In mathematics, a Poisson–Lie group is a Poisson manifold that is also a Lie group, with the group multiplication being compatible with the Poisson algebra structure on the manifold. The infinitesimal counterpart of a Poisson–Lie group is a Lie bialgebra, in analogy to Lie algebras as the infinitesimal counterparts of Lie groups. Many quantum groups are quantizations of the Poisson algebra of functions on a Poisson–Lie group. Definition A Poisson–Lie group is a Lie group equipped with a Poisson bracket for which the group multiplication with is a Poisson map, where the manifold has been given the structure of a product Poisson manifold. Explicitly, the following identity must hold for a Poisson–Lie group: where and are real-valued, smooth functions on the Lie group, while and are elements of the Lie group. Here, denotes left-multiplication and denotes right-multiplication. If denotes the corresponding Poisson bivector on , the condition above can be equivalently stated as In particular, taking one obtains , or equivalently . Applying Weinstein splitting theorem to one sees that non-trivial Poisson-Lie structure is never symplectic, not even of constant rank. Poisson-Lie groups - Lie bialgebra correspondence The Lie algebra of a Poisson–Lie group has a natural structure of Lie coalgebra given by linearising the Poisson tensor at the identity, i.e. is a comultiplication. Moreover, the algebra and the coalgebra structure are compatible, i.e. is a Lie bialgebra, The classical Lie group–Lie algebra correspondence, which gives an equivalence of categories between simply connected Lie groups and finite-dimensional Lie algebras, was extended by Drinfeld to an equivalence of categories between simply connected Poisson–Lie groups and finite-dimensional Lie bialgebras. Thanks to Drinfeld theorem, any Poisson–Lie group has a dual Poisson–Lie group, defined as the Poisson–Lie group integrating the dual of its bialgebra. Homomorphisms A Poisson–Lie group homomorphism is defined to be both a Lie group homomorphism and a Poisson map. Although this is the "obvious" definition, neither left translations nor right translations are Poisson maps. Also, the inversion map taking is not a Poisson map either, although it is an anti-Poisson map: for any two smooth functions on . Examples Trivial examples Any trivial Poisson structure on a Lie group defines a Poisson–Lie group structure, whose bialgebra is simply with the trivial comultiplication. The dual of a Lie algebra, together with its linear Poisson structure, is an additive Poisson–Lie group. These two example are dual of each other via Drinfeld theorem, in