Jacobi identity

 ( Lie Algebras ) 

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, 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 in his 1862 paper on differential equations.C. G. J. Jacobi (1862), §26, Theorem V.T. Hawkins (1991)

The cross product $a\backslash times\; b$ and the Lie algebra $a,b$ both satisfy the Jacobi identity. In analytical mechanics, the Jacobi identity is satisfied by the . In quantum mechanics, it is satisfied by operator commutators on a Hilbert space and equivalently in the phase space formulation of quantum mechanics by the Moyal bracket.

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Definition Let $+$ and $\backslash times$ be two , and let $0$ be the identity element for $+$. The is

$x\; \backslash times\; (y\; \backslash times\; z)\; \backslash \; +\backslash \; y\; \backslash times\; (z\; \backslash times\; x)\; \backslash \; +\backslash \; z\; \backslash times\; (x\; \backslash times\; y)\backslash \; =\backslash \; 0.$

Notice the pattern in the variables on the left side of this identity. In each subsequent expression of the form $a\; \backslash times\; (b\; \backslash times\; c)$, the variables $x$, $y$ and $z$ are permuted according to the cycle $x\; \backslash mapsto\; y\; \backslash mapsto\; z\; \backslash mapsto\; x$. Alternatively, we may observe that the ordered triples $(x,y,z)$, $(y,z,x)$ and $(z,x,y)$, are the even permutations of the ordered triple $(x,y,z)$.

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Commutator bracket form The simplest informative example of a Lie algebra is constructed from the (associative) ring of $n\backslash times\; n$ 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 $X\backslash times\; Y$, the Lie bracket notation is used:

$X,Y=XY-YX.$

In that notation, the Jacobi identity is:

$X,\; ]\; +\; Y,\; ]\; +\; Z,\; ]\; \backslash \; =\backslash \; 0$

That is easily checked by computation.

More generally, if ' is an associative algebra and ' is a subspace of ' that is closed under the bracket operation: $X,Y=XY-YX$ belongs to ' for all $X,Y\backslash in\; V$, the Jacobi identity continues to hold on . Example 3.3 Thus, if a binary operation $X,Y$ satisfies the Jacobi identity, it may be said that it behaves as if it were given by $XY-YX$ in some associative algebra even if it is not actually defined that way.

Using the antisymmetry property $X,Y=-Y,X$, the Jacobi identity may be rewritten as a modification of the associativity:

$[X,,\; Z]\; =\; X,]\; -\; Y,]~.$

If $X,Z$ is the action of the infinitesimal motion ' on ', that can be stated as:

There is also a plethora of graded Jacobi identities involving $\backslash \{X,Y\backslash \}$, such as:

$$

\{X,Y\},Z+ \{Y,Z\},X+\{Z,X\},Y =0,\qquad \{X,Y\},Z+ \{Z,Y,X\}+\{[Z, X],Y\} =0.

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Adjoint form Most common examples of the Jacobi identity come from the bracket multiplication $x,y$ on and . The Jacobi identity is written as:

$x,[y,z]\; +\; z,[x,y]\; +\; y,[z,x]\; =\; 0.$

Because the bracket multiplication is antisymmetric, the Jacobi identity admits two equivalent reformulations. Defining the adjoint operator $\backslash operatorname\{ad\}\_x:\; y\; \backslash mapsto\; x,y$, the identity becomes:

$\backslash operatorname\{ad\}\_xy,z=\backslash operatorname\{ad\}\_xy,z+y,\backslash operatorname\{ad\}\_xz.$

Thus, the Jacobi identity for Lie algebras states that the action of any element on the algebra is a 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:

$\backslash operatorname\{ad\}\_\{x,y\}=\backslash operatorname\{ad\}\_x,\backslash operatorname\{ad\}\_y.$

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 $\backslash mathrm\{ad\}$ map sending each element to its adjoint action is a Lie algebra homomorphism.

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Related identities

• The Hall–Witt identity is the analogous identity for the commutator operation in a group.

• The following higher order Jacobi identity holds in arbitrary Lie algebra:

$[[x\_1,x\_2,x\_3],x\_4]+[[x\_2,x\_1,x\_4],x\_3]+[[x\_3,x\_4,x\_1],x\_2]+[[x\_4,x\_3,x\_2],x\_1]\; =\; 0.$

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

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See also

• Structure constants
• Super Jacobi identity
• Three subgroups lemma (Hall–Witt identity)
• Quintuple product identity

• .

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External links

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