Co-Poisson structures on polynomial Hopf algebras

2016/01/17 by Qi Lou, Lou, Qi, Quanshui Wu +1
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1601.04269

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, it is not true in general. There is no nontrivial Poisson Hopf structure on the universal enveloping algebra of a non-abelian Lie algebra. The Poisson Hopf structures on A=k[x₁, x₂, ⋯, x_d], viewed as the universal enveloping algebra of a finite-dimensional abelian Lie algebra, are exactly linear Poisson structures on A. The co-Poisson structures on polynomial Hopf algebra A are characterized. Some correspondences between co-Poisson and Poisson structures are also established.

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