• 대한수학회논문집
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• 제20권4호
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• pp.649-655
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• 2005

The universal mapping property and the Gelfand- Kirillov dimension of a skew enveloping algebra are studied and it is proved that every Poisson enveloping algebra is a homomorphic image of a skew enveloping algebra.

• 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.

• 충청수학회지
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• 제23권1호
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• pp.177-184
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• 2010

Let G be an abelian group, $\alpha$ an antisymmetric bicharacter on G and g a (G, $\alpha$)-Lie algebra. Here we give a complete proof for that the associated graded algebra determined by a natural filtration in the universal enveloping algebra U(g) is a (G, $\alpha$)-Poisson Hopf algebra.

• 충청수학회지
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• 제21권1호
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• pp.57-60
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• 2008

Let $U(sp_4)$ be the universal enveloping algebra of the symplectic Lie algebra $sp_4$. Then the restricted dual $U(sp_4)^{\circ}$ becomes a Poisson Hopf algebra with the Sklyanin Poisson bracket determined by the standard classical r-matrix. Here we illustrate a method to obtain the Lie bialgebra from a Poisson bialgebra $U(sp_4)^{\circ}$.

• 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.

• 대한수학회지
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• 제47권4호
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• pp.705-718
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• 2010

It gives a method to obtain a natural Lie bialgebra from a Poisson bialgebra by an algebraic point of view. Let g be a coboundary Lie bialgebra associated to a Poission Lie group G. As an application, we obtain a Lie bialgebra from a sub-Poisson bialgebra of the restricted dual of the universal enveloping algebra U(g).