Automorphism group of representation ring of the weak Hopf algebra $\widetilde{H_8}$

Dong Su; Shilin Yang

Displaying similar documents to “Automorphism group of representation ring of the weak Hopf algebra $\tilde{H_{8}}$ ”

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.

A note on Skolem-Noether algebras

Juncheol Han, Tsiu-Kwen Lee, Sangwon Park (2021)

Czechoslovak Mathematical Journal

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The paper was motivated by Kovacs’ paper (1973), Isaacs’ paper (1980) and a recent paper, due to Brešar et al. (2018), concerning Skolem-Noether algebras. Let $K$ be a unital commutative ring, not necessarily a field. Given a unital $K$ -algebra $S$ , where $K$ is contained in the center of $S$ , $n \in ℕ$ , the goal of this paper is to study the question: when can a homomorphism $φ : M_{n} (K) \to M_{n} (S)$ be extended to an inner automorphism of $M_{n} (S)$ ? As an application of main results presented in the paper, it is proved that if $S$ is...

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.

Annihilators of skew derivations with Engel conditions on prime rings

Taylan Pehlivan, Emine Albas (2020)

Czechoslovak Mathematical Journal

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Let $R$ be a noncommutative prime ring of characteristic different from 2, with its two-sided Martindale quotient ring $Q$ , $C$ the extended centroid of $R$ and $a \in R$ . Suppose that $δ$ is a nonzero $σ$ -derivation of $R$ such that $a {[δ (x^{n}), x^{n}]}_{k} = 0$ for all $x \in R$ , where $σ$ is an automorphism of $R$ , $n$ and $k$ are fixed positive integers. Then $a = 0$ .

(Generalized) filter properties of the amalgamated algebra

Yusof Azimi (2022)

Archivum Mathematicum

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Let $R$ and $S$ be commutative rings with unity, $f : R \to S$ a ring homomorphism and $J$ an ideal of $S$ . Then the subring $R ⋈^{f} J : = {(a, f (a) + j) ∣ a \in R$ and $j \in J}$ of $R \times S$ is called the amalgamation of $R$ with $S$ along $J$ with respect to $f$ . In this paper, we determine when $R ⋈^{f} J$ is a (generalized) filter ring.

The subspace of weak $P$ -points of $ℕ^{*}$

Salvador García-Ferreira, Y. F. Ortiz-Castillo (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let $W$ be the subspace of $ℕ^{*}$ consisting of all weak $P$ -points. It is not hard to see that $W$ is a pseudocompact space. In this paper we shall prove that this space has stronger pseudocompact properties. Indeed, it is shown that $W$ is a $p$ -pseudocompact space for all $p \in ℕ^{*}$ .

Augmentation quotients for Burnside rings of generalized dihedral groups

Shan Chang (2016)

Czechoslovak Mathematical Journal

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Let $H$ be a finite abelian group of odd order, $𝒟$ be its generalized dihedral group, i.e., the semidirect product of $C_{2}$ acting on $H$ by inverting elements, where $C_{2}$ is the cyclic group of order two. Let $Ω (𝒟)$ be the Burnside ring of $𝒟$ , $Δ (𝒟)$ be the augmentation ideal of $Ω (𝒟)$ . Denote by $Δ^{n} (𝒟)$ and $Q_{n} (𝒟)$ the $n$ th power of $Δ (𝒟)$ and the $n$ th consecutive quotient group $Δ^{n} (𝒟) / Δ^{n + 1} (𝒟)$ , respectively. This paper provides an explicit $ℤ$ -basis for $Δ^{n} (𝒟)$ and determines the isomorphism class of $Q_{n} (𝒟)$ for each positive integer $n$ .

Automorphisms and generalized skew derivations which are strong commutativity preserving on polynomials in prime and semiprime rings

Vincenzo de Filippis (2016)

Czechoslovak Mathematical Journal

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Let $R$ be a prime ring of characteristic different from 2, $Q_{r}$ its right Martindale quotient ring and $C$ its extended centroid. Suppose that $F$ , $G$ are generalized skew derivations of $R$ with the same associated automorphism $α$ , and $p (x_{1}, ..., x_{n})$ is a non-central polynomial over $C$ such that $[F (x), α (y)] = G ([x, y])$ for all $x, y \in {p (r_{1}, ..., r_{n}) : r_{1}, ..., r_{n} \in R}$ . Then there exists $λ \in C$ such that $F (x) = G (x) = λ α (x)$ for all $x \in R$ .

The clean elements of the ring $ℛ (L)$

Ali Akbar Estaji, Maryam Taha (2024)

Czechoslovak Mathematical Journal

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We characterize clean elements of $ℛ (L)$ and show that $α \in ℛ (L)$ is clean if and only if there exists a clopen sublocale $U$ in $L$ such that $𝔠_{L} (coz (α - 1)) \subseteq U \subseteq 𝔬_{L} (coz (α))$ . Also, we prove that $ℛ (L)$ is clean if and only if $ℛ (L)$ has a clean prime ideal. Then, according to the results about $ℛ (L),$ we immediately get results about $𝒞_{c} (L) .$

Strongly 2-nil-clean rings with involutions

Huanyin Chen, Marjan Sheibani Abdolyousefi (2019)

Czechoslovak Mathematical Journal

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A $*$ -ring $R$ is strongly 2-nil- $*$ -clean if every element in $R$ is the sum of two projections and a nilpotent that commute. Fundamental properties of such $*$ -rings are obtained. We prove that a $*$ -ring $R$ is strongly 2-nil- $*$ -clean if and only if for all $a \in R$ , $a^{2} \in R$ is strongly nil- $*$ -clean, if and only if for any $a \in R$ there exists a $*$ -tripotent $e \in R$ such that $a - e \in R$ is nilpotent and $e a = a e$ , if and only if $R$ is a strongly $*$ -clean SN ring, if and only if $R$ is abelian, $J (R)$ is nil and $R / J (R)$ is $*$ -tripotent. Furthermore, we explore...

Displaying similar documents to “Automorphism group of representation ring of the weak Hopf algebra H 8 ˜ ”

Displaying similar documents to “Automorphism group of representation ring of the weak Hopf algebra $\tilde{H_{8}}$ ”