Solutions of the Classical Yang–Baxter Equation and Noncommutative Deformations

Abstract

We show that given a finite-dimensional real Lie algebra $${\mathcal{G}}$$ acting on a smooth manifold P then, for any solution of the classical Yang–Baxter equation on $${\mathcal{G}}$$ , there is a canonical Poisson tensor on P and an associated canonical torsion-free and flat contravariant connection. Moreover, we prove that the metacurvature of this contravariant connection vanishes if the isotropy Lie subalgebras of the action are trivial. Those results permit to get a large class of smooth manifolds satisfying the necessary conditions, introduced by Eli Hawkins, to the existence of noncommutative deformations.