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Concept
Jacobi identity
Summary
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.
Let and be two binary operations, and let be the neutral element for . The is
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 .
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.
Official source
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