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

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

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:

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

Defining the adjoint map

$\operatorname {ad} _{x}:y\mapsto [x,y],$

permits two equivalent formulations of the Jacobi identity. After a rearrangement, 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 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

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

which implies that the map sending each element to its adjoint action is a Lie algebra homomorphism into the algebra of derivations of the algebra. This latter property gives rise to the adjoint representation.

A similar identity called the Hall-Witt identity exists for commutators of 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