Hopf $ {\pi } $-crossed biproduct and related coquasitriangular structures

Tianshui Ma; Yanan Song

Displaying similar documents to “Hopf $π$ -crossed biproduct and related coquasitriangular structures”

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.

A new way to iterate Brzeziński crossed products

Leonard Dăuş, Florin Panaite (2016)

Colloquium Mathematicae

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If $A \otimes_{R, σ} V$ and $A \otimes_{P, ν} W$ are two Brzeziński crossed products and Q: W⊗ V → V⊗ W is a linear map satisfying certain properties, we construct a Brzeziński crossed product $A \otimes_{S, θ} (V \otimes W)$ . This construction contains as a particular case the iterated twisted tensor product of algebras.

Yetter-Drinfeld-Long bimodules are modules

Daowei Lu, Shuan Hong Wang (2017)

Czechoslovak Mathematical Journal

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Let $H$ be a finite-dimensional bialgebra. In this paper, we prove that the category $ℒℛ (H)$ of Yetter-Drinfeld-Long bimodules, introduced by F. Panaite, F. Van Oystaeyen (2008), is isomorphic to the Yetter-Drinfeld category $_{H \otimes H^{*}}^{H \otimes H^{*}} 𝒴𝒟$ over the tensor product bialgebra $H \otimes H^{*}$ as monoidal categories. Moreover if $H$ is a finite-dimensional Hopf algebra with bijective antipode, the isomorphism is braided. Finally, as an application of this category isomorphism, we give two results.

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.

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 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.

Characterization of automorphisms of Radford's biproduct of Hopf group-coalgebra

Xing Wang, Daowei Lu, Ding-Guo Wang (2024)

Czechoslovak Mathematical Journal

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We study certain subgroups of the Hopf group-coalgebra automorphism group of Radford’s $π$ -biproduct. Firstly, we discuss the endomorphism monoid ${End}_{π -Hopf} (A \times H, p)$ and the automorphism group ${Aut}_{π -Hopf} (A \times H, p)$ of Radford’s $π$ -biproduct $A \times H = {A \times H_{α}}_{α \in π}$ , and prove that the automorphism has a factorization closely related to the factors $A$ and $H = {H_{α}}_{α \in π}$ . What’s more interesting is that a pair of maps $(F_{L}, F_{R})$ can be used to describe a family of mappings $F = {F_{α}}_{α \in π}$ . Secondly, we consider the relationship between the automorphism group ${Aut}_{π -Hopf} (A \times H, p)$ and the automorphism group...

$ω_{0}$ -categoricity of generalized products

Jan Waszkiewicz (1973)

Colloquium Mathematicae

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Classification of ideals of $8$ -dimensional Radford Hopf algebra

Yu Wang (2022)

Czechoslovak Mathematical Journal

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Let $H_{m, n}$ be the $m n^{2}$ -dimensional Radford Hopf algebra over an algebraically closed field of characteristic zero. We give the classification of all ideals of $8$ -dimensional Radford Hopf algebra $H_{2, 2}$ by generators.

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.

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.

Automorphism group of green algebra of weak Hopf algebra corresponding to Sweedler Hopf algebra

Liufeng Cao, Dong Su, Hua Yao (2023)

Czechoslovak Mathematical Journal

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Let $r (𝔴_{2}^{0})$ be the Green ring of the weak Hopf algebra $𝔴_{2}^{0}$ corresponding to Sweedler’s 4-dimensional Hopf algebra $H_{2}$ , and let $Aut (R (𝔴_{2}^{0}))$ be the automorphism group of the Green algebra $R (𝔴_{2}^{0}) = r (𝔴_{2}^{0}) \otimes_{ℤ} ℂ$ . We show that the quotient group $Aut (R (𝔴_{2}^{0})) / C_{2} ≅ S_{3}$ , where $C_{2}$ contains the identity map and is isomorphic to the infinite group $(ℂ^{*}, \times)$ and $S_{3}$ is the symmetric group of order 6.

Corrigendum to “Braided coproduct, antipode and adjoint action for $U_{q} (s l_{2})$ ” [Arch. Math. (Brno) 60(5) (2024), 365–376, DOI: 10.5817/AM2024-5-365]

(2025)

Archivum Mathematicum

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Automorphism group of representation ring of the weak Hopf algebra $\tilde{H_{8}}$

Dong Su, Shilin Yang (2018)

Czechoslovak Mathematical Journal

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Let $H_{8}$ be the unique noncommutative and noncocommutative eight dimensional semi-simple Hopf algebra. We first construct a weak Hopf algebra $\tilde{H_{8}}$ based on $H_{8}$ , then we investigate the structure of the representation ring of $\tilde{H_{8}}$ . Finally, we prove that the automorphism group of $r (\tilde{H_{8}})$ is just isomorphic to $D_{6}$ , where $D_{6}$ is the dihedral group with order 12.

Relating quantum and braided Lie algebras

X. Gomez, S. Majid (2003)

Banach Center Publications

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We outline our recent results on bicovariant differential calculi on co-quasitriangular Hopf algebras, in particular that if $_{Γ}$ is a quantum tangent space (quantum Lie algebra) for a CQT Hopf algebra A, then the space $k \oplus_{Γ}$ is a braided Lie algebra in the category of A-comodules. An important consequence of this is that the universal enveloping algebra $U (_{Γ})$ is a bialgebra in the category of A-comodules.

Covariantization of quantized calculi over quantum groups

Seyed Ebrahim Akrami, Shervin Farzi (2020)

Mathematica Bohemica

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We introduce a method for construction of a covariant differential calculus over a Hopf algebra $A$ from a quantized calculus $d a = [D, a]$ , $a \in A$ , where $D$ is a candidate for a Dirac operator for $A$ . We recover the method of construction of a bicovariant differential calculus given by T. Brzeziński and S. Majid created from a central element of the dual Hopf algebra $A^{\circ}$ . We apply this method to the Dirac operator for the quantum $SL (2)$ given by S. Majid. We find that the differential calculus obtained by our...

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

Philip Olin (1975)

Colloquium Mathematicae

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Separable functors for the category of Doi Hom-Hopf modules

Shuangjian Guo, Xiaohui Zhang (2016)

Colloquium Mathematicae

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Let $̃ (_{k}) {(H)}_{A}^{C}$ be the category of Doi Hom-Hopf modules, $̃ {(_{k})}_{A}$ be the category of A-Hom-modules, and F be the forgetful functor from $̃ (_{k}) {(H)}_{A}^{C}$ to $̃ {(_{k})}_{A}$ . The aim of this paper is to give a necessary and suffcient condition for F to be separable. This leads to a generalized notion of integral. Finally, applications of our results are given. In particular, we prove a Maschke type theorem for Doi Hom-Hopf modules.

Two results of $n$ -exangulated categories

Jian He, Jing He, Panyue Zhou (2024)

Czechoslovak Mathematical Journal

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M. Herschend, Y. Liu, H. Nakaoka introduced $n$ -exangulated categories, which are a simultaneous generalization of $n$ -exact categories and $(n + 2)$ -angulated categories. This paper consists of two results on $n$ -exangulated categories: (1) we give an equivalent characterization of axiom (EA2); (2) we provide a new way to construct a closed subfunctor of an $n$ -exangulated category.

One-sided $n$ -suspended categories

Jing He, Yonggang Hu, Panyue Zhou (2024)

Czechoslovak Mathematical Journal

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For an integer $n \geq 3$ , we introduce a simultaneous generalization of $(n - 2)$ -exact categories and $n$ -angulated categories, referred to as one-sided $n$ -suspended categories. Notably, one-sided $n$ -angulated categories are specific instances of this structure. We establish a framework for transitioning from these generalized categories to their $n$ -angulated counterparts. Additionally, we present a method for constructing $n$ -angulated quotient categories from Frobenius $n$ -prile categories. Our results unify...

The Group of Invertible Elements of the Algebra of Quaternions

Irina A. Kuzmina, Marie Chodorová (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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We have, that all two-dimensional subspaces of the algebra of quaternions, containing a unit, are 2-dimensional subalgebras isomorphic to the algebra $ℂ$ of complex numbers. It was proved in the papers of N. E. Belova. In the present article we consider a 2-dimensional subalgebra $(i)$ of complex numbers with basis $1, i$ and we construct the principal locally trivial bundle which is isomorphic to the Hopf fibration.

Displaying similar documents to “Hopf π -crossed biproduct and related coquasitriangular structures”

Displaying similar documents to “Hopf $π$ -crossed biproduct and related coquasitriangular structures”