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In [[mathematics]] the '''Jacobi identity''' is a property a [[binary operation]] can have that determines how the order of evaluation behaves for the given operation. Unlike for [[associativity|associative operations]], order of evaluation is significant for operations satisfying Jacobi identity.  It is named after the [[Germany|German]] [[mathematician]] [[Carl Gustav Jakob Jacobi]].
 
==Definition==
A binary operation ''×'' on a [[Set (mathematics)|set]] ''S'' possessing a binary operation ''+'' with additive identity denoted 0 satisfies the Jacobi identity if
 
: <math>a \times (b \times c) + c \times (a \times b) + b \times (c \times a) = 0\quad \forall{a,b,c}\in S.</math>
 
That is, the sum of all even permutations of (a,(b,c)) must be zero. (Where the permutation is done by leaving the parentheses fixed and interchanging letters an even number of times.)
 
==Interpretation==
In a [[Lie algebra]], the objects that obey the Jacobi identity are infinitesimal motions. When acting on an operator with an infinitesimal motion, the change in the operator is the [[commutator]].
 
The Jacobi Identity
::<math>
[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0
\,</math>
which can be changed into the following form by [[Bilinearity]] and [[Alternating]].
::<math>
[ [A , B] , C  ] = [A , [B , C]] - [ B , [A , C]]
\,</math>
 
This formula can be expatiated with plain words: "the infinitesimal motion of B followed by the infinitesimal motion of A ([A,[B,⋅]]), minus the infinitesimal motion of A followed by the infinitesimal motion of B ([B,[A,⋅]]), is the infinitesimal motion of [A,B] ([[A,B],⋅]), when acting on any arbitrary infinitesimal motion C (thus, these are equal)".
 
==Examples==
The Jacobi identity is satisfied by the multiplication (bracket) operation on [[Lie Algebra|Lie algebras]] and [[Lie ring]]s and these provide the majority of examples of operations satisfying the Jacobi identity in common use. Because of this the Jacobi identity is often expressed using Lie bracket notation:
 
: <math>[x,[y,z]] + [z,[x,y]] + [y,[z,x]] = 0.</math>
 
If the multiplication is [[anticommutativity|antisymmetric]], the Jacobi identity admits two equivalent reformulations. Defining the [[adjoint representation of a Lie algebra|adjoint map]]
:<math>\operatorname{ad}_x: y \mapsto [x,y],</math>
after a rearrangement, the identity becomes
:<math>\operatorname{ad}_x[y,z]=[\operatorname{ad}_xy,z]+[y,\operatorname{ad}_xz].</math>
Thus, the Jacobi identity for Lie algebras simply becomes the assertion that the action of any element on the algebra is a [[derivation (abstract algebra)|derivation]]. This form of the Jacobi identity is also used to define the notion of [[Leibniz algebra]].
 
Another rearrangement shows that the Jacobi identity is equivalent to the following identity between the operators of the [[Adjoint representation of a Lie algebra|adjoint representation]]:
:<math>\operatorname{ad}_{[x,y]}=[\operatorname{ad}_x,\operatorname{ad}_y].</math>
This identity implies that the map sending each element to its adjoint action is a [[Lie algebra homomorphism]] of the original algebra into the Lie algebra of its derivations.
 
A similar identity, called the [[Commutators#Identities|Hall–Witt identity]], exists for the [[commutator]]s in [[group (mathematics)|groups]].
 
In [[analytical mechanics]], Jacobi identity is satisfied by [[Poisson bracket]]s. In the [[Copenhagen interpretation]] of [[quantum mechanics]], it is satisfied by operator [[Commutator#Ring theory|commutator]]s on [[Hilbert space]] and, equivalently, in the [[phase space formulation]] of quantum mechanics by the [[Moyal bracket]].
 
==See also==
* [[Super Jacobi identity]]
* [[Hall-Witt identity]]
 
== External links ==
*{{MathWorld|JacobiIdentities|Jacobi Identities}}
 
{{DEFAULTSORT:Jacobi Identity}}
[[Category:Lie algebras]]
[[Category:Mathematical identities]]
[[Category:Non-associative algebra]]

Latest revision as of 22:18, 29 December 2014

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