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Letters in Mathematical Physics

, Volume 54, Issue 4, pp 263–277 | Cite as

Pre-Poisson Algebras

  • Marcelo Aguiar
Article

Abstract

A definition of pre-Poisson algebras is proposed, combining structures of pre-Lie and zinbiel algebra on the same vector space. It is shown that a pre-Poisson algebra gives rise to a Poisson algebra by passing to the corresponding Lie and commutative products. Analogs of basic constructions of Poisson algebras (through deformations of commutative algebras, or from filtered algebras whose associated graded algebra is commutative) are shown to hold for pre-Poisson algebras. The Koszul dual of pre-Poisson algebras is described. It is explained how one may associate a pre-Poisson algebra to any Poison algebra equipped with a Baxter operator, and a dual pre-Poisson algebra to any Poisson algebra equipped with an averaging operator. Examples of this construction are given. It is shown that the free zinbiel algebra (the shuffle algebra) on a pre-Lie algebra is a pre-Poisson algebra. A connection between the graded version of this result and the classical Yang–Baxter equation is discussed.

Poisson algebras pre-Lie algebras zinbiel algebras dendriform algebras Baxter operators shuffle algebra Gerstenhaber algebras classical Yang–Baxter equation associative Yang–Baxter equation 
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Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Marcelo Aguiar
    • 1
  1. 1.Centre de Recherches MathématiquesUniversité de MontréalMontréalCanada

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