Jacobi identity

In mathematics the Jacobi identity is a property of a binary operation which 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 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 Jakob Jacobi.

The cross product $a\times b$ and the Lie bracket operation $[a,b]$ both satisfy the Jacobi identity. In analytical mechanics, the Jacobi identity is satisfied by the Poisson brackets. 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.

Definition

Consider a set A with two binary operations + and × , with an additive identity 0. This satisfies the Jacobi identity if:

$x\times (y\times z)\ +\ y\times (z\times x)\ +\ z\times (x\times y)\ =\ 0\quad \forall \ {x,y,z}\in A.$

The left side is the sum of all even permutations of x × (y × z): that is, we leave the parentheses fixed and interchange letters an even number of times.

Commutator bracket form

The simplest example of a Lie algebra is constructed from the (associative) ring of $n\times n$  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 $X\times Y$ , one uses the Lie bracket notation:

$[X,Y]=XY-YX.$

In this notation, the Jacobi identity is:

$[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]\ =\ 0.$

This is easily checked by a computation.

More generally, suppose A is an associative algebra and V is a subspace of A which is closed under the bracket operation: $[X,Y]=XY-YX$  belongs to V for all $X,Y\in V$ . Then the Jacobi identity continues to hold on V. Thus, if a binary operation $[X,Y]$  satisfies the Jacobi identity, we may say 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 can be rewritten as a modification of the associative property:

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

Considering $[X,Z]$  as the action of the infinitesimal motion X on Z, this can be stated as:

The action of Y followed by X (operator $[X,[Y,\cdot \ ]]$ ), minus the action of X followed by Y (operator $([Y,[X,\cdot \ ]]$ ), is equal to the action of $[X,Y]$ , (operator $[[X,Y],\cdot \ ]$ ).

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

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

The majority of common examples of the Jacobi identity come from the bracket multiplication $[x,y]$  on Lie algebras and Lie rings. 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 $\operatorname {ad} _{x}:y\mapsto [x,y]$ , the identity becomes:

$\operatorname {ad} _{x}[y,z]=[\operatorname {ad} _{x}y,z]+[y,\operatorname {ad} _{x}z].$

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

$\operatorname {ad} _{[x,y]}=[\operatorname {ad} _{x},\operatorname {ad} _{y}].$

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

Related identities

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

The following identitity follows from anticommutativity and Jacobi identity and holds in arbitrary Lie algebra:

$[x,[y,[z,w]]]+[y,[x,[w,z]]]+[z,[w,[x,y]]]+[w,[z,[y,x]]]=0.$