Jacobi identity

In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. Unlike for associative operations, order of evaluation is significant for operations satisfying Jacobi identity.

Definition

A binary operation $*$ on a set $S$ possessing a commutative binary operation $+$, satisfies the Jacobi identity if

: $a*(b*c)\; +\; c*(a*b)\; +\; b*(c*a)\; =\; 0quad\; forall\{a,b,c\}in\; S.$

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 can then be translated into words:

::$[\; [A\; ,\; B]\; ,\; C\; ]\; =\; [A\; ,\; [B\; ,\; C]\; -\; [\; B\; ,\; [A\; ,\; C]\; ,$

meaning "the infinitesimal motion of B followed by the infinitesimal motion of A ($[A,\; [B,cdot]$), minus the infinitesimal motion of A followed by the infinitesimal motion of B ($[B,\; [A,cdot]$), is the infinitesimal motion of [A,B] ($[A,B]\; ,cdot]$), 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 algebras and Lie rings 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:

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

If the multiplication is antisymmetric, the Jacobi identity admits two equivalent reformulations. Defining the adjoint map:$operatorname\{ad\}\_x:\; y\; mapsto\; [x,y]\; ,$after a rearrangement, the identity becomes:$operatorname\{ad\}\_x\; [y,z]\; =\; [operatorname\{ad\}\_xy,z]\; +\; [y,operatorname\{ad\}\_xz]\; .$Thus, the Jacobi identity for Lie algebras simply becomes the assertion 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::$operatorname\{ad\}\_\{\; [x,y]\; \}=\; [operatorname\{ad\}\_x,operatorname\{ad\}\_y]\; .$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 Hall-Witt identity, exists for the commutators in groups.

In analytical mechanics, Jacobi identity is satisfied by Poisson brackets, while in quantum mechanics it is satisfied by operator commutators.

See also

* Super Jacobi identity
* Hall-Witt identity