  We connect generalizations of Poisson algebras with the classical and
associative Yang-Baxter equations. In particular, we prove that solutions of
the classical Yang-Baxter equation on a vector space V are equivalent to
``twisted'' Poisson algebra structures on the tensor algebra TV. Here,
``twisted'' refers to working in the category of graded vector spaces equipped
with S_n-actions in degree n. We show that the associative Yang-Baxter equation
is similarly related to the double Poisson algebras of Van den Bergh. We
generalize to L-infinity-algebras and define ``infinity'' versions of
Yang-Baxter equations and double Poisson algebras. The proofs are based on the
observation that Lie is essentially unique among quadratic operads having a
certain distributivity property over the commutative operad; we also give a
L-infinity generalization. In the appendix, we prove a generalized version of
Schur-Weyl duality, which is related to the use of nonstandard S_n-module
structures on the n-th tensor power of V.
