• 제목/요약/키워드: Hopf algebras

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INDUCED HOPF CORING STRUCTURES

  • Saramago, Rui Miguel
    • 대한수학회지
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    • 제48권3호
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    • pp.627-639
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    • 2011
  • Hopf corings are dened in this work as coring objects in the category of algebras over a commutative ring R. Using the Dieudonn$\'{e}$ equivalences from [7] and [19], one can associate coring structures built from the Hopf algebra $F_p[x_0,x_1,{\ldots}]$, p a prime, with Hopf ring structures with same underlying connected Hopf algebra. We have that $F_p[x_0,x_1,{\ldots}]$ coring structures classify thus Hopf ring structures for a given Hopf algebra. These methods are applied to dene new ring products in the Hopf algebras underlying known Hopf rings that come from connective Morava ${\kappa}$-theory.

ORE EXTENSIONS OF HOPF GROUP COALGEBRAS

  • Wang, Dingguo;Lu, Daowei
    • 대한수학회지
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    • 제51권2호
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    • pp.325-344
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    • 2014
  • The aim of this paper is to generalize the theory of Hopf-Ore extension on Hopf algebras to Hopf group coalgebras. First the concept of Hopf-Ore extension of Hopf group coalgebra is introduced. Then we will give the necessary and sufficient condition for the Ore extensions to become a Hopf group coalgebra, and certain isomorphism between Ore extensions of Hopf group coalgebras are discussed.

OPPOSITE SKEW COPAIRED HOPF ALGEBRAS

  • Park, Junseok;Kim, Wansoon
    • 충청수학회지
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    • 제17권1호
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    • pp.85-101
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    • 2004
  • Let A be a Hopf algebra with a linear form ${\sigma}:k{\rightarrow}A{\otimes}A$, which is convolution invertible, such that ${\sigma}_{21}({\Delta}{\otimes}id){\tau}({\sigma}(1))={\sigma}_{32}(id{\otimes}{\Delta}){\tau}({\sigma}(1))$. We define Hopf algebras, ($A_{\sigma}$, m, u, ${\Delta}_{\sigma}$, ${\varepsilon}$, $S_{\sigma}$). If B and C are opposite skew copaired Hopf algebras and $A=B{\otimes}_kC$ then we find Hopf algebras, ($A_{[{\sigma}]}$, $m_B{\otimes}m_C$, $u_B{\otimes}u_C$, ${\Delta}_{[{\sigma}]}$, ${\varepsilon}B{\otimes}{\varepsilon}_C$, $S_{[{\sigma}]}$). Let H be a finite dimensional commutative Hopf algebra with dual basis $\{h_i\}$ and $\{h_i^*\}$, and let $A=H^{op}{\otimes}H^*$. We show that if we define ${\sigma}:k{\rightarrow}H^{op}{\otimes}H^*$ by ${\sigma}(1)={\sum}h_i{\otimes}h_i^*$ then ($A_{[{\sigma}]}$, $m_A$, $u_A$, ${\Delta}_{[{\sigma}]}$, ${\varepsilon}_A$, $S_{[{\sigma}]}$) is the dual space of Drinfeld double, $D(H)^*$, as Hopf algebra.

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TWISTING COPRODUCTS ON HOPF ALGEBRAS

  • Min, Kang Ju;Park, Jun Seok
    • 충청수학회지
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    • 제11권1호
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    • pp.99-113
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    • 1998
  • Let (H, K) be a paired Hopf algebras and let A be arbitrary left H-module coalgebra. We construct twisting coproduct on $A{\otimes}K$. We show that the well known construction of the smash coproduct can be viewed as a particular case of the construction above.

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SKEW COPAIRED BIALGEBRAS

  • Park, Jun Seok;Cho, Myung Sang
    • 충청수학회지
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    • 제16권1호
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    • pp.81-96
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    • 2003
  • Let ${\sigma}:k{\rightarrow}A{\otimes}B$ be a skew copairing on (A, B), where A and B are Hopf algebras of the same dimension n. Skew dual bases of A and B are introduced. If ${\sigma}$ is an invertible skew copairing then we can give a 2-cocycle bilinear form [${\sigma}$] on $A{\otimes}B$ and define a new Hopf algebra.

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HOPF STRUCTURE FOR POISSON ENVELOPING ALGEBRAS

  • Min, Kangju;Oh, Sei-Qwon
    • 충청수학회지
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    • 제13권2호
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    • pp.29-39
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    • 2001
  • For a Poisson Hopf algebra A, we find a natural Hopf structure in the Poisson enveloping algebra U(A) of A. As an application, we show that the Poisson enveloping algebra U(S(L)), where S(L) is the symmetric algebra of a Lie algebra L, has a Hopf structure such that a sub-Hopf algebra of U(S(L)) is Hopf-isomorphic to the universal enveloping algebra of L.

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REPRESENTATIONS OVER GREEN ALGEBRAS OF WEAK HOPF ALGEBRAS BASED ON TAFT ALGEBRAS

  • Liufeng Cao
    • 대한수학회보
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    • 제60권6호
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    • pp.1687-1695
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    • 2023
  • In this paper, we study the Green ring r(𝔴0n) of the weak Hopf algebra 𝔴0n based on Taft Hopf algebra Hn(q). Let R(𝔴0n) := r(𝔴0n) ⊗ ℂ be the Green algebra corresponding to the Green ring r(𝔴0n). We first determine all finite dimensional simple modules of the Green algebra R(𝔴0n), which is based on the observations of the roots of the generating relations associated with the Green ring r(𝔴0n). Then we show that the nilpotent elements in r(𝔴0n) can be written as a sum of finite dimensional indecomposable projective 𝔴0n-modules. The Jacobson radical J(r(𝔴0n)) of r(𝔴0n) is a principal ideal, and its rank equals n - 1. Furthermore, we classify all finite dimensional non-simple indecomposable R(𝔴0n)-modules. It turns out that R(𝔴0n) has n2 - n + 2 simple modules of dimension 1, and n non-simple indecomposable modules of dimension 2.

ACTIONS OF FINITE-DIMENSIONAL SEMISIMPLE HOPF ALGEBRAS AND INVARIANT ALGEBRAS

  • Min, Kang-Ju;Park, Jun-Seok
    • 대한수학회논문집
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    • 제13권2호
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    • pp.225-232
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    • 1998
  • Let H be a finite dimensional Hopf algebra over a field k, and A be an H-module algebra over k which the H-action on A is D-continuous. We show that $Q_{max}(A)$, the maximal ring or quotients of A, is an H-module algebra. This is used to prove that if H is a finite dimensional semisimple Hopf algebra and A is a semiprime right(left) Goldie algebra than $A#H$ is a semiprime right(left) Goldie algebra. Assume that Asi a semiprime H-module algebra Then $A^H$ is left Artinian if and only if A is left Artinian.

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Department of Mathematics, Dongeui University

  • Yoon, Suk-Bong
    • 대한수학회보
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    • 제38권3호
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    • pp.527-541
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    • 2001
  • We find the necessary and sufficient conditions for the smash product algebra structure and the crossed coproduct coalgebra structure with th dual cocycle $\alpha$ to afford a Hopf algebra (A equation,※See Full-text). If B and H are finite algebra and Hopf algebra, respectively, then the linear dual (※See Full-text) is also a Hopf algebra. We show that the weak coaction admissible mapping system characterizes the new Hopf algebras (※See Full-text).

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