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

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Published: 17 September 2021

Last updated: 18 February 2022

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1

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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2

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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3

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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4

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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5

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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6

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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7

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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8

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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9

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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10

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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11

BULACU, DANIEL, and STEFAAN CAENEPEEL. "A MONOIDAL STRUCTURE ON THE CATEGORY OF RELATIVE HOPF MODULES." Journal of Algebra and Its Applications 11, no. 02 (April 2012): 1250026. http://dx.doi.org/10.1142/s0219498811005506.

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Let B be a bialgebra, and A be a left B-comodule algebra in a braided monoidal category [Formula: see text], and assume that A is also a coalgebra, with a not-necessarily associative or unital left B-action. Then we can define a right A-action on the tensor product of two relative Hopf modules, and this defines a monoidal structure on the category of relative Hopf modules if and only if A is a bialgebra in the category of left Yetter–Drinfeld modules over B. Some examples are given.

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12

Cristian Iovanov, Miodrag. "Generalized Frobenius Algebras and Hopf Algebras." Canadian Journal of Mathematics 66, no. 1 (February 2014): 205–40. http://dx.doi.org/10.4153/cjm-2012-060-7.

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Abstract“Co-Frobenius” coalgebras were introduced as dualizations of Frobenius algebras. We previously showed that they admit left-right symmetric characterizations analogous to those of Frobenius algebras. We consider the more general quasi-co-Frobenius (QcF) coalgebras. The first main result in this paper is that these also admit symmetric characterizations: a coalgebra is QcF if it is weakly isomorphic to its (left, or right) rational dual Rat(C*) in the sense that certain coproduct or product powers of these objects are isomorphic. Fundamental results of Hopf algebras, such as the equivalent characterizations of Hopf algebras with nonzero integrals as left (or right) co-Frobenius, QcF, semiperfect or with nonzero rational dual, as well as the uniqueness of integrals and a short proof of the bijectivity of the antipode for such Hopf algebras all follow as a consequence of these results. This gives a purely representation theoretic approach to many of the basic fundamental results in the theory of Hopf algebras. Furthermore, we introduce a general concept of Frobenius algebra, which makes sense for infinite dimensional and for topological algebras, and specializes to the classical notion in the finite case. This will be a topological algebra A that is isomorphic to its complete topological dual Aν. We show that A is a (quasi)Frobenius algebra if and only if A is the dual C* of a (quasi)co-Frobenius coalgebra C. We give many examples of co-Frobenius coalgebras and Hopf algebras connected to category theory, homological algebra and the newer q-homological algebra, topology or graph theory, showing the importance of the concept.

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13

CROSSLEY, MARTIN. "GROUP BIALGEBRAS AND PERMUTATION BIALGEBRAS." Glasgow Mathematical Journal 55, no. 3 (February 25, 2013): 639–43. http://dx.doi.org/10.1017/s0017089512000808.

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AbstractMalvenuto and Reutenauer (C. Malvenuto and C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra177 (1995), 967–982) showed how the total symmetric group ring ⊕nZΣn could be made into a Hopf algebra with a very nice structure which admitted the Solomon descent algebra as a sub-Hopf algebra. To do this they replaced the group multiplication by a convolution product, thus distancing their structure from the group structure of Σn. In this paper we examine what is possible if we keep to the group multiplication, and we also consider the question for more general families of groups. We show that a Hopf algebra structure is not possible, but cocommutative and non-cocommutative counital bialgebras can be obtained, arising from certain diagrams of group homomorphisms. In the case of the symmetric groups we note that all such structures are weak in the sense that the dual algebras have many zero-divisors, but structures which respect descent sums can be found.

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14

Rangipour, Bahram. "Cyclic Cohomology of Corings." Journal of K-Theory 4, no. 1 (November 14, 2008): 193–207. http://dx.doi.org/10.1017/is008007024jkt060.

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AbstractWe define cyclic cohomology of corings over not necessarily commutative algebras. Our method is a generalization of Hopf-cyclic cohomology obtained by replacing coalgebras and Hopf algebras with corings and para-Hopf algebroids, respectively. We also study the dual of this theory whose cyclic cohomology, in contrast with the case of algebras and coalgebras, is not trivial.

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15

Caenepeel, S., S. D[acaron]scalescu, and Ş. Raianu. "Cosemisimple hopf algebras coacting on coalgebras." Communications in Algebra 24, no. 5 (January 1996): 1649–77. http://dx.doi.org/10.1080/00927879608825661.

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16

Parker, Darren B. "Forms of Coalgebras and Hopf Algebras." Journal of Algebra 239, no. 1 (May 2001): 1–34. http://dx.doi.org/10.1006/jabr.2000.8678.

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17

Masuoka, Akira. "Extensions of Hopf Algebras and Lie Bialgebras." Transactions of the American Mathematical Society 352, no. 8 (March 24, 2000): 3837–79. http://dx.doi.org/10.1090/s0002-9947-00-02394-1.

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18

HASEGAWA, MASAHITO. "A quantum double construction in Rel." Mathematical Structures in Computer Science 22, no. 4 (May 18, 2012): 618–50. http://dx.doi.org/10.1017/s0960129511000703.

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We study bialgebras and Hopf algebras in the compact closed categoryRelof sets and binary relations. Various monoidal categories with extra structure arise as the categories of (co)modules of bialgebras and Hopf algebras inRel. In particular, for any groupG, we derive a ribbon category of crossedG-sets as the category of modules of a Hopf algebra inRelthat is obtained by the quantum double construction. This category of crossedG-sets serves as a model of the braided variant of propositional linear logic.

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19

Van Oystaeyen, Fred, and Yinhuo Zhang. "Finite-dimensional Hopf algebras coacting on coalgebras." Bulletin of the Belgian Mathematical Society - Simon Stevin 5, no. 1 (1998): 1–14. http://dx.doi.org/10.36045/bbms/1103408962.

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20

Agore, A. L. "Free Poisson Hopf algebras generated by coalgebras." Journal of Mathematical Physics 55, no. 8 (August 2014): 083502. http://dx.doi.org/10.1063/1.4889936.

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21

Kochetov, M. "On identities for coalgebras and hopf algebras." Communications in Algebra 28, no. 3 (January 2000): 1211–21. http://dx.doi.org/10.1080/00927870008826890.

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22

Akbarpour, R., and M. Khalkhali. "Cyclic Homology of Hopf Comodule Algebras and Hopf Module Coalgebras." Communications in Algebra 31, no. 11 (January 10, 2003): 5653–71. http://dx.doi.org/10.1081/agb-120023979.

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23

Schmitt, William R. "Hopf Algebras of Combinatorial Structures." Canadian Journal of Mathematics 45, no. 2 (April 1, 1993): 412–28. http://dx.doi.org/10.4153/cjm-1993-021-5.

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AbstractA generalization of the definition of combinatorial species is given by considering functors whose domains are categories of finite sets, with various classes of relations as moronisms. Two cases in particular correspond to species for which one has notions of restriction and quotient of structures. Coalgebras and/or Hopf algebras can be associated to such species, the duals of which provide an algebraic framework for studying invariants of structures.

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24

Cao, Haijun. "S-Hopf Quivers and Semilattice Graded Weak Hopf Algebras." Algebra Colloquium 20, no. 01 (January 16, 2013): 109–22. http://dx.doi.org/10.1142/s1005386713000102.

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The main work of this article is to give a construction of semilattice graded weak Hopf algebras via S-Hopf quivers, mainly based on the work of Cibils, Rosso, and Montgomery. This provides a class of non-commutative and non-cocommutative pointed semilattice graded weak Hopf algebras: they have the natural bases consisting of paths and the underlying coalgebra structures are path coalgebras. It also provides a new way of verifying old results and testing new ideas on pointed Hopf algebras, such as the decomposition of pointed semilattice graded weak Hopf algebras according to the result of Montgomery.

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25

Song, Guang’ai, and Xiaoqing Yue. "Dual Lie Bialgebra Structures of Twisted Schrödinger-Virasoro Type." Algebra Colloquium 25, no. 04 (December 2018): 627–52. http://dx.doi.org/10.1142/s1005386718000445.

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In this paper, the structures of dual Lie bialgebras of twisted Schrödinger-Virasoro type are investigated. By studying the maximal good subspaces, we determine the dual Lie coalgebras of the twisted Schrödinger-Virasoro algebras. Then based on this, we construct the dual Lie bialgebra structures of this type. As by-products, four new infinite dimensional Lie algebras are obtained.

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26

Laugwitz, Robert. "Comodule algebras and 2-cocycles over the (Braided) Drinfeld double." Communications in Contemporary Mathematics 21, no. 04 (May 31, 2019): 1850045. http://dx.doi.org/10.1142/s0219199718500451.

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We show that for dually paired bialgebras, every comodule algebra over one of the paired bialgebras gives a comodule algebra over their Drinfeld double via a crossed product construction. These constructions generalize to working with bialgebra objects in a braided monoidal category of modules over a quasitriangular Hopf algebra. Hence two ways to provide comodule algebras over the braided Drinfeld double (the double bosonization) are provided. Furthermore, a map of second Hopf algebra cohomology spaces is constructed. It takes a pair of 2-cocycles over dually paired Hopf algebras and produces a 2-cocycle over their Drinfeld double. This construction also has an analogue for braided Drinfeld doubles.

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27

MANCHON, DOMINIQUE. "ON BIALGEBRAS AND HOPF ALGEBRAS OF ORIENTED GRAPHS." Confluentes Mathematici 04, no. 01 (March 2012): 1240003. http://dx.doi.org/10.1142/s1793744212400038.

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28

Yanagihara, Hiroshi. "On homomorphisms of cocommutative coalgebras and Hopf algebras." Hiroshima Mathematical Journal 17, no. 2 (1987): 433–46. http://dx.doi.org/10.32917/hmj/1206130079.

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29

Montgomery, Susan. "Indecomposable coalgebras, simple comodules, and pointed Hopf algebras." Proceedings of the American Mathematical Society 123, no. 8 (August 1, 1995): 2343. http://dx.doi.org/10.1090/s0002-9939-1995-1257119-3.

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30

Block, Richard E. "Commutative Hopf algebras, lie coalgebras, and divided powers." Journal of Algebra 96, no. 1 (September 1985): 275–306. http://dx.doi.org/10.1016/0021-8693(85)90050-x.

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31

García-Martínez, Xabier, and Tim Van der Linden. "A note on split extensions of bialgebras." Forum Mathematicum 30, no. 5 (September 1, 2018): 1089–95. http://dx.doi.org/10.1515/forum-2017-0016.

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AbstractWe prove a universal characterization of Hopf algebras among cocommutative bialgebras over an algebraically closed field: a cocommutative bialgebra is a Hopf algebra precisely when every split extension over it admits a join decomposition. We also explain why this result cannot be extended to a non-cocommutative setting.

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32

Leroux, Philippe. "Construction of Nijenhuis operators and dendriform trialgebras." International Journal of Mathematics and Mathematical Sciences 2004, no. 49 (2004): 2595–615. http://dx.doi.org/10.1155/s0161171204402117.

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We construct Nijenhuis operators from particular bialgebras called dendriform-Nijenhuis bialgebras. It turns out that such Nijenhuis operators commute withTD-operators, a kind of Baxter-Rota operators, and are therefore closely related to dendriform trialgebras. This allows the construction of associative algebras, called dendriform-Nijenhuis algebras, made out of nine operations and presenting an exotic combinatorial property. We also show that the augmented free dendriform-Nijenhuis algebra and its commutative version have a structure of connected Hopf algebras. Examples are given.

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33

Jia, Ling. "The structure theorem and duality theorem for endomorphism algebras of weak Hopf group coalgebras." Journal of Algebra and Its Applications 16, no. 11 (October 4, 2017): 1750208. http://dx.doi.org/10.1142/s0219498817502085.

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In this paper, we investigate the HOM-functor and state the structure theorem for endomorphism algebras of weak two-sided [Formula: see text]-Hopf [Formula: see text]-modules in order to explore homological algebras for weak Hopf [Formula: see text]-modules, and present the duality theorem for weak group “big” Smash products which extends the result of Menini and Raianu [Morphisms of relative Hopf modules, Smash products and duality, J. Algebra 219 (1999) 547–570] in the setting of weak Hopf group coalgebras.

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34

FOISSY, LOÏC, and FRÉDÉRIC PATRAS. "NATURAL ENDOMORPHISMS OF SHUFFLE ALGEBRAS." International Journal of Algebra and Computation 23, no. 04 (June 2013): 989–1009. http://dx.doi.org/10.1142/s0218196713400183.

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We show that there exist two natural endomorphism algebras for shuffle bialgebras such as Sh (X), where X is a graded set. One of these endomorphism algebras is a natural extension of the Malvenuto–Reutenauer Hopf algebra and is defined using graded permutations. The other one, the dendriform descent algebra, is a subalgebra of the first defined by mimicking the definition of the descent algebras by convolution from the graded projections in the tensor algebra. We study these algebras for their own, show that they carry bidendriform structures and establish freeness properties, study their generators, dimensions, bases, and also feature their relations to the internal structure of shuffle algebras. As an application of these ideas, we give a new proof of Chapoton's rigidity theorem for shuffle bialgebras.

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35

Chen, Quan-Guo, and Ding-Guo Wang. "Partial group (co)actions of Hopf group coalgebras." Filomat 28, no. 1 (2014): 131–51. http://dx.doi.org/10.2298/fil1401131c.

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We will develop partial group (co)actions of a Hopf group coalgebra on a family of algebras by introducing partial group entwining structure. Then we give necessary and sufficient conditions for a family of functors from the category of partial group entwining modules to the category of modules over a suitable algebra to be separable. Also, the applications of our results are considered.

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IOVANOV, MIODRAG CRISTIAN. "ABSTRACT ALGEBRAIC INTEGRALS AND FROBENIUS CATEGORIES." International Journal of Mathematics 24, no. 10 (September 2013): 1350081. http://dx.doi.org/10.1142/s0129167x1350081x.

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We generalize results on the connection between existence and uniqueness of integrals and representation theoretic properties for Hopf algebras and compact groups. For this, given a coalgebra C, we study analogues of the existence and uniqueness properties for the integral functor Hom C(C, -), which generalizes the notion of integral in a Hopf algebra. We show that the coalgebra C is co-Frobenius if and only if dim ( Hom C(C, M)) = dim (M) for all finite dimensional right (left) comodules M. As applications, we give a few new categorical characterizations of co-Frobenius, quasi-co-Frobenius (QcF) coalgebras and semiperfect coalgebras, and re-derive classical results of Lin, Larson, Sweedler and Sullivan on Hopf algebras. We show that a coalgebra is QcF if and only if the category of left (right) comodules is Frobenius, generalizing results from finite dimensional algebras, and we show that a one-sided QcF coalgebra is two-sided semiperfect. We also construct a class of examples derived from quiver coalgebras to show that the results of the paper are the best possible. Finally, we examine the case of compact groups, and note that algebraic integrals can be interpreted as certain skew-invariant measure theoretic integrals on the group.

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You, Miman, Nan Zhou, and Shuanhong Wang. "Hom-Hopf group coalgebras and braided T-categories obtained from Hom-Hopf algebras." Journal of Mathematical Physics 56, no. 11 (November 2015): 112302. http://dx.doi.org/10.1063/1.4935527.

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38

Böhm, Gabriella. "Comodules over weak multiplier bialgebras." International Journal of Mathematics 25, no. 05 (May 2014): 1450037. http://dx.doi.org/10.1142/s0129167x14500372.

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This is a sequel paper of [Weak multiplier bialgebras, Trans. Amer. Math. Soc., in press] in which we study the comodules over a regular weak multiplier bialgebra over a field, with a full comultiplication. Replacing the usual notion of coassociative coaction over a (weak) bialgebra, a comodule is defined via a pair of compatible linear maps. Both the total algebra and the base (co)algebra of a regular weak multiplier bialgebra with a full comultiplication are shown to carry comodule structures. Kahng and Van Daele's integrals [The Larson–Sweedler theorem for weak multiplier Hopf algebras, in preparation] are interpreted as comodule maps from the total to the base algebra. Generalizing the counitality of a comodule to the multiplier setting, we consider the particular class of so-called full comodules. They are shown to carry bi(co)module structures over the base (co)algebra and constitute a monoidal category via the (co)module tensor product over the base (co)algebra. If a regular weak multiplier bialgebra with a full comultiplication possesses an antipode, then finite-dimensional full comodules are shown to possess duals in the monoidal category of full comodules. Hopf modules are introduced over regular weak multiplier bialgebras with a full comultiplication. Whenever there is an antipode, the Fundamental Theorem of Hopf Modules is proven. It asserts that the category of Hopf modules is equivalent to the category of firm modules over the base algebra.

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39

MASUOKA, AKIRA, and MAKOTO YANAGAWA. "×R-BIALGEBRAS ASSOCIATED WITH ITERATIVE q-DIFFERENCE RINGS." International Journal of Mathematics 24, no. 04 (April 2013): 1350030. http://dx.doi.org/10.1142/s0129167x13500304.

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Realizing the possibility suggested by Hardouin [Iterative q-difference Galois theory, J. Reine Angew. Math.644 (2010) 101–144], we show that her own Picard–Vessiot (PV) theory for iterative q-difference rings is covered by the (consequently, more general) framework, settled by Amano and Masuoka [Picard–Vessiot extensions of artinian simple module algebras, J. Algebra285 (2005) 743–767], of artinian simple module algebras over a cocommutative pointed Hopf algebra. An essential point is to represent iterative q-difference modules over an iterative q-difference ring R, by modules over a certain cocommutative ×R-bialgebra. Recall that the notion of ×R-bialgebras was defined by Sweedler [Groups of simple algebras, Publ. Math. Inst. Hautes Études Sci.44 (1974) 79–189], as a generalization of bialgebras.

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40

Beggs, E. J., and E. J. Taft. "Products of Linear Maps on Bialgebras, with Applications to Left Hopf Algebras and Hopf Algebroids." Communications in Algebra 34, no. 10 (October 2006): 3511–23. http://dx.doi.org/10.1080/00927870600796102.

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41

Nikonov, I., and G. Sharygin. "Pairings in Hopf cyclic cohomology of algebras and coalgebras with coefficients." Journal of K-Theory 5, no. 2 (March 15, 2010): 289–348. http://dx.doi.org/10.1017/is010003006jkt076.

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AbstractThis paper is concerned with the theory of cup products in the Hopf cyclic cohomology of algebras and coalgebras. We show that the cyclic cohomology of a coalgebra can be obtained from a construction involving the noncommutative Weil algebra. Then we introduce the notion of higher -twisted traces and use a generalization of the Quillen and Crainic constructions (see [14] and [3]) to define the cup product. We discuss the relation of the cup product above and S-operations on cyclic cohomology. We show that the product we define can be realized as a combination of the composition product in bivariant cyclic cohomology and a map from the cyclic cohomology of coalgebras to bivariant cohomology. In the last section, we briefly discuss the relation of our constructions with that in [9]. More precisely, we propose still another construction of such pairings which can be regarded as an intermediate step between the “Crainic” pairing and that of [9]. We show that it coincides with what in [9] and as far its relation to Crainic's construction is concerned, we reduce the question to a discussuion of a certain map in cohomology (see the question at the end of section 5).The results of the current paper were announced in [12].

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42

Carlier, Louis, and Joachim Kock. "Antipodes of monoidal decomposition spaces." Communications in Contemporary Mathematics 22, no. 02 (December 5, 2018): 1850081. http://dx.doi.org/10.1142/s0219199718500815.

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We introduce a notion of antipode for monoidal (complete) decomposition spaces, inducing a notion of weak antipode for their incidence bialgebras. In the connected case, this recovers the usual notion of antipode in Hopf algebras. In the non-connected case, it expresses an inversion principle of more limited scope, but still sufficient to compute the Möbius function as [Formula: see text], just as in Hopf algebras. At the level of decomposition spaces, the weak antipode takes the form of a formal difference of linear endofunctors [Formula: see text], and it is a refinement of the general Möbius inversion construction of Gálvez–Kock–Tonks, but exploiting the monoidal structure.

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43

Phùng, Hô Hai. "On an injectivity lemma in the proof of Tannakian duality." Journal of Algebra and Its Applications 15, no. 09 (August 22, 2016): 1650167. http://dx.doi.org/10.1142/s021949881650167x.

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In this short work we give a very short and elementary proof of the injectivity lemma, which plays an important role in the Tannakian duality for Hopf algebras over a field. Based on this we provide some generalizations of this fact to the case of flat coalgebras over a Noetherian domain.

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TJIN, T. "INTRODUCTION TO QUANTIZED LIE GROUPS AND ALGEBRAS." International Journal of Modern Physics A 07, no. 25 (October 10, 1992): 6175–213. http://dx.doi.org/10.1142/s0217751x92002805.

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We give a self-contained introduction to the theory of quantum groups according to Drinfeld, highlighting the formal aspects as well as the applications to the Yang-Baxter equation and representation theory. Introductions to Hopf algebras, Poisson structures and deformation quantization are also provided. After defining Poisson Lie groups we study their relation to Lie bialgebras and the classical Yang-Baxter equation. Then we explain in detail the concept of quantization for them. As an example the quantization of sl2 is explicitly carried out. Next we show how quantum groups are related to the Yang-Baxter equation and how they can be used to solve it. Using the quantum double construction we explicitly construct the universal R matrix for the quantum sl2 algebra. In the last section we deduce all finite-dimensional irreducible representations for q a root of unity. We also give their tensor product decomposition (fusion rules), which is relevant to conformal field theory.

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45

"Polyadic Hopf Algebras and Quantum Groups." 2, no. 2 (2021). http://dx.doi.org/10.26565/2312-4334-2021-2-01.

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This article continues the study of concrete algebra-like structures in our polyadic approach, where the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions (“arity freedom principle”). In this way, generalized associative algebras, coassociative coalgebras, bialgebras and Hopf algebras are defined and investigated. They have many unusual features in comparison with the binary case. For instance, both the algebra and its underlying field can be zeroless and nonunital, the existence of the unit and counit is not obligatory, and the dimension of the algebra is not arbitrary, but “quantized”. The polyadic convolution product and bialgebra can be defined, and when the algebra and coalgebra have unequal arities, the polyadic version of the antipode, the querantipode, has different properties. As a possible application to quantum group theory, we introduce the polyadic version of braidings, almost co-commutativity, quasitriangularity and the equations for the R-matrix (which can be treated as a polyadic analog of the Yang-Baxter equation). We propose another concept of deformation which is governed not by the twist map, but by the medial map, where only the latter is unique in the polyadic case. We present the corresponding braidings, almost co-mediality and M-matrix, for which the compatibility equations are found.

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Fontes, Eneilson, Grasiela Martini, and Graziela Fonseca. "Partial actions of weak Hopf algebras on coalgebras." Journal of Algebra and Its Applications, October 24, 2020, 2250012. http://dx.doi.org/10.1142/s0219498822500128.

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In this work, the notions of a partial action of a weak Hopf algebra on a coalgebra and of a partial action of a groupoid on a coalgebra will be introduced, together with some important properties. An equivalence between these notions will be presented. Finally, a dual relation between the structures of a partial action on a coalgebra and of a partial action on an algebra will be established, as well as a globalization theorem for partial module coalgebras will be presented.

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Krähmer, Ulrich, and Friedrich Wagemann. "Racks, Leibniz algebras and Yetter–Drinfel'd modules." Georgian Mathematical Journal 22, no. 4 (January 1, 2015). http://dx.doi.org/10.1515/gmj-2015-0049.

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AbstractA Hopf algebra object in Loday and Pirashvili's category of linear maps entails an ordinary Hopf algebra and a Yetter–Drinfel'd module. We equip the latter with a structure of a braided Leibniz algebra. This provides a unified framework for examples of racks in the category of coalgebras discussed recently by Carter, Crans, Elhamdadi and Saito.

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Marberg, Eric. "Linear Compactness and Combinatorial Bialgebras." Electronic Journal of Combinatorics 28, no. 3 (July 2, 2021). http://dx.doi.org/10.37236/9459.

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We present an expository overview of the monoidal structures in the category of linearly compact vector spaces. Bimonoids in this category are the natural duals of infinite-dimensional bialgebras. We classify the relations on words whose equivalence classes generate linearly compact bialgebras under shifted shuffling and deconcatenation. We also extend some of the theory of combinatorial Hopf algebras to bialgebras that are not connected or of finite graded dimension. Finally, we discuss several examples of quasi-symmetric functions, not necessarily of bounded degree, that may be constructed via terminal properties of combinatorial bialgebras.

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Makhlouf A, Chebel Z. "Kaplansky's Type Constructions for Weak Bialgebras and Weak Hopf Algebras." Journal of Generalized Lie Theory and Applications s1 (2015). http://dx.doi.org/10.4172/1736-4337.s1-008.

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Azevedo, Danielle, Grasiela Martini, Antonio Paques, and Leonardo Silva. "Hopf algebras arising from partial (co)actions." Journal of Algebra and Its Applications, September 24, 2020, 2140006. http://dx.doi.org/10.1142/s0219498821400065.

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In this paper, extending the idea presented by Takeuchi in [M. Takeuchi, Matched pairs of groups and bismash products of Hopf algebras, Comm. Algebra 9 (1981) 841–882.] and more generally by Majid in [S. Majid, Physics for algebraists: Non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction, J. Algebra 130(1) (1990) 17–64.], we introduce the notion of partial matched pair [Formula: see text] involving the concepts of partial action and partial coaction between two bialgebras [Formula: see text] and [Formula: see text]. Furthermore, we present sufficient conditions for the corresponding bismash product [Formula: see text] to generate a new Hopf algebra and, as illustration, a family of examples is provided. Moreover, under some hypotheses such sufficient conditions are also necessary conditions.

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