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Poisson AlgebraA Poisson algebra is an associative algebra together with a Lie bracket, satisfying Leibniz' law. More precisely, a Poisson algebra is a vector space over a field K equipped with two bilinear products, and , such that forms an associative K-algebra and ,, called the Poisson bracket, forms a Lie algebra, and for any three elements x, y and z, yz = yz + yz (i.e. the Poisson bracket acts as a derivation). Examples - The space of smooth functions over a symplectic manifold.
- If A is a noncommutative associative algebra, then the commutator x,y≡xy−yx turns it into a Poisson algebra.
See also Poisson manifold, Poisson superalgebra, antibracket algebra
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