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 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 Jacob Jacobi.
He derived the Jacobi identity for Poisson brackets in his 1862 paper on differential equations.
and the Lie bracket operation
[3] 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.
The Jacobi identity is Notice the pattern in the variables on the left side of this identity.
In each subsequent expression of the form
Alternatively, we may observe that the ordered triples
, are the even permutations of the ordered triple
The simplest informative example of a Lie algebra is constructed from the (associative) ring of
matrices, which may be thought of as infinitesimal motions of an n-dimensional vector space.
, the Lie bracket notation is used: In that notation, the Jacobi identity is: That is easily checked by computation.
More generally, if A is an associative algebra and V is a subspace of A that is closed under the bracket operation:
, the Jacobi identity continues to hold on V.[4] Thus, if a binary operation
satisfies the Jacobi identity, it may be said that it behaves as if it were given by
in some associative algebra even if it is not actually defined that way.
, the Jacobi identity may be rewritten as a modification of the associative property: If
), minus the action of X followed by Y (operator
There is also a plethora of graded Jacobi identities involving anticommutators
, such as: Most common examples of the Jacobi identity come from the bracket multiplication
The Jacobi identity is written as: Because the bracket multiplication is antisymmetric, the Jacobi identity admits two equivalent reformulations.
Defining the adjoint operator
, the identity becomes: 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: 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
map sending each element to its adjoint action is a Lie algebra homomorphism.