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Dual Lie bialgebra structures of Poisson types

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Abstract

Let \(\mathcal{A} = \mathbb{F}[x,y]\) be the polynomial algebra on two variables x, y over an algebraically closed field \(\mathbb{F}\) of characteristic zero. Under the Poisson bracket, \(\mathcal{A}\) is equipped with a natural Lie algebra structure. It is proven that the maximal good subspace of \(\mathcal{A}*\) induced from the multiplication of the associative commutative algebra \(\mathcal{A}\) coincides with the maximal good subspace of \(\mathcal{A}*\) induced from the Poisson bracket of the Poisson Lie algebra \(\mathcal{A}\). Based on this, structures of dual Lie bialgebras of the Poisson type are investigated. As by-products, five classes of new infinite-dimensional Lie algebras are obtained.

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Authors and Affiliations

  1. College of Mathematics and Information Science, Shandong Institute of Business and Technology, Yantai, 264005, China

    Guang’Ai Song

  2. Department of Mathematics, Tongji University, Shanghai, 200092, China

    YuCai Su

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  1. Guang’Ai Song
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  2. YuCai Su
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Correspondence to YuCai Su.

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Song, G., Su, Y. Dual Lie bialgebra structures of Poisson types. Sci. China Math. 58, 1151–1162 (2015). https://doi.org/10.1007/s11425-015-4991-7

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  • DOI: https://doi.org/10.1007/s11425-015-4991-7

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