Jacobi identity

In mathematics the Jacobi identity is a property of a binary operation which 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 Jakob Jacobi. [ X , [ Y , Z ] ] + [ Y , [ Z , X ] ] + [ Z , [ X , Y ] ]   =   0. {displaystyle ]+]+] = 0.} The action of Y followed by X (operator [ X , [ Y , ⋅   ] ] {displaystyle ]} ), minus the action of X followed by Y (operator ( [ Y , [ X , ⋅   ] ] {displaystyle (]} ), is equal to the action of [ X , Y ] {displaystyle } , (operator [ [ X , Y ] , ⋅   ] {displaystyle ,cdot ]} ). In mathematics the Jacobi identity is a property of a binary operation which 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 Jakob Jacobi. The cross product a × b {displaystyle a imes b} and the Lie bracket operation [ a , b ] {displaystyle } 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. Consider a set A with two binary operations + and × , with an additive identity 0. This satisfies the Jacobi identity if: The left side is the sum of all even permutations of x × (y × z): that is, we leave the parentheses fixed and interchange letters an even number of times. The simplest example of a Lie algebra is constructed from the (associative) ring of n × n {displaystyle n imes n} 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 X × Y {displaystyle X imes Y} , one uses the Lie bracket notation: