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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. ==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''×''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, ''Lg'' denotes left-multiplication and ''Rg'' denotes right-multiplication. If denotes the corresponding Poisson bivector on ''G'', the condition above can be equivalently stated as : Note that for Poisson-Lie group always , or equivalently . This means that non-trivial Poisson-Lie structure is never symplectic, not even of constant rank. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Poisson–Lie group」の詳細全文を読む スポンサード リンク
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