• 대한수학회보
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• 제48권1호
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• pp.17-21
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• 2011

Let A be a co-Poisson Hopf algebra with Poisson co-bracket $\delta$. Here it is shown that the Hopf dual $A^{\circ}$ is a Poisson Hopf algebra with Poisson bracket {f, g}(x) = < $\delta(x)$, $f\;{\otimes}\;g$ > for any f, g $\in$ $A^{\circ}$ and x $\in$ A if A is an almost normalizing extension over the ground field. Moreover we get, as a corollary, the fact that the Hopf dual of the universal enveloping algebra U(g) for a finite dimensional Lie bialgebra g is a Poisson Hopf algebra.

• 충청수학회지
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• 제8권1호
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• pp.173-181
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• 1995

Let H and G be finite dimensional semisimple Hopf algebras and let A and B be left H and G-module algebras respectively. We use smash product algebras to show that 1) if A is right Artinian then $A^H$ is right Artinian, 2) $Soc\;V_A{\subset}Soc\;V_{A^H}$ and rad $V_A{\supset}\;radV_{A^H}$, 3) $K\;dim\;_BV_A=K\;dim\;_{B^G}V_{A^H}$.

• 충청수학회지
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• 제14권1호
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• pp.85-103
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• 2001

Suppose that Hand K are paired Hopf algebras and that A is an H - K - bicomodule algebra with multiplication which is a left H-comodule map and is a right K-comodule map. We define a new twisted algebra, $A^{\tau}$ and define $M^{\tau}$ for $M{\in}M_A^K$. We find an equivalent condition for $M^{\tau}{\in}M_{A^{\tau}}^K$. We show that the above defined twisted multiplication is the special case of Beattie's twist multiplication. We show that if K is commutative, then A is an H-module algebra and show that if $H^*$ is cocommutative then the construction of smash product appears as a special case of the new twist product.

• 대한수학회보
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• 제54권2호
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• pp.463-484
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• 2017

We construct a weak Hopf algebra $wX_q(A_1)$ corresponding to non-standard quantum group $X_q(A_1)$. The PBW basis of $wX_q(A_1)$ is described and all the highest weight modules of $wX_q(A_1)$ are classified. Finally we give the Clebsch-Gordan decomposition of the tensor product of two highest weight modules of $wX_q(A_1)$.

• 대한수학회보
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• 제13권2호
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• pp.149.2-149.2
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• 1976

• 대한수학회보
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• 제47권6호
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• pp.1275-1283
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• 2010

If the Wiener-Hopf $C^*$-algebra W(G,M) for a discrete group G with a semigroup M has the uniqueness property, then the structure of it is to some extent independent of the choice of isometries on a Hilbert space. In this paper we show that if the Wiener-Hopf $C^*$-algebra W(G,M) of a partially ordered group G with the positive cone M has the uniqueness property, then (G,M) is weakly unperforated. We also prove that the Wiener-Hopf $C^*$-algebra W($\mathbb{Z}$, M) of subsemigroup generating the integer group $\mathbb{Z}$ is isomorphic to the Toeplitz algebra, but W($\mathbb{Z}$, M) does not have the uniqueness property except the case M = $\mathbb{N}$.

• 대한수학회보
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• 제52권5호
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• pp.1587-1606
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• 2015

Comonoid, bi-algebra, and Hopf algebra structures are studied within the universal-algebraic context of entropic varieties. Attention focuses on the behavior of setlike and primitive elements. It is shown that entropic $J{\acute{o}}nsson$-Tarski varieties provide a natural universal-algebraic setting for primitive elements and group quantum couples (generalizations of the group quantum double). Here, the set of primitive elements of a Hopf algebra forms a Lie algebra, and the tensor algebra on any algebra is a bi-algebra. If the tensor algebra is a Hopf algebra, then the underlying $J{\acute{o}}nsson$-Tarski monoid of the generating algebra is cancellative. The problem of determining when the $J{\acute{o}}nsson$-Tarski monoid forms a group is open.

• 충청수학회지
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• 제23권2호
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• pp.323-333
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• 2010

The biproduct bialgebra has been generalized to generalized biproduct bialgebra $B{\times}^L_H\;D$ in [5]. Let (D, B) be an admissible pair and let D be a bialgebra. We show that if generalized biproduct bialgebra $B{\times}^L_H\;D$ is a Hopf algebra with antipode s, then D is a Hopf algebra and the identity $id_B$ has an inverse in the convolution algebra $Hom_k$(B, B). We show that if D is a Hopf algebra with antipode $s_D$ and $s_B$ in $Hom_k$(B, B) is an inverse of $id_B$ then $B{\times}^L_H\;D$ is a Hopf algebra with antipode s described by $s(b{\times}^L_H\;d)={\Sigma}(1_B{\times}^L_H\;s_D(b_{-1}{\cdot}d))(s_B(b_0){\times}^L_H\;1_D)$. We show that the mapping system $B{\leftrightarrows}^{{\Pi}_B}_{j_B}\;B{\times}^L_H\;D{\rightleftarrows}^{{\pi}_D}_{i_D}\;D$ (where $j_B$ and $i_D$ are the canonical inclusions, ${\Pi}_B$ and ${\pi}_D$ are the canonical coalgebra projections) characterizes $B{\times}^L_H\;D$. These generalize the corresponding results in [6].

• 충청수학회지
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• 제21권3호
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• pp.301-320
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• 2008

The smash product algebra has been generalized to general smash product algebra in [3] and we can generalize the smash coproduct coalgebra to obtain the general smash coproduct coalgebra. It is natural to replace the smash product and smash coproduct by the generalized smash product and generalized smash coproduct and consider the condition under which the generalized smash product algebra structure and the generalized smash coproduct coalgebra structure will inherit a bialgebra structure or a Hopf algebra structure. We derive necessary sufficient conditions for the problem. This generalizes the corresponding results in [7] and [4].

• 대한수학회보
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• 제52권2호
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• pp.557-570
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• 2015

Let $\mathbb{G}$ be a von Neumann algebraic locally compact quantum group, in the sense of Kustermans and Vaes. In this paper, as a consequence of a notion of amenability for actions of Lau algebras, we show that $\hat{\mathbb{G}}$, the dual of $\mathbb{G}$, is co-amenable if and only if there is a state $m{\in}L^{\infty}(\hat{\mathbb{G}})^*$ which is invariant under a left module action of $L^1(\mathbb{G})$ on $L^{\infty}(\hat{\mathbb{G}})^*$. This is the quantum group version of a result by Stokke [17]. We also characterize amenable action of Lau algebras by several properties such as fixed point property. This yields in particular, a fixed point characterization of amenable groups and H-amenable representation of groups.