Quasitriangular Hom-Hopf algebras

Yuanyuan Chen; Zhongwei Wang; Liangyun Zhang

Displaying similar documents to “Quasitriangular Hom-Hopf algebras”

Classifying bicrossed products of two Sweedler's Hopf algebras

Costel-Gabriel Bontea (2014)

Czechoslovak Mathematical Journal

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We continue the study started recently by Agore, Bontea and Militaru in “Classifying bicrossed products of Hopf algebras” (2014), by describing and classifying all Hopf algebras $E$ that factorize through two Sweedler’s Hopf algebras. Equivalently, we classify all bicrossed products $H_{4} ⋈ H_{4}$ . There are three steps in our approach. First, we explicitly describe the set of all matched pairs $(H_{4}, H_{4}, ▹, ◃)$ by proving that, with the exception of the trivial pair, this set is parameterized by the ground field...

Cobraided smash product Hom-Hopf algebras

Tianshui Ma, Haiying Li, Tao Yang (2014)

Colloquium Mathematicae

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Let (A,α) and (B,β) be two Hom-Hopf algebras. We construct a new class of Hom-Hopf algebras: R-smash products $(A ♮_{R} B, α \otimes β)$ . Moreover, necessary and sufficient conditions for $(A ♮_{R} B, α \otimes β)$ to be a cobraided Hom-Hopf algebra are given.

Lazy 2-cocycles over monoidal Hom-Hopf algebras

Xiaofan Zhao, Xiaohui Zhang (2016)

Colloquium Mathematicae

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We introduce the notion of a lazy 2-cocycle over a monoidal Hom-Hopf algebra and determine all lazy 2-cocycles for a class of monoidal Hom-Hopf algebras. We also study the extension of lazy 2-cocycles to a Radford Hom-biproduct.

Hopf algebras and dendriform structures arising from parking functions

Jean-Christophe Novelli, Jean-Yves Thibon (2007)

Fundamenta Mathematicae

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We introduce a graded Hopf algebra based on the set of parking functions (hence of dimension ${(n + 1)}^{n - 1}$ in degree n). This algebra can be embedded into a noncommutative polynomial algebra in infinitely many variables. We determine its structure, and show that it admits natural quotients and subalgebras whose graded components have dimensions respectively given by the Schröder numbers (plane trees), the Catalan numbers, and powers of 3. These smaller algebras are always bialgebras and belong to...

Parametric representations of BiHom-Hopf algebras

Xiaohui Zhang, Wei Wang, Juzhen Chen (2024)

Czechoslovak Mathematical Journal

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The main purpose of the present paper is to study representations of BiHom-Hopf algebras. We first introduce the notion of BiHom-Hopf algebras, and then discuss BiHom-type modules, Yetter-Dinfeld modules and Drinfeld doubles with parameters. We get some new $n$ -monoidal categories via the category of BiHom-(co)modules and the category of BiHom-Yetter-Drinfeld modules. Finally, we obtain a center construction type theorem on BiHom-Hopf algebras.

A construction of the Hom-Yetter-Drinfeld category

Haiying Li, Tianshui Ma (2014)

Colloquium Mathematicae

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In continuation of our recent work about smash product Hom-Hopf algebras [Colloq. Math. 134 (2014)], we introduce the Hom-Yetter-Drinfeld category $_{H}^{H}$ via the Radford biproduct Hom-Hopf algebra, and prove that Hom-Yetter-Drinfeld modules can provide solutions of the Hom-Yang-Baxter equation and $_{H}^{H}$ is a pre-braided tensor category, where (H,β,S) is a Hom-Hopf algebra. Furthermore, we show that $(A ♮_{⋄} H, α \otimes β)$ is a Radford biproduct Hom-Hopf algebra if and only if (A,α) is a Hom-Hopf algebra in the category...

The strong Morita equivalence for coactions of a finite-dimensional C-Hopf algebra on unital C-algebras

Kazunori Kodaka, Tamotsu Teruya (2015)

Studia Mathematica

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Following Jansen and Waldmann, and Kajiwara and Watatani, we introduce notions of coactions of a finite-dimensional C*-Hopf algebra on a Hilbert C*-bimodule of finite type in the sense of Kajiwara and Watatani and define their crossed product. We investigate their basic properties and show that the strong Morita equivalence for coactions preserves the Rokhlin property for coactions of a finite-dimensional C*-Hopf algebra on unital C*-algebras.

The coribbon structures of some finite dimensional braided Hopf algebras generated by 2×2-matrix coalgebras

Michihisa Wakui (2003)

Banach Center Publications

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We determine the coribbon structures of some finite dimensional braided Hopf algebras generated by 2×2-matrix coalgebras constructed by S. Suzuki. As a consequence, we see that such a Hopf algebra has a coribbon structure if and only if it is of Kac-Paljutkin type.

Schwartz kernel theorem in algebras of generalized functions

Vincent Valmorin (2010)

Banach Center Publications

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A new approach to the generalization of Schwartz’s kernel theorem to Colombeau algebras of generalized functions is given. It is based on linear maps from algebras of classical functions to algebras of generalized ones. In particular, this approach enables one to give a meaning to certain hypotheses in preceding similar work on this theorem. Results based on the properties of $G^{\infty}$ -generalized functions class are given. A straightforward relationship between the classical and the generalized...

Actions of Hopf algebras on pro-semisimple noetherian algebras and their invariants

Andrzej Tyc (2001)

Colloquium Mathematicae

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Let H be a Hopf algebra over a field k such that every finite-dimensional (left) H-module is semisimple. We give a counterpart of the first fundamental theorem of the classical invariant theory for locally finite, finitely generated (commutative) H-module algebras, and for local, complete H-module algebras. Also, we prove that if H acts on the k-algebra A = k[[X₁,...,Xₙ]] in such a way that the unique maximal ideal in A is invariant, then the algebra of invariants $A^{H}$ is a noetherian Cohen-Macaulay...

Smash (co)products and skew pairings.

José N. Alonso Alvarez, José Manuel Fernández Vilaboa, Ramón González Rodríguez (2001)

Publicacions Matemàtiques

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Let τ be an invertible skew pairing on (B,H) where B and H are Hopf algebras in a symmetric monoidal category C with (co)equalizers. Assume that H is quasitriangular. Then we obtain a new algebra structure such that B is a Hopf algebra in the braided category γD and there exists a Hopf algebra isomorphism w: B ∞ H → B [×] H in C, where B ∞ H is a Hopf algebra with (co)algebra structure the smash (co)product and B [×] H is the Hopf algebra defined by Doi and Takeuchi. ...

Coalgebra deformations of bialgebras by Harrison cocycles, copairings of Hopf algebras and double crosscoproducts.

Caenepeel, S., Dăscălescu, S., Militaru, G., Panaite, F. (1997)

Bulletin of the Belgian Mathematical Society - Simon Stevin

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Generalized Post algebras and their application to some infinitary many-valued logics

Cat-Ho Nguyen

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CONTENTSIntroduction............................................................................................................................................................................... 5Part I. A generalization of Post algebras............................................................................................................................. 7 1. Definition and characterization of generalized Post algebras............................................. 7 2. Post...

On a Construction of ModularGMS-algebras

Abd El-Mohsen Badawy (2015)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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In this paper we investigate the class of all modular GMS-algebras which contains the class of MS-algebras. We construct modular GMS-algebras from the variety ${\underset{̲}{𝐊}}_{2}$ by means of ${\underset{̲}{K}}_{2}$ -quadruples. We also characterize isomorphisms of these algebras by means of ${\underset{̲}{K}}_{2}$ -quadruples.

A geometric approach to full Colombeau algebras

R. Steinbauer (2010)

Banach Center Publications

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We present a geometric approach to diffeomorphism invariant full Colombeau algebras which allows a particularly clear view of the construction of the intrinsically defined algebra $\hat{} (M)$ on the manifold M given in [gksv].

On Singer's algebra and coalgebra structures

Luciano A. Lomonaco (2006)

Bollettino dell'Unione Matematica Italiana

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Recently W. M. Singer has introduced the notion of algebra with coproducts (and the dual notion of coalgebra with products) by somehow weakening the notion of Hopf algebra (see [6]). In this paper we consider certain algebras of invariants and show that they are, in fact, further examples of algebras with coproducts and coalgebras with products. Moreover, we discuss the close relation between such algebras and the structures considered in Singer's paper.

Deformed mesh algebras of Dynkin type ℂₙ

Jerzy Białkowski, Karin Erdmann, Andrzej Skowroński (2012)

Colloquium Mathematicae

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In our recent paper (J. Algebra 345 (2011)) we prove that the deformed preprojective algebras of generalized Dynkin type ₙ (in the sense of our earlier work in Trans. Amer Math. Soc. 359 (2007)) are exactly (up to isomorphism) the stable Auslander algebras of simple plane singularities of Dynkin type $_{2 n}$ . In this article we complete the picture by showing that the deformed mesh algebras of Dynkin type ℂₙ are isomorphic to the canonical mesh algebras of type ℂₙ, and hence to the stable...

Multiloop algebras, iterated loop algebras and extended affine Lie algebras of nullity 2

Bruce Allison, Stephen Berman, Arturo Pianzola (2014)

Journal of the European Mathematical Society

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Let $𝕄_{n}$ be the class of all multiloop algebras of finite dimensional simple Lie algebras relative to $n$ -tuples of commuting finite order automorphisms. It is a classical result that $𝕄_{1}$ is the class of all derived algebras modulo their centres of affine Kac-Moody Lie algebras. This combined with the Peterson-Kac conjugacy theorem for affine algebras results in a classification of the algebras in $𝕄_{1}$ . In this paper, we classify the algebras in $𝕄_{2}$ , and further determine the relationship between...

Standardly stratified split and lower triangular algebras

Eduardo do N. Marcos, Hector A. Merklen, Corina Sáenz (2002)

Colloquium Mathematicae

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In the first part, we study algebras A such that A = R ⨿ I, where R is a subalgebra and I a two-sided nilpotent ideal. Under certain conditions on I, we show that A is standardly stratified if and only if R is standardly stratified. Next, for $A = [\begin{matrix} U & 0 \\ M & V \end{matrix}]$ , we show that A is standardly stratified if and only if the algebra R = U × V is standardly stratified and $_{V} M$ is a good V-module.

The fundamental theorem and Maschke's theorem in the category of relative Hom-Hopf modules

Yuanyuan Chen, Zhongwei Wang, Liangyun Zhang (2016)

Colloquium Mathematicae

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We introduce the concept of relative Hom-Hopf modules and investigate their structure in a monoidal category $̃ (ℳ_{k})$ . More particularly, the fundamental theorem for relative Hom-Hopf modules is proved under the assumption that the Hom-comodule algebra is cleft. Moreover, Maschke’s theorem for relative Hom-Hopf modules is established when there is a multiplicative total Hom-integral.

A new proof of the noncommutative Banach-Stone theorem

David Sherman (2006)

Banach Center Publications

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Surjective isometries between unital C*-algebras were classified in 1951 by Kadison [K]. In 1972 Paterson and Sinclair [PS] handled the nonunital case by assuming Kadison’s theorem and supplying some supplementary lemmas. Here we combine an observation of Paterson and Sinclair with variations on the methods of Yeadon [Y] and the author [S1], producing a fundamentally new proof of the structure of surjective isometries between (nonunital) C*-algebras. In the final section we indicate...

Cosemisimple Hopf algebras with antipode of arbitrary finite order.

Bichon, Julien (2002)

The New York Journal of Mathematics [electronic only]

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Metric generalizations of Banach algebras

W. Żelazko

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CONTENTSPRELIMINARIES§ 0. Introduction.......................................................................................................................................................................3§ 1. Definitions and notation.................................................................................................................................................5Chapter ILOCALLY BOUNDED ALGEBRAS§ 2. Basic facts and examples..............................................................................................................................................6§...