Poisson algebras and Yang-Baxter equations

Details Poisson algebras and Yang-Baxter equations Poisson algebras and Yang-Baxter equations

2006-12-18

arxiv.org/abs/math/0612493v2

Mathematics

Quantum Algebra

12 pages; completely rewritten from v1

Scientific paper

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.

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