A Poisson-Lie group is a Lie group that has a compatible Poisson structure, meaning it possesses both the algebraic structure of a Lie group and the geometric structure of a Poisson manifold. This combination allows for the interaction between Lie group theory and classical mechanics, especially in describing Hamiltonian systems. The Poisson structure is typically expressed in terms of a bivector field on the group, which encodes the relationships between coordinates and momenta.
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