Poisson–Lie group

Definition

A Poisson–Lie group is a Lie group G equipped with a Poisson bracket for which the group multiplication μ : G × G → G with μ ( g 1 , g 2 ) = g 1 g 2 is a Poisson map, where the manifold G×G has been given the structure of a product Poisson manifold.

Explicitly, the following identity must hold for a Poisson–Lie group:

where f₁ and f₂ 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 P denotes the corresponding Poisson bivector on G, the condition above can be equivalently stated as

Note that for Poisson-Lie group always { f , g } ( e ) = 0 , or equivalently P ( e ) = 0 . This means that non-trivial Poisson-Lie structure is never symplectic, not even of constant rank.

Homomorphisms

A Poisson–Lie group homomorphism ϕ : G → 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 ι : G → G taking ι ( g ) = g − 1 is not a Poisson map either, although it is an anti-Poisson map:

for any two smooth functions f 1 , f 2 on G.