Summary: PREPOISSON ALGEBRAS
Abstract. A definition of prepoisson algebras is proposed, combining structures of prelie and zinbiel
algebra on the same vector space. It is shown that a prepoisson 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 prepoisson algebras. The Koszul
dual of prepoisson algebras is described. It is explained how one may associate a prepoisson algebra
to any Poisson algebra equipped with a Baxter operator, and a dual prepoisson algebra to any Poisson
algebra equippedwith an averaging operator. Examples of this construction are given. It is shown that
the free zinbiel algebra (the shuffle algebra) on a prelie algebra is a prepoisson algebra. A connection
between the graded version of this result and the classical YangBaxter equation is discussed.
Acknowledgements. The author thanks Steve Chase, Andr'e Joyal, Muriel Livernet and JeanMichel
Oudom for stimulating conversations and suggestions.
1. Prelie and zinbiel algebras
Throughout the paper, we deal with the left version of each type of algebras. All definitions and
results admit a right version.
A left prelie algebra is a vector space A together with a bilinear map ffi : A \Theta A ! A such that
x ffi (y ffi z) \Gamma (x ffi y) ffi z = y ffi (x ffi z) \Gamma (y ffi x) ffi z :