trouvée par moyen d'une change de base―en utilisant une inversion de Möbius dans certains posets construits sur le $1$-squelette de $\mathcal{M}_{\bullet}$. Notre analyse utilise la notion d'$\textit{interval retract}$, qui devrait être intéressante par soi-même dans la théorie des ensembles partiellement ordonnés. Notre analyse donne lieu également à des nouvelles familles des polytopes, et même une nouvelle factorisation d'une projection connue des associahèdres aux hypercubes.

Variants of bosonization in parabosonic algebra: the Hopf and super-Hopf structures in parabosonic algebra

Parabosonic algebra in finite or infinite degrees of freedom is considered as a -graded associative algebra, and is shown to be a -graded (or super) Hopf algebra. The super-Hopf algebraic structure of the parabosonic algebra is established directly without appealing to its relation to the osp(1/2n) Lie superalgebraic structure. The notion of super-Hopf algebra is equivalently described as a Hopf algebra in the braided monoidal category . The bosonization technique for switching a Hopf algebra in the braided monoidal category (where H is a quasitriangular Hopf algebra) into an ordinary Hopf algebra is reviewed. In this paper, we prove that for the parabosonic algebra PB, beyond the application of the bosonization technique to the original super-Hopf algebra, a bosonization-like construction is also achieved using two operators, related to the parabosonic total number operator. Both techniques switch the same super-Hopf algebra PB to an ordinary Hopf algebra, thus producing two different variants of PB, with an ordinary Hopf structure.

Hopf structures on ambiskew polynomial rings

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 ( s l 2 ) , U q ( s l 2 ) and the enveloping algebra of the three-dimensional Heisenberg Lie algebra. In a torsion-free case we describe the finite-dimensional simple modules, in particular their dimensions, and prove a Clebsch–Gordan decomposition theorem for the tensor product of two simple modules. We construct a Casimir type operator and prove that any finite-dimensional weight module is semisimple.

SOME HOPF ALGEBRAS OF WORDS

A number of integral Hopf algebras have been studied that have, as their underlying modules, the free are all of this type. Some of these are known to be isomorphic or dual to each other, some are known only to be rationally isomorphic, some have been stated in the literature to be isomorphic when they are only rationally isomorphic.This paper is, in part, an attempt to find order in this chaos of word Hopf algebras. We consider three multiplications on such modules, and their dual comultiplications, and clarify which of these operations can be combined to obtain Hopf structures. We discuss when the results are isomorphic, integrally or rationally, and study the resulting structures. We are not attempting a classification of Hopf algebras of words, merely an organization of some of the Hopf algebras of this type that have been studied in the literature.

Twists in and their quantizations

The solution of the Drinfeld equation corresponding to the full set of different carrier subalgebras in sl(3) are explicitly constructed. The obtained Hopf structures are studied. It is demonstrated that the presented twist deformations can be considered as limits of the corresponding quantum analogues (q-twists) defined for the q-quantized algebras.

Geometric structures of vectorial type

We study geometric structures of W 4 -type in the sense of A. Gray on a Riemannian manifold. If the structure group G ⊂ SO ( n ) preserves a spinor or a non-degenerate differential form, its intrinsic torsion Γ is a closed 1-form (Proposition 2.1 and Theorem 2.1). Using a G -invariant spinor we prove a splitting theorem (Proposition 2.2). The latter result generalizes and unifies a recent result obtained in [S. Ivanov, M. Parton, P. Piccinni, Locally conformal parallel G 2 - and Spin ( 7 ) -structures, mathdg/0509038], where this splitting has been proved in dimensions n = 7 , 8 only. Finally we investigate geometric structures of vectorial type and admitting a characteristic connection ∇ c . An interesting class of geometric structures generalizing Hopf structures are those with a ∇ c -parallel intrinsic torsion Γ . In this case, Γ induces a Killing vector field (Proposition 4.1) and for some special structure groups it is even parallel.

Monomial Hopf algebras over fields of positive characteristic

In this paper, we study the structures of monomial Hopf algebras over a field of positive characteristic. A necessary and sufficient condition for the monomial coalgebra C d (n) to admit Hopf structures is given here, and if it is the case, all graded Hopf structures on C d (n) are completely classified. Moreover, we construct a Hopf algebras filtration on C d (n) which will help us to discuss a conjecture posed by Andruskiewitsch and Schneider. Finally combined with a theorem by Montgomery, we give the structure theorem for all monomial Hopf algebras.

Monomial Hopf algebras

Let K be a field of characteristic 0 containing all roots of unity. We classified all the Hopf structures on monomial K-coalgebras, or, in dual version, on monomial K-algebras.

Particle in the Brusselator model with flow

We consider the interaction of a small moving particle with a stationary space-periodic pattern in a chemical reaction–diffusion system with a flow. The pattern is produced by a one-dimensional Brusselator model that is perturbed by a constant displacement from the equilibrium state at the inlet. By partially blocking the flow, the particle gives rise to a local increment of the flow rate. For certain parameter values a response with intermittent Hopf and Turing type structures is observed. In other regimes a wave of substitution of missing peaks runs across the pattern.

Yangians, quantum groups and solutions of the quantum dynamical Yang-Baxter equation

We construct Drinfel’d twists that define deformed Hopf structures. In particular, we obtain deformed double Yangians and dynamical double Yangians.