• Journal of the Chungcheong Mathematical Society
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• v.21 no.1
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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 $U(sp_4)^{\circ}$.

• Journal of the Korean Mathematical Society
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• v.47 no.4
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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).

• Communications of the Korean Mathematical Society
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• v.22 no.3
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• pp.353-357
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• 2007

Let $(g,\;\delta)$ be a Lie bialgebra. Here we give an explicit formula for the Poisson bracket on a subalgebra of $U(g)^{\circ}$ induced by the given cobracket $\delta$.

• Journal of the Chungcheong Mathematical Society
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• v.28 no.2
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• pp.307-310
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• 2015

Let H be a color Poisson bialgebra. Here we find a canonical nondegenerate bilinear form $\mathfrak{g}(H){\times}\mathfrak{m/m^2}$, where $\mathfrak{g}(H)$ and $\mathfrak{m/m^2}$ are certain color Lie algebras induced by H.

• Bulletin of the Korean Mathematical Society
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• v.48 no.1
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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.