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. See also Lie coalgebra Manin triple References H.-D. Doebner, J.-D. Hennig, eds, Quantum groups, Proceedings of the 8th International Workshop on Mathematical Physics, Arnold Sommerfeld Institute, Claausthal, FRG, 1989, Springer-Verlag Berlin, . Vyjayanthi Chari and Andrew Pressley, A Guide to Quantum Groups, (1994), Cambridge University Press, Cambridge . Lie algebras Coalgebras Symplectic geometry