mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Jacobi identity is a property of a

binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, 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 In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule of replaceme ...

, 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 bracket In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. Th ...

s in his 1862 paper on differential equations. C. G. J. Jacobi (1862), §26, Theorem V. T. Hawkins (1991) The

cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...

a\times b

and the Lie bracket operation

,b /math> both satisfy the Jacobi identity. In

analytical mechanics In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related formulations of classical mechanics. Analytical mechanics uses '' scalar'' properties of motion representing the sy ...

, the Jacobi identity is satisfied by the

s. In

quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...

, it is satisfied by operator

commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...

s on a

Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...

and equivalently in the phase space formulation of quantum mechanics by the Moyal bracket.

Definition

Let

+

and

\times

be two

s, and let

0

be the

neutral element In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...

for

+

. The is :

x \times (y \times z) \ +\ y \times (z \times x)  \ +\ z \times (x \times y)\ =\ 0.

Notice the pattern in the variables on the left side of this identity. In each subsequent expression of the form

a \times (b \times c)

, the variables

x

y

and

z

are permuted according to the cycle

x \mapsto y \mapsto z \mapsto x

. Alternatively, we may observe that the ordered triples

(x,y,z)

(y,z,x)

and

(z,x,y)

, are the even permutations of the ordered triple

(x,y,z)

Commutator bracket form

The simplest informative example of a

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

is constructed from the (associative) ring of

n\times n

matrices, which may be thought of as infinitesimal motions of an ''n''-dimensional vector space. The × operation is the

, which measures the failure of commutativity in matrix multiplication. Instead of

X\times Y

, the Lie bracket notation is used: :

,Y XY-YX.

In that notation, the Jacobi identity is: :

">,_Z.html" ;"title=", [Y, Z">, [Y, Z+ [Y, [Z, X">,_Z">,_[Y,_Z<_a>.html" ;"title=",_Z.html" ;"title=", [Y, Z">, [Y, Z">,_Z.html" ;"title=", [Y, Z">, [Y, Z+ [Y, [Z, X] + [Z, [X, Y] ] \ =\ 0

That is easily checked by computation. More generally, if is an associative algebra and is a subspace of that is closed under the bracket operation:

,Y XY-YX

belongs to for all

X,Y\in V

, the Jacobi identity continues to hold on . Thus, if a binary operation

,Y /math> satisfies the Jacobi identity, it may be said that it behaves as if it were given by XY-YX in some associative algebra even if it is not actually defined that way.

Using the antisymmetry property,Y -,X /math>, the Jacobi identity may be rewritten as a modification of the

: :

X, Y Z] =, [Y, Z - [Y, [X, Z">,_Z.html" ;"title=", [Y, Z">, [Y, Z - [Y, [X, Z~.

[X,Z">,_Z">,_[Y,_Z<_a>_-_[Y,_[X,_Z.html" ;"title=",_Z.html" ;"title=", [Y, Z">, [Y, Z - [Y, [X, Z">,_Z.html" ;"title=", [Y, Z">, [Y, Z - [Y, [X, Z~.

[X,Z/math> is the action of the infinitesimal motion  on , that can be stated as: 


There is also a plethora of graded Jacobi identities involving anticommutator">Lie superalgebra#properties">graded Jacobi identities involving anticommutators \, such as:

: [\,Z]+ [\,X]+[\,Y] =0,\qquad
[\,Z]+ \+\ =0.

Adjoint form

Most common examples of the Jacobi identity come from the bracket multiplication

[x,y]

s and

Lie ring In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...

s. The Jacobi identity is written as: :

,[y,z + [z,[x,y">,z.html" ;"title=",[y,z">,[y,z + [z,[x,y + [y,[z,x">,z">,[y,z<_a>_+_[z,[x,y.html" ;"title=",z.html" ;"title=",[y,z">,[y,z +,z.html" ;"title=",[y,z">,[y,z + [z,[x,y + [y,[z,x = 0.

Because the bracket multiplication is anticommutativity">antisymmetric, the Jacobi identity admits two equivalent reformulations. Defining the adjoint operator

\operatorname_x: y \mapsto adjoint representation of a Lie algebra">adjoint operator \operatorname_x: y \mapsto [x,y/math>, the identity becomes:
: \operatorname_x,y">adjoint representation of a Lie algebra">adjoint operator \operatorname_x: y \mapsto [x,y/math>, the identity becomes:
: \operatorname_x[y,z[\operatorname_xy,z">,z">,y">adjoint representation of a Lie algebra">adjoint operator \operatorname_x: y \mapsto [x,y/math>, the identity becomes:
: \operatorname_x[y,z[\operatorname_xy,z[y,\operatorname_xz]. Thus, the Jacobi identity for Lie algebras states that the action of any element on the algebra is a derivation (abstract algebra), 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:
: \operatorname_= operatorname_x,\operatorname_y 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 \mathrm map sending each element to its adjoint action is a Lie algebra homomorphism .

Related identities

* The Hall–Witt identity is the analogous identity for the

operation in a group. * The following higher order Jacobi identity holds in arbitrary Lie algebra: :

[x_1,x_2 x_3">[x_1,x_2.html" ;"title="[x_1,x_2">[x_1,x_2x_3x_4">_1,x_2">[x_1,x_2<_a>x_3.html" ;"title="[x_1,x_2.html" ;"title="[x_1,x_2">[x_1,x_2x_3">[x_1,x_2.html" ;"title="[x_1,x_2">[x_1,x_2x_3x_4[x_2,x_1],x_4],x_3]+[x_3,x_4],x_1],x_2]+[x_4,x_3],x_2],x_1] = 0.

* The Jacobi identity is equivalent to the Product Rule, with the Lie bracket acting as both a product and a derivative:

,[Y,Z = X,Y">,Z.html" ;"title=",[Y,Z">,[Y,Z = X,Y Z">,Z">,[Y,Z<_a>_=_X,Y.html" ;"title=",Z.html" ;"title=",[Y,Z">,[Y,Z = X,Y">,Z.html" ;"title=",[Y,Z">,[Y,Z = X,Y Z+ [Y, [X,Z

. If

X, Y

are vector fields, then

,Y /math> is literally a derivative operator acting on Y, namely the Lie derivative \mathcal_X Y .

References

* . * *

External links