Poisson algebra
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A 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, <math>\cdot<math> and [,] such that <math>\cdot<math> forms an associative K-algebra and [,], called the Poisson bracket, forms a Lie algebra, and for any three elements x, y and z, [x, yz] = [x, y]z + y[x, z] (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.
- For a vertex operator algebra <math>(V,Y, \omega, 1)<math>, the space <math>V/C_2(V)<math> is a poission algebra with <math>\{a,b\}=a_0b<math> and <math>a \cdot b =a_{-1}b<math>. For certain vertex operator algebras, these Poisson alegbras are finite dimensional.
See also
Poisson manifold, Poisson superalgebra, antibracket algebraTemplate:Math-stub