Poisson-Lie group

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Definition

A Poisson-Lie group $G$ is a differential manifold $G$ equipped with the following two additional structures:

• The structure of a Lie group
• The structure of a Poisson manifold (in other words, a Poisson algebra to the set $C^{\alpha }(G)$)

satisfying the following compatibility conditions: the group multiplication $G\times G\to G$ is a Poisson map from $G\times G$ with the Poisson product structure, to $G$.

Relation with other structures

Weaker structures

• Poisson manifold
• Lie group