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 G equipped with a Poisson bracket for which the group multiplication with is a Poisson map, where the manifold G\times G has been given the structure of a product Poisson manifold. Explicitly, the following identity must hold for a Poisson–Lie group: where f_1 and f_2 are real-valued, smooth functions on the Lie group, while g and g' are elements of the Lie group. Here, L_g denotes left-multiplication and R_g denotes right-multiplication. If \mathcal{P} denotes the corresponding Poisson bivector on G, the condition above can be equivalently stated as In particular, taking g = g' = e one obtains, or equivalently. Applying Weinstein splitting theorem to e 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 G has a dual Poisson–Lie group, defined as the Poisson–Lie group integrating the dual of its bialgebra.

Homomorphisms

A Poisson–Lie group homomorphism \phi:G\to H 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 f_1, f_2 on G.

Examples

Trivial examples

These two example are dual of each other via Drinfeld theorem, in the sense explained above.

Other examples

Let G be any semisimple Lie group. Choose a maximal torus T\subset G and a choice of positive roots. Let be the corresponding opposite Borel subgroups, so that and there is a natural projection. Then define a Lie group which is a subgroup of the product, and has the same dimension as G. The standard Poisson–Lie group structure on G is determined by identifying the Lie algebra of G^* with the dual of the Lie algebra of G, as in the standard Lie bialgebra example. This defines a Poisson–Lie group structure on both G and on the dual Poisson Lie group G^. This is the "standard" example: the Drinfeld-Jimbo quantum group is a quantization of the Poisson algebra of functions on the group G^. Note that G^* is solvable, whereas G is semisimple.