Solutions of the classical Yang-Baxter equation and noncommutative deformations

Details Solutions of the classical Yang-Baxter equation and noncommutative deformations Solutions of the classical Yang-Baxter equation and noncommutative deformations

2005-11-20

arxiv.org/abs/math/0511500v1

Mathematics

Symplectic Geometry

17 pages

Scientific paper

In this paper, I will show that, if a Lie algebra $\G$ acts on a manifold $P$, any solution of the classical Yang-Baxter equation on $\G$ gives arise to a Poisson tensor on $P$ and a torsion-free and flat contravariant connection (with respect to the Poisson tensor). Moreover, if the action is locally free, the matacurvature of the above contravariant connection vanishes. This will permit to get a large class of manifolds which satisfy the necessary conditions, presented by Hawkins in math.QA/0504232, to the existence of a noncommutative deformation.

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