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Relevant bibliographies by topics / Coalgebras, bialgebras, Hopf algebras

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Academic literature on the topic 'Coalgebras, bialgebras, Hopf algebras'

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Published: 17 September 2021

Last updated: 18 February 2022

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Journal articles on the topic "Coalgebras, bialgebras, Hopf algebras"

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Benson, David B. "Bialgebras: Some Foundations for Distributed and Concurrent Computation1." Fundamenta Informaticae 12, no. 4 (October 1, 1989): 427–86. http://dx.doi.org/10.3233/fi-1989-12402.

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A categorical bimonoid consists of a monoid and a comonoid which act homomorphically on one another. In applications bimonoids are typically called bialgebras or Hopf algebras. The definitions are given at a level suitable to computer science applications and examples are included. The elements of the theory of graded bialgebras are developed and we provide a universal recursion theorem for morphisms between algebras and between coalgebras. With this theory we explicate refinements of concurrent programs and give a treatment of traces. The concluding section characterizes all coalgebras which interact with the match algebra to form a bialgebra and characterizes all algebras which interact with the fork (diagonal) coalgebra to form a bialgebra.

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Agore, A. L. "Limits of coalgebras, bialgebras and Hopf algebras." Proceedings of the American Mathematical Society 139, no. 03 (March 1, 2011): 855. http://dx.doi.org/10.1090/s0002-9939-2010-10542-7.

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KRÄHMER, ULRICH, and LUCIA ROTHERAY. "(WEAK) INCIDENCE BIALGEBRAS OF MONOIDAL CATEGORIES." Glasgow Mathematical Journal 63, no. 1 (March 16, 2020): 139–57. http://dx.doi.org/10.1017/s0017089520000075.

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AbstractIncidence coalgebras of categories in the sense of Joni and Rota are studied, specifically cases where a monoidal product on the category turns these into (weak) bialgebras. The overlap with the theory of combinatorial Hopf algebras and that of Hopf quivers is discussed, and examples including trees, skew shapes, Milner’s bigraphs and crossed modules are considered.

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MAKHLOUF, ABDENACER, and SERGEI SILVESTROV. "HOM-ALGEBRAS AND HOM-COALGEBRAS." Journal of Algebra and Its Applications 09, no. 04 (August 2010): 553–89. http://dx.doi.org/10.1142/s0219498810004117.

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The aim of this paper is to develop the theory of Hom-coalgebras and related structures. After reviewing some key constructions and examples of quasi-deformations of Lie algebras involving twisted derivations and giving rise to the class of quasi-Lie algebras incorporating Hom–Lie algebras, we describe the notion and some properties of Hom-algebras and provide examples. We introduce Hom-coalgebra structures, leading to the notions of Hom-bialgebra and Hom–Hopf algebras, and prove some fundamental properties and give examples. Finally, we define the concept of Hom–Lie admissible Hom-coalgebra and provide their classification based on subgroups of the symmetric group.

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JANSSEN, K., and J. VERCRUYSSE. "MULTIPLIER BI- AND HOPF ALGEBRAS." Journal of Algebra and Its Applications 09, no. 02 (April 2010): 275–303. http://dx.doi.org/10.1142/s0219498810003926.

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We propose a categorical interpretation of multiplier Hopf algebras, in analogy to usual Hopf algebras and bialgebras. Since the introduction of multiplier Hopf algebras by Van Daele [Multiplier Hopf algebras, Trans. Amer. Math. Soc.342(2) (1994) 917–932] such a categorical interpretation has been missing. We show that a multiplier Hopf algebra can be understood as a coalgebra with antipode in a certain monoidal category of algebras. We show that a (possibly nonunital, idempotent, nondegenerate, k-projective) algebra over a commutative ring k is a multiplier bialgebra if and only if the category of its algebra extensions and both the categories of its left and right modules are monoidal and fit, together with the category of k-modules, into a diagram of strict monoidal forgetful functors.

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Gálvez-Carrillo, Imma, Joachim Kock, and Andrew Tonks. "Decomposition Spaces and Restriction Species." International Mathematics Research Notices 2020, no. 21 (September 12, 2018): 7558–616. http://dx.doi.org/10.1093/imrn/rny089.

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Abstract We show that Schmitt’s restriction species (such as graphs, matroids, posets, etc.) naturally induce decomposition spaces (a.k.a. unital $2$-Segal spaces), and that their associated coalgebras are an instance of the general construction of incidence coalgebras of decomposition spaces. We introduce directed restriction species that subsume Schmitt’s restriction species and also induce decomposition spaces. Whereas ordinary restriction species are presheaves on the category of finite sets and injections, directed restriction species are presheaves on the category of finite posets and convex maps. We also introduce the notion of monoidal (directed) restriction species, which induce monoidal decomposition spaces and hence bialgebras, most often Hopf algebras. Examples of this notion include rooted forests, directed graphs, posets, double posets, and many related structures. A prominent instance of a resulting incidence bialgebra is the Butcher–Connes–Kreimer Hopf algebra of rooted trees. Both ordinary and directed restriction species are shown to be examples of a construction of decomposition spaces from certain cocartesian fibrations over the category of finite ordinals that are also cartesian over convex maps. The proofs rely on some beautiful simplicial combinatorics, where the notion of convexity plays a key role. The methods developed are of independent interest as techniques for constructing decomposition spaces.

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FIGUEROA, HÉCTOR, and JOSÉ M. GRACIA-BONDÍA. "COMBINATORIAL HOPF ALGEBRAS IN QUANTUM FIELD THEORY I." Reviews in Mathematical Physics 17, no. 08 (September 2005): 881–976. http://dx.doi.org/10.1142/s0129055x05002467.

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This paper stands at the interface between combinatorial Hopf algebra theory and renormalization theory. Its plan is as follows: Sec. 1.1 is the introduction, and contains an elementary invitation to the subject as well. The rest of Sec. 1 is devoted to the basics of Hopf algebra theory and examples in ascending level of complexity. Section 2 turns around the all-important Faà di Bruno Hopf algebra. Section 2.1 contains a first, direct approach to it. Section 2.2 gives applications of the Faà di Bruno algebra to quantum field theory and Lagrange reversion. Section 2.3 rederives the related Connes–Moscovici algebras. In Sec. 3, we turn to the Connes–Kreimer Hopf algebras of Feynman graphs and, more generally, to incidence bialgebras. In Sec. 3.1, we describe the first. Then in Sec. 3.2, we give a simple derivation of (the properly combinatorial part of) Zimmermann's cancellation-free method, in its original diagrammatic form. In Sec. 3.3, general incidence algebras are introduced, and the Faà di Bruno bialgebras are described as incidence bialgebras. In Sec. 3.4, deeper lore on Rota's incidence algebras allows us to reinterpret Connes–Kreimer algebras in terms of distributive lattices. Next, the general algebraic-combinatorial proof of the cancellation-free formula for antipodes is ascertained. The structure results for commutative Hopf algebras are found in Sec. 4. An outlook section very briefly reviews the coalgebraic aspects of quantization and the Rota–Baxter map in renormalization.

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Caenepeel, S., S. Dăscălescu, G. Militaru, and F. Panaite. "Coalgebra deformations of bialgebras by Harrison cocycles, copairings of Hopf algebras and double crosscoproducts." Bulletin of the Belgian Mathematical Society - Simon Stevin 4, no. 5 (1997): 647–71. http://dx.doi.org/10.36045/bbms/1105737769.

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Ma, Tianshui, Linlin Liu, and Shaoxian Xu. "Twisted tensor biproduct monoidal Hom–Hopf algebras." Asian-European Journal of Mathematics 10, no. 01 (March 2017): 1750011. http://dx.doi.org/10.1142/s1793557117500115.

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Let [Formula: see text] be a monoidal Hom-bialgebra, [Formula: see text] a monoidal Hom-algebra and a monoidal Hom-coalgebra. Let [Formula: see text] and [Formula: see text] be two linear maps. First, we construct the [Formula: see text]-smash product monoidal Hom-algebra [Formula: see text] and [Formula: see text]-smash coproduct monoidal Hom-coalgebra [Formula: see text]. Second, the necessary and sufficient conditions for [Formula: see text] and [Formula: see text] to be a monoidal Hom-bialgebra are obtained, which generalizes the results in [8, 11]. Lastly, we give some examples and applications.

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Li, Jinqi. "Dual Quasi-Hopf Algebras and Antipodes." Algebra Colloquium 13, no. 01 (March 2006): 111–18. http://dx.doi.org/10.1142/s1005386706000137.

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Let H be a coalgebra. In this paper, we show that H is a dual quasi-bialgebra if and only if the category [Formula: see text] of comodules is a tensor category; and H is a braided dual quasi-bialgebra if and only if [Formula: see text] is a braided tensor category. If H is a braided dual quasi-Hopf algebra, it is shown that the antipode of H is inner, i.e., s2(h) = ∑ τ (h1)h2τ-1(h3).

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Dissertations / Theses on the topic "Coalgebras, bialgebras, Hopf algebras"

1

Ragozzine, Charles B. "On Hopf Algebras generated by coalgebras /." The Ohio State University, 2000. http://rave.ohiolink.edu/etdc/view?acc_num=osu1488192119262242.

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Grabowski, Jan E. "Inductive constructions for Lie bialgebras and Hopf algebras." Thesis, Queen Mary, University of London, 2006. http://qmro.qmul.ac.uk/xmlui/handle/123456789/1744.

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In recent years, two generalisations of the theory of Lie algebras have become prominent, namely the "semi-classical" theory of Lie bialgebras and the "quantum" theory of Bopf algebras, including the quantized enveloping algebras. I develop an inductive approach to the study of these objects. An important tool is a construction called double-bosonisation defined by Majid for both Lie bialgebras and Hopf algebras, inspired by the triangular decomposition of a Lie algebra into positive and negative roots and a Cartan subalgebra. We describe two specific applications. The first uses double-bosonisation to add positive and negative roots and considers the relationship between two algebras when there is an inclusion of the associated Dynkin diagrams. In this setting, which we call Lie induction, doublebosonisation realises the addition of nodes to Dynkin diagrams. We use our methods to obtain necessary conditions for such an induction to be simple, using representation theory, providing a different perspective on the classification of simple Lie algebras. We consider the corresponding scheme for quantized enveloping algebras, based on inclusions of the associated root data. We call this quantum Lie induction. We prove that we have a double-bosonisation associated to these inclusions and investigate the structure of the resulting objects, which are Hopf algebras in braided categories, that is, covariant Bopf algebras. The second application generalises one of the most important constructions in this field, namely the Drinfel'd double of a Lie bialgebra, which has dimension twice that of the underlying algebra. Our construction, the triple, has dimension three times that of the input algebra. Our main result is that when the input algebra is factorisable, this is isomorphic to the triple direct sum as an algebra and a twisting as a coalgebra. We also indicate a number of ways in which the triple is related to the double.

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Rivezzi, Andrea. "Lie bialgebras and Etingof-Kazhdan quantization." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2020. http://amslaurea.unibo.it/21784/.

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In questa tesi viene presentata la soluzione data da Pavel Etingof e David Kazhdan al problema della quantizzazione delle bialgebre di Lie, formulato da Vladimir Drinfeld nel 1992. Il problema consiste nel trovare un funtore che, data una bialgebra di Lie, costruisca una algebra di Hopf che la quantizzi. Nel primo capitolo vengono presentati gli aspetti di teoria delle categorie necessarie per la lettura. Nel secondo capitolo, introduciamo le nozioni di algebra, coalgebra, bialgebra e algebra di Hopf, con particolare attenzione alla loro teoria delle rappresentazioni. Nel terzo capitolo, presentiamo le nozioni base della teoria delle algebre di Lie, per poi definire le nozioni di coalgebra di Lie e di bialgebra di Lie. Vengono quindi definite le triple di Manin e il doppio di Drinfeld di una bialgebra di Lie. Nel quarto capitolo definiamo la nozione di quantizzazione di una bialgebra di Lie, e presentiamo i quantum groups di Drinfeld e Jimbo, che ne sono un esempio nel caso delle algebre di Kac-Moody simmetrizzabili. Infine, nel quinto ed ultimo capitolo presentiamo la costruzione della quantizzazione di Etingof e Kazhdan. Tale tecnica di quantizzazione si suddivide in diversi passi, ed è basata sulla dualità di Tannaka-Krein. In un primo momento, analizziamo il caso in cui la bialgebra di Lie è di dimensione finita. In seguito, adattiamo la costruzione del caso finito dimensionale al caso infinito dimensionale.

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Abdou, Damdji Ahmed Zahari. "Etude et Classification des algèbres Hom-associatives." Thesis, Mulhouse, 2017. http://www.theses.fr/2017MULH0158/document.

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La thèse comporte six chapitres. Dans le premier chapitre, on rappelle les bases de la théorie et on étudie la structure des algèbres Hom-associatives ainsi que les différentes constructions comme la composition avec des endomorphismes qui nous permet de construire de nouveaux objets et d’établir certaines nouvelles propriétés. Parmi les résultats originaux, on peut signaler l’étude des algèbres Hom-associatives simples ainsi que leurs constructions. On a montré que toutes les algèbres Hom-associatives multiplicatives simples s’obtiennent par composition d’algèbres simples et d’automorphismes. Dans le deuxième chapitre, on commence par étudier les propriétés des changements de base dans ces structures algébriques. On a calculé la base de Gröbner de l’idéal engendrant la variété algébrique des algèbres Hom-associatives de dimension 2 où la multiplication µ et l’application linéaire α sont identifiées à leurs constantes de structure relativement à une base donnée. La classification, à isomorphisme près, des algèbres Hom-associatives unitaires et non unitaires est établie en dimension 2 et 3. On a aussi décrit les algèbres de type associatif en se basant sur le théorème de twist de Yau. Dans le troisième chapitre, on étudie certaines propriétés et invariants comme les dérivations, αk-dérivations où k est un entier positif. Dans le quatrième chapitre, on établit la cohomologie de ces algèbres. On a pu lister les algèbres rigides grâce à leur classe de cohomologie puis on s'est 'intéressé aux déformations infinitésimales et dégénérations. D’une part, la cohomologie et déformation de ces algèbres nous a permis d’identifier les algèbres rigides dont le deuxième groupe de cohomologie est nulle, et d’autre part de caractérisation de composante irréductible. Dans le cinquième chapitre, on s’intéresse aux structures Rota-Baxter de poids λ ϵK de ces algèbres. Enfin, dans le dernier chapitre, on a travaillé sur les structures Hom-bialgèbres et leurs invariants
The purpose of this thesis is to study the structure of Hom-associative algebras and provide classifications. Among the results obtained in this thesis, we provide 2-dimensional and 3-dimensional Hom-associative algebras and give a characterization of multiplicative simple Hom-associative algebras. Moreover we compute some invariants and discuss irreducible components of the corresponding algebraic varieties. The thesis is organized as follows. In the first chapter we give the basics about Hom-associative algebras and provide some new properties. Moreover, we discuss unital Hom-associative algebras. Chapter 2 deals with simple multiplicative Hom-associative algebras. We present one of the main results of this paper, that is a characterization of simple multiplicative Hom-associative algebras. Indeed, we show that they are all obtained by twistings of simple associative algebras. Chapter 3 is dedicated to describe algebraic varieties of Hom-associative algebras and provide classifications, up to isomorphism, of 2-dimensional and 3-dimensional Hom-associative algebras. In chapter 4, we compute their derivations and twisted derivations, whereas in chapter 5, we compute their Hom-Type Hochschild cohomology. In the last section of this chapter, we consider the geometric classification problem using one-parameter formel deformations, and describe the irreducible components. In chapter 6, we compute Rota-Baxter structures of weight k of Hom-associative algebras appearing in our classification. In chapter 7, We work out Hom-bialgebras structures as well as their invariants. Properties and classifications, as well as the calculation of certain invariants such as the first and second cohomology groups, were studied

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Vieira, Larissa Hagedorn. "PARES ADMISSÍVEIS, SISTEMAS ADMISSÍVEIS E BIÁLGEBRAS NA CATEGORIA DOS MÓDULOS DE YETTER-DRINFELD." Universidade Federal de Santa Maria, 2014. http://repositorio.ufsm.br/handle/1/9989.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
The purpose of this work is to study the relationships between admissible pairs, systems admissible and bialgebras in the category of Yetter-Drinfeld modules, as well as some properties of the Hopf algebra associated (via bosonization) to an admissible pair. We end this dissertation with a family of examples of admissible pairs.
O objetivo deste trabalho é estudar as relações entre pares admissíveis, sistemas admissíveis e biálgebras na categoria dos módulos de Yetter-Drinfeld, bem como algumas propriedades da álgebra de Hopf associada (via bosonização) a um par admissível. Finalizamos esta dissertação com uma família de exemplos de pares admissíveis.

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Fantino, Fernando Amado. "Algebras de Hopf punteadas sobre grupos no abelianos /." Doctoral thesis, 2008. http://hdl.handle.net/11086/117.

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Tesis (Doctor en Matemática)--Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física, 2008.
Sea G un grupo finito no abeliano. Esta tesis trata acerca del problema de clasificación de las álgebras de Hopf punteadas complejas de dimensión finita H con grupo de elementos de tipo grupo G(H) isomorfo a G. Se analizan criterios que permiten dar condiciones suficientes para que el álgebra de Nichols B(O,f) tenga dimensión infinita estudiando subracks de O, donde O es una clase de conjugación de G y f es una representación irreducible de G^s, el centralizador de un elemento fijo s en O.
Fernando Amado Fantino ; director Nicolás Andruskiewitsch.

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Rossi, Bertone Fiorela. "Álgebras cuánticas de potencias divididas." Doctoral thesis, 2016. http://hdl.handle.net/11086/3943.

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Tesis (Doctor en Matemática)--Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía, Física y Computación, 2016.
Se definen las álgebras de Lusztig y sus dobles de Drinfeld, las cuánticas de potencias divididas, asociadas a álgebras de Nichols de tipo diagonal de dimensión finita. En ambos casos se prueba una presentación por generadores y relaciones y algunas propiedades básicas. Además, para trenzas de rango 2 y trenzas (súper) de tipo A, se asocia un álgebra de Lie semisimple tal que un cociente del álgebra de Lusztig es isomorfo al álgebra universal de la parte positiva de dicho álgebra de Lie. Por otro lado, se prueban resultados conocidos sobre álgebras de Hopf co-Frobenius en el contexto trenzado.
We define the so called Lusztig algebras and their Drinfeld doubles, the quantum divided powers algebras, associated to finite dimensional Nichols algebras of diagonal type. We present them by generators and relations and prove some basic properties. Also, for braidings of rank 2 and braidings of super type A, we associate a semisimple Lie algebra such that there is a quotient of the Lusztig algebra which is isomorphic to the universal algebra of the positive part of this Lie algebra. On the other hand, we prove versions of known results about co-Frobenius Hopf algebras for braided Hopf algebras.

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Ochoa, Arango Jesús Alonso. "Grupoides y algebroides dobles de Lie /." Doctoral thesis, 2010. http://hdl.handle.net/11086/144.

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Tesis (Doctor en Matemática)--Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física, 2010.
En este trabajo demostramos que todo grupoide doble de Lie con acción medular propia esta completamente determinado por una factorización de un cierto grupoide de Lie diagonal canónicamente definido. Tambien, estudiamos la versión infinitesimal de este concepto, la de algebroide doble de Lie y como resultado introducimos una nueva clase de ejemplos construidos a partir de ciertos diagramas de álgebras de Lie. En la parte final, proponemos los conceptos de biálgebra infinitesimál de multiplicadores y de bialgebra de Lie de derivadores. Presentamos algunos ejemplos y como resultado principal demostramos, bajo ciertas condiciones, como obtener a partir de una biálgebra infinitesimál de multiplicadores una biálgebra de Lie de derivadores.
Jesús Alonso Ochoa Arango.

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Parker, Darren B. "Hopf Galois Extensions and forms of coalgebras and Hopf algebras." 1998. http://catalog.hathitrust.org/api/volumes/oclc/40734586.html.

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Thesis (Ph. D.)--University of Wisconsin--Madison, 1998.
Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 105-107).

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Rotheray, Lucia Alessandra. "Incidence Bialgebras of Monoidal Categories." 2020. https://tud.qucosa.de/id/qucosa%3A74695.

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Incidence coalgebras of categories as defined by Joni and Rota are studied, specifically in cases where a strict monoidal product on the underlying category turns the incidence coalgebra into a bialgebra or weak bialgebra. Examples of these incidence bialgebras in combinatorics are given, and include rooted trees and forests, skew shapes and bigraphs. The relations between incidence bialgebras of monoidal categories, incidence bialgebras of operads and posets, combinatorial Hopf algebras and the quiver Hopf algebras of Cibils and Rosso are discussed. Building on a result of Bergbauer and Kreimer, incidence bialgebras are seen as a useful setting in which to study aspects of combinatorial Dyson-Schwinger equations. The possibility of defining a grafting operator B+ and combinatorial DysonSchwinger equations for general incidence bialgebras is explored through the example of skew shapes.:1. Introduction 2. Background 3. Incidence bialgebras of monoidal categories and multicategories 4. Combinatorial Dyson-Schwinger equations

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Books on the topic "Coalgebras, bialgebras, Hopf algebras"

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Loday, Jean-Louis. Generalized bialgebras and triples of operads. Paris, France: Société mathématique de France, 2008.

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Generalized bialgebras and triples of operads. Paris, France: Société mathématique de France, 2008.

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Loday, Jean-Louis. Generalized bialgebras and triples of operads. Paris, France: Société mathématique de France, 2008.

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Conference on Hopf Algebras and Tensor Categories (2011 University of Almeria). Hopf algebras and tensor categories: International conference, July 4-8, 2011, University of Almería, Almería, Spain. Edited by Andruskiewitsch Nicolás 1958-, Cuadra Juan 1975-, and Torrecillas B. (Blas) 1958-. Providence, Rhode Island: American Mathematical Society, 2013.

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Book chapters on the topic "Coalgebras, bialgebras, Hopf algebras"

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Manin, Yuri I. "Bialgebras and Hopf Algebras." In Quantum Groups and Noncommutative Geometry, 11–17. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-97987-8_3.

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Brown, Ken A., and Ken R. Goodearl. "Bialgebras and Hopf Algebras." In Lectures on Algebraic Quantum Groups, 81–91. Basel: Birkhäuser Basel, 2002. http://dx.doi.org/10.1007/978-3-0348-8205-7_9.

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Lambe, Larry A., and David E. Radford. "Quasitriangular Algebras, Bialgebras, Hopf Algebras and The Quantum Double." In Introduction to the Quantum Yang-Baxter Equation and Quantum Groups: An Algebraic Approach, 161–95. Boston, MA: Springer US, 1997. http://dx.doi.org/10.1007/978-1-4615-4109-7_6.

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Hazewinkel, Michiel, Nadiya Gubareni, and V. Kirichenko. "Bialgebras and Hopf algebras. Motivation, definitions, and examples." In Mathematical Surveys and Monographs, 131–73. Providence, Rhode Island: American Mathematical Society, 2010. http://dx.doi.org/10.1090/surv/168/03.

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"Coalgebras, bialgebras, and Hopf algebras." In A Tour of Representation Theory, 427–63. Providence, Rhode Island: American Mathematical Society, 2018. http://dx.doi.org/10.1090/gsm/193/09.

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"Bialgebras." In Hopf Algebras, 165–202. WORLD SCIENTIFIC, 2011. http://dx.doi.org/10.1142/9789814338660_0005.

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"Coalgebras." In Hopf Algebras, 19–75. WORLD SCIENTIFIC, 2011. http://dx.doi.org/10.1142/9789814338660_0002.

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"Representations of coalgebras." In Hopf Algebras, 77–122. WORLD SCIENTIFIC, 2011. http://dx.doi.org/10.1142/9789814338660_0003.

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"Algebras and coalgebras." In Hopf Algebra, 19–82. CRC Press, 2000. http://dx.doi.org/10.1201/9781482270747-6.

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"Quasitriangular bialgebras and Hopf algebras." In Hopf Algebras, 387–411. WORLD SCIENTIFIC, 2011. http://dx.doi.org/10.1142/9789814338660_0012.

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Conference papers on the topic "Coalgebras, bialgebras, Hopf algebras"

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Wakui, Michihisa. "The coribbon structures of some finite dimensional braided Hopf algebras generated by 2×2-matrix coalgebras." In Noncommutative Geometry and Quantum Groups. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2003. http://dx.doi.org/10.4064/bc61-0-20.

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