Cobraided smash product Hom-Hopf algebras

Tianshui Ma; Haiying Li; Tao Yang

Displaying similar documents to “Cobraided smash product Hom-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 duality theorem for twisted smash products of Hopf algebras and its applications

Zhongwei Wang, Liangyun Zhang (2015)

Colloquium Mathematicae

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Let $A_{T} H$ denote the twisted smash product of an arbitrary algebra A and a Hopf algebra H over a field. We present an analogue of the celebrated Blattner-Montgomery duality theorem for $A_{T} H$ , and as an application we establish the relationship between the homological dimensions of $A_{T} H$ and A if H and its dual H* are both semisimple.

Monomorphisms of coalgebras

A. L. Agore (2010)

Colloquium Mathematicae

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We prove new necessary and sufficient conditions for a morphism of coalgebras to be a monomorphism, different from the ones already available in the literature. More precisely, φ: C → D is a monomorphism of coalgebras if and only if the first cohomology groups of the coalgebras C and D coincide if and only if $\sum_{i \in I} ε (a^{i}) b^{i} = \sum_{i \in I} a^{i} ε (b^{i})$ for all $\sum_{i \in I} a^{i} \otimes b^{i} \in C ◻_{D} C$ . In particular, necessary and sufficient conditions for a Hopf algebra map to be a monomorphism are given.

Examples of polynomial identities distinguishing the Galois objects over finite-dimensional Hopf algebras

Christian Kassel (2013)

Annales mathématiques Blaise Pascal

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We define polynomial $H$ -identities for comodule algebras over a Hopf algebra $H$ and establish general properties for the corresponding $T$ -ideals. In the case $H$ is a Taft algebra or the Hopf algebra $E (n)$ , we exhibit a finite set of polynomial $H$ -identities which distinguish the Galois objects over $H$ up to isomorphism.

$ℵ_{0}$ -categoricity of products of 1-unary algebras

Philip Olin (1975)

Colloquium Mathematicae

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Quasitriangular Hom-Hopf algebras

Yuanyuan Chen, Zhongwei Wang, Liangyun Zhang (2014)

Colloquium Mathematicae

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A twisted generalization of quasitriangular Hopf algebras called quasitriangular Hom-Hopf algebras is introduced. We characterize these algebras in terms of certain morphisms. We also give their equivalent description via a braided monoidal category $̃ (_{H} ℳ)$ . Finally, we study the twisting structure of quasitriangular Hom-Hopf algebras by conjugation with Hom-2-cocycles.

The structures of Hopf $*$ -algebra on Radford algebras

Hassan Suleman Esmael Mohammed, Hui-Xiang Chen (2019)

Czechoslovak Mathematical Journal

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We investigate the structures of Hopf $*$ -algebra on the Radford algebras over $ℂ$ . All the $*$ -structures on $H$ are explicitly given. Moreover, these Hopf $*$ -algebra structures are classified up to equivalence.

The bicrossed products of $H_{4}$ and $H_{8}$

Daowei Lu, Yan Ning, Dingguo Wang (2020)

Czechoslovak Mathematical Journal

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Let $H_{4}$ and $H_{8}$ be the Sweedler’s and Kac-Paljutkin Hopf algebras, respectively. We prove that any Hopf algebra which factorizes through $H_{8}$ and $H_{4}$ (equivalently, any bicrossed product between the Hopf algebras $H_{8}$ and $H_{4}$ ) must be isomorphic to one of the following four Hopf algebras: $H_{8} \otimes H_{4}, H_{32, 1}, H_{32, 2}, H_{32, 3}$ . The set of all matched pairs $(H_{8}, H_{4}, ▹, ◃)$ is explicitly described, and then the associated bicrossed product is given by generators and relations.

Projective abelian Hopf algebras over a field

Andrzej Skowroński

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CONTENTSIntroduction............................................................................................................................................51. Cohen schemes.................................................................................................................................72. Projective abelian Hopf algebras......................................................................................................113. The structure of groups $H o m_{ℋ ₁} (^{m} P,^{n} P)$ ..............................................................................174....

Hopf $π$ -crossed biproduct and related coquasitriangular structures

Tianshui Ma, Yanan Song (2013)

Rendiconti del Seminario Matematico della Università di Padova

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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...

On generalized Beurling’s theorem and symmetry of $L_{1}$ -group algebras

Zenobia Anusiak (1971)

Colloquium Mathematicae

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A representation theorem for tense $n \times m$ -valued Łukasiewicz-Moisil algebras

Aldo Victorio Figallo, Gustavo Pelaitay (2015)

Mathematica Bohemica

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In 2000, Figallo and Sanza introduced $n \times m$ -valued Łukasiewicz-Moisil algebras which are both particular cases of matrix Łukasiewicz algebras and a generalization of $n$ -valued Łukasiewicz-Moisil algebras. Here we initiate an investigation into the class $_{n \times m}$ of tense $n \times m$ -valued Łukasiewicz-Moisil algebras (or tense LM $_{n \times m}$ -algebras), namely $n \times m$ -valued Łukasiewicz-Moisil algebras endowed with two unary operations called tense operators. These algebras constitute a generalization of tense...

A class of quantum doubles of pointed Hopf algebras of rank one

Hua Sun, Yueming Li (2023)

Czechoslovak Mathematical Journal

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We construct a class of quantum doubles $D (H_{D_{n}})$ of pointed Hopf algebras of rank one $H_{𝒟}$ . We describe the algebra structures of $D (H_{D_{n}})$ by generators with relations. Moreover, we give the comultiplication $Δ_{D}$ , counit $ε_{D}$ and the antipode $S_{D}$ , respectively.