Jacobi identity

In mathematics the Jacobi identity is a property a binary operation can have that 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. It is named after the German mathematician Carl Gustav Jakob Jacobi.

Definition

A binary operation × on a set S possessing a binary operation + with additive identity denoted 0 satisfies the Jacobi identity if

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

That is, the sum of all even permutations of (a,(b,c)) must be zero. (Where the permutation is done by leaving the parentheses fixed and interchanging letters an even number of times.)
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 ring commutator.

The Jacobi Identity

\( [A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0 \, \)

which may be rewritten in the following form by antisymmetry,

\( [ [A , B] , C ] = [A , [B , C]] - [ B , [A , C]]~. \)

This formula then can be expatiated on in plain words,

"The infinitesimal motion of B followed by the infinitesimal motion of A ([A,[B,⋅]]), minus the infinitesimal motion of A followed by the infinitesimal motion of B ([B,[A,⋅]]), is the infinitesimal motion of [A,B] ([[A,B],⋅]), 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, 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.
See also

Super Jacobi identity
Three subgroups lemma (Hall-Witt identity)

External links

Weisstein, Eric W., "Jacobi Identities", MathWorld.

Mathematics Encyclopedia