Editing Jacobi identity

{{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), §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]].

== Definition ==
Let <math>+</math> and <math>\times</math> be two [[binary operation]]s, and let <math>0</math> be the [[neutral element]] for <math>+</math>. The '''{{visible anchor|Jacobi identity}}''' is

:<math>x \times (y \times z) \ +\ y \times (z \times x)  \ +\ z \times (x \times y)\ =\ 0.</math>

Notice the pattern in the variables on the left side of this identity. In each subsequent expression of the form <math>a \times (b \times c)</math>, the variables <math>x</math>, <math>y</math> and <math>z</math> are permuted according to the cycle <math>x \mapsto y \mapsto z \mapsto x</math>. Alternatively, we may observe that the ordered triples <math>(x,y,z)</math>, <math>(y,z,x)</math> and <math>(z,x,y)</math>, are the [[Permutation#Parity of a permutation|even permutations]] of the ordered triple <math>(x,y,z)</math>.

==Commutator bracket form==
The simplest informative example of a [[Lie algebra]] is constructed from the (associative) ring of <math>n\times n</math> matrices, which may be thought of as infinitesimal motions of an ''n''-dimensional vector space. The × operation is the [[commutator]], which measures the failure of commutativity in matrix multiplication. Instead of <math>X\times Y</math>, the Lie bracket notation is used:
:<math>[X,Y]=XY-YX.</math>

In that notation, the Jacobi identity is:
:<math>[X, [Y, Z] ] + [Y, [Z, X] ] + [Z, [X, Y] ] \ =\  0</math>
That is easily checked by computation.

More generally, if '''{{math|A}}''' is an associative algebra and '''{{mvar|V}}''' is a subspace of '''{{math|A}}''' that is closed under the bracket operation: <math>[X,Y]=XY-YX</math> belongs to '''{{mvar|V}}''' for all <math>X,Y\in V</math>, the Jacobi identity continues to hold on '''{{mvar|V}}'''.<ref>{{harvnb|Hall|2015}} Example 3.3</ref> Thus, if a binary operation <math>[X,Y]</math> satisfies the Jacobi identity, it may be said that it behaves as if it were given by <math>XY-YX</math> in some associative algebra even if it is not actually defined that way.

Using the [[anticommutativity|antisymmetry property]] <math>[X,Y]=-[Y,X]</math>, the Jacobi identity may be rewritten as a modification of the [[associativity|associative property]]:

:<math>[[X, Y], Z] = [X, [Y, Z]] - [Y, [X, Z]]~.</math>

If <math>[X,Z]</math> is the action of the infinitesimal motion '''{{mvar|X}}''' on '''{{mvar|Z}}''', that can be stated as: 
{{blockquote
 | The action of ''Y'' followed by ''X'' (operator <math>[X,[Y,\cdot\ ] ]</math>), minus the action of ''X'' followed by ''Y'' (operator <math>([Y,[X,\cdot\ ] ]</math>), is equal to the action of <math>[X,Y]</math>, (operator <math>[ [X,Y],\cdot\ ]</math>).
|sign=|source=}}

There is also a plethora of [[Lie superalgebra#properties|graded Jacobi identities]] involving [[anticommutator]]s <math>\{X,Y\}</math>, such as:

:<math>
[\{X,Y\},Z]+ [\{Y,Z\},X]+[\{Z,X\},Y] =0,\qquad
[\{X,Y\},Z]+ \{[Z,Y],X\}+\{[Z,
X],Y\} =0.
</math>
{{See also|Lie bracket of vector fields|Baker–Campbell–Hausdorff formula}}

==Adjoint form==
Most common examples of the Jacobi identity come from the bracket multiplication <math>[x,y]</math> on [[Lie algebra]]s and [[Lie ring]]s. The Jacobi identity is written as:

: <math>[x,[y,z]] + [z,[x,y]] + [y,[z,x]] = 0.</math>

Because the bracket multiplication is [[anticommutativity|antisymmetric]], the Jacobi identity admits two equivalent reformulations. Defining the [[adjoint representation of a Lie algebra|adjoint operator]] <math>\operatorname{ad}_x: y \mapsto [x,y]</math>, 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 states that the action of any element on the algebra is a [[derivation (abstract algebra)|derivation]]. That 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:
:<math>\operatorname{ad}_{[x,y]}=[\operatorname{ad}_x,\operatorname{ad}_y].</math>

There, the bracket on the left side is the operation of the original algebra, the bracket on the right is the commutator of the composition of operators, and the identity states that the <math>\mathrm{ad}</math> map sending each element to its adjoint action is a [[Lie algebra homomorphism]].

== Related identities ==

* The [[Commutator#Identities (group theory)|Hall–Witt identity]] is the analogous identity for the [[commutator]] operation in a [[group (mathematics)|group]].

* The following higher order Jacobi identity holds in arbitrary Lie algebra:<ref>{{cite arXiv
 | first1=Ilya |last1=Alekseev | first2=Sergei O. |last2=Ivanov |eprint=1604.05281
 | title       = Higher Jacobi Identities |date=18 April 2016
 |class=math.GR }}</ref>

: <math>[[[x_1,x_2],x_3],x_4]+[[[x_2,x_1],x_4],x_3]+[[[x_3,x_4],x_1],x_2]+[[[x_4,x_3],x_2],x_1] = 0.</math>

* The Jacobi identity is equivalent to the [[Product Rule]], with the Lie bracket acting as both a product and a derivative:  <math>[X,[Y,Z]] = [[X,Y], Z] + [Y, [X,Z]]</math>.   If <math>X, Y</math> are vector fields, then <math>[X,Y]</math> is literally a derivative operator acting on <math>Y</math>, namely the [[Lie derivative]] <math>\mathcal{L}_X Y</math>.

==See also==
* [[Structure constants]]
* [[Super Jacobi identity]]
* [[Three subgroups lemma]] (Hall–Witt identity)
* [[Quintuple product identity]]

==References==
{{Reflist}}
* {{citation|first=Brian C.|last=Hall|title=Lie Groups, Lie Algebras, and Representations: An Elementary Introduction|edition= 2nd|series=Graduate Texts in Mathematics|volume=222 |publisher=Springer|year=2015|isbn=978-3319134666}}.
*{{Cite journal|first1=C. G. J.|last1=Jacobi|authorlink1=Carl Gustav Jacob Jacobi|title=Nova methodus, aequationes differentiales partiales primi ordinis inter numerum variabilium quemcunque propositas integrandi|journal=Journal für die reine und angewandte Mathematik|year=1862|volume=60|page=1-181|url=https://eudml.org/doc/147847|ref=jacobi1862}}
*{{Cite journal|first1=Thomas|last1=Hawkins|authorlink1=Thomas W. Hawkins Jr.|title=Jacobi and the Birth of Lie's Theory of Groups|journal=Arch. Hist. Exact Sci.|year=1991|volume=42|issue=3 |page=187-278|doi=10.1007/BF00375135|ref=hawkins1991}}

== External links ==
*{{MathWorld|JacobiIdentities|Jacobi Identities}}

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[[Category:Lie algebras]]
[[Category:Mathematical identities]]
[[Category:Non-associative algebra]]
[[Category:Properties of binary operations]]