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In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz' law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central in the study of quantum groups. Manifolds with a Poisson algebra structure are known as Poisson manifolds, of which the symplectic manifolds and the Poisson-Lie groups are a special case. The algebra is named in honour of Siméon Denis Poisson. ==Definition== A Poisson algebra is a vector space over a field ''K'' equipped with two bilinear products, ⋅ and , having the following properties: * The product ⋅ forms an associative ''K''-algebra. * The product , called the Poisson bracket, forms a Lie algebra, and so it is anti-symmetric, and obeys the Jacobi identity. * The Poisson bracket acts as a derivation of the associative product ⋅, so that for any three elements ''x'', ''y'' and ''z'' in the algebra, one has = ⋅ ''z'' + ''y'' ⋅ . The last property often allows a variety of different formulations of the algebra to be given, as noted in the examples below. 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 algebra」の詳細全文を読む スポンサード リンク
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