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Jacobi identity
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{{Short description|Property of some binary operations}} In [[mathematics]], the '''Jacobi identity''' is a property of a [[binary operation]] that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the [[Associativity|associative property]], any order of evaluation gives the same result (parentheses in a multiple product are not needed). The identity is named after the German mathematician [[Carl Gustav Jacob Jacobi]]. He derived the Jacobi identity for [[Poisson bracket]]s in his 1862 paper on differential equations.<ref name="Poisson1809">[[#jacobi1862|C. G. J. Jacobi (1862), Β§26, Theorem V.]]</ref><ref name="Hawkins1991">[[#hawkins1991|T. Hawkins (1991)]]</ref> The [[cross product]] <math>a\times b</math> and the [[Lie algebra|Lie bracket operation]] <math>[a,b]</math> both satisfy the Jacobi identity.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Jacobi Identities |url=https://mathworld.wolfram.com/JacobiIdentities.html |access-date=2025-01-31 |website=mathworld.wolfram.com |language=en |quote=The elements of a Lie algebra satisfy this identity.}}</ref> In [[analytical mechanics]], the Jacobi identity is satisfied by the [[Poisson bracket]]s. In [[quantum mechanics]], it is satisfied by operator [[Commutator#Ring theory|commutator]]s on a [[Hilbert space]] and equivalently in the [[phase space formulation]] of quantum mechanics by the [[Moyal bracket]].
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