Duality of co-Poisson Hopf algebras
Bull. Korean Math. Soc. 2011 Vol. 48, No. 1, 17-21
https://doi.org/10.4134/BKMS.2011.48.1.17
Published online January 1, 2011
Sei-Qwon Oh and Hyung-Min Park
Chungnam National University, Chungnam National University
Abstract : 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 $\{f,g\}(x)=\langle\delta(x), f\otimes g\rangle$ 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 $U(\frak g)$ for a finite dimensional Lie bialgebra $\frak g$ is a Poisson Hopf algebra.
Keywords : co-Poisson Hopf algebra, Poisson Hopf algebra
MSC numbers : 17B62, 17B63, 16W30
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