Abstract
The Hopf dual H° of any Poisson Hopf algebra H is proved to be a co-Poisson Hopf algebra provided H is noetherian. Without noetherian assumption, unlike it is claimed in literature, the statement does not hold. It is proved that there is no nontrivial Poisson Hopf structure on the universal enveloping algebra of a non-abelian Lie algebra. So the polynomial Hopf algebra, viewed as the universal enveloping algebra of a finite-dimensional abelian Lie algebra, is considered. The Poisson Hopf structures on polynomial Hopf algebras are exactly linear Poisson structures. The co-Poisson structures on polynomial Hopf algebras are characterized. Some correspondences between co-Poisson and Poisson structures are also established.
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Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 11331006 and 11171067). The authors thank Ruipeng Zhu for useful discussions.
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Lou, Q., Wu, Q. Co-Poisson structures on polynomial Hopf algebras. Sci. China Math. 61, 813–830 (2018). https://doi.org/10.1007/s11425-016-9075-6
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DOI: https://doi.org/10.1007/s11425-016-9075-6