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.
The cross product and the Lie bracket operation 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.
Notice the pattern in the variables on the left side of this identity. In each subsequent expression of the form , the variables , and are permuted according to the cycle . Alternatively, we may observe that the ordered triples , and , are the even permutations of the ordered triple .
Commutator bracket form
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 × operation is the commutator, which measures the failure of commutativity in matrix multiplication. Instead of , 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: belongs to V for all , the Jacobi identity continues to hold on V. 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.
If is the action of the infinitesimal motion X on Z, that can be stated as:
The action of Y followed by X (operator ), minus the action of X followed by Y (operator ), is equal to the action of , (operator ).
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.
- The following identity follows from anticommutativity and Jacobi identity and holds in arbitrary Lie algebra:
- The Jacobi identity is equivalent to the Product Rule, with the Lie bracket acting as both a product and a derivative: . If are vector fields, then is literally a derivative operator acting on , namely the Lie derivative .
- Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666.