Relation To Poisson-Lie Groups
Let G be a Poisson-Lie group, with being two smooth functions on the group manifold. Let be the differential at the identity element. Clearly, . The Poisson structure on the group then induces a bracket on, as
where is the Poisson bracket. Given be the Poisson bivector on the manifold, define to be the right-translate of the bivector to the identity element in G. Then one has that
The cocommutator is then the tangent map:
so that
is the dual of the cocommutator.
Read more about this topic: Lie Bialgebra
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