Poisson-Lie group

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 algebra of a Poisson–Lie group is a Lie bialgebra.

A Poisson–Lie group is a Lie group G equipped with a Poisson bracket for which the group multiplication ${\displaystyle \mu :G\times G\to G}$ with ${\displaystyle \mu (g_{1},g_{2})=g_{1}g_{2}}$ 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.

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