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 {\displaystyle G} equipped with a Poisson bracket for which the group multiplication μ : G × G → G {\displaystyle \mu :G\times G\to G} with μ ( g 1 , g 2 ) = g 1 g 2 {\displaystyle \mu (g_{1},g_{2})=g_{1}g_{2}} is a Poisson map, where the manifold G × G {\displaystyle G\times G} has been given the structure of a product Poisson manifold. Explicitly, the following identity must hold for a Poisson–Lie group:
{ f 1 , f 2 } ( g g ′ ) = { f 1 ∘ L g , f 2 ∘ L g } ( g ′ ) + { f 1 ∘ R g ′ , f 2 ∘ R g ′ } ( g ) {\displaystyle \{f_{1},f_{2}\}(gg')=\{f_{1}\circ L_{g},f_{2}\circ L_{g}\}(g')+\{f_{1}\circ R_{g^{\prime }},f_{2}\circ R_{g'}\}(g)}
where f 1 {\displaystyle f_{1}} and f 2 {\displaystyle f_{2}} are real-valued, smooth functions on the Lie group, while g {\displaystyle g} and g ′ {\displaystyle g'} are elements of the Lie group. Here, L g {\displaystyle L_{g}} denotes left-multiplication and R g {\displaystyle R_{g}} denotes right-multiplication. If P {\displaystyle {\mathcal {P}}} denotes the corresponding Poisson bivector on G {\displaystyle G} , the condition above can be equivalently stated as
P ( g g ′ ) = L g ∗ ( P ( g ′ ) ) + R g ′ ∗ ( P ( g ) ) {\displaystyle {\mathcal {P}}(gg')=L_{g\ast }({\mathcal {P}}(g'))+R_{g'\ast }({\mathcal {P}}(g))}
In particular, taking g = g ′ = e {\displaystyle g=g'=e} one obtains P ( e ) = 0 {\displaystyle {\mathcal {P}}(e)=0} , or equivalently { f , g } ( e ) = 0 {\displaystyle \{f,g\}(e)=0} . Applying Weinstein splitting theorem to e {\displaystyle e} one sees that non-trivial Poisson-Lie structure is never symplectic, not even of constant rank.
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