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. The cross product $a\times b$ and the Lie bracket operation $[a,b]$ both satisfy the Jacobi identity.

Definition

A binary operation × on a set S possessing a binary operation + with an additive identity denoted by 0 satisfies the Jacobi identity if:

a\times (b\times c)\ +\ b\times (c\times a)\ +\ c\times (a\times b)\ =\ 0\quad \forall \ {a,b,c}\in S.

That is, if the sum of all even permutations of (a,(b,c)) is zero (where the permutation is performed by leaving the parentheses fixed and interchanging letters an even number of times).

Interpretation

The simplest example of a Lie algebra 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 Lie bracket operation is then defined as the commutator, which measures the failure of commutativity in matrix multiplication:

[A,B]=AB-BA.

It is then easy to check the Jacobi identity:

$[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0.$

More generally, suppose A is an associative algebra and V is a subspace of A with the property that for all A and B in A, the element $AB-BA$ belongs to V. Then the Jacobi identity holds on V for the bracket operator given by $[A,B]=AB-BA$ .^[1] Thus, if a binary operation satisfies the Jacobi identity, we may say that it behaves as if it were given by $AB-BA$ in some associative algebra, even if it is not actually defined that way.

Using the antisymmetry property $[A,B]=-[B,A]$ , the Jacobi identity can be rewritten as a modification of the associative property:

[[A,B],C]=[A,[B,C]]-[B,[A,C]]~.

Considering $[A,C]$ as the action of the infinitesimal motion A on C, this can be stated as:

The action of B followed by A (operator $[A,[B,\cdot \ ]]$ ), minus the action of A followed by B (operator $([B,[A,\cdot \ ]]$ ), is equal to the action of $[A,B]$ , (operator $[[A,B],\cdot \ ]$ ).

There is also a plethora of mixed analogs involving anticommutators, such as

[\{A,B\},C]+[\{B,C\},A]+[\{C,A\},B]=0,\qquad [\{A,B\},C]+\{[C,B],A\}+\{[C,A],B\}=0,

etc. (Graded Jacobi identities)

Examples

The majority of common examples of the Jacobi identity come from the bracket multiplication on Lie algebras and Lie rings. Because of this the Jacobi identity is often expressed using Lie bracket notation:

[x,[y,z]] + [z,[x,y]] + [y,[z,x]] = 0.

Because the bracket multiplication is antisymmetric, the Jacobi identity admits two equivalent reformulations. Defining the adjoint operator $\operatorname {ad} _{x}:y\mapsto [x,y]$ , the identity becomes:

\operatorname{ad}_x[y,z]=[\operatorname{ad}_xy,z]+[y,\operatorname{ad}_xz].

Thus, the Jacobi identity for Lie algebras simply states that the action of any element on the algebra is a derivation. This 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{ad}_{[x,y]}=[\operatorname{ad}_x,\operatorname{ad}_y].

This identity implies that the map sending each element to its adjoint action is a Lie algebra homomorphism of the original algebra into the Lie algebra of its derivations.

The Hall–Witt identity is the analogous identity for the commutator operation in a group.

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 following identitity follows from anticommutativity and Jacobi identity and holds in arbitrary Lie algebra:^[2]

[x,[y,[z,w]]]+[y,[x,[w,z]]]+[z,[w,[x,y]]]+[w,[z,[y,x]]]=0.

References

↑ Hall 2015 Example 3.3
↑ Alekseev, Ilya; Ivanov, Sergei O. (18 April 2016). "Higher Jacobi Identities". arXiv:1604.05281.

Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer, ISBN 978-3319134666 .

External links

Weisstein, Eric W. "Jacobi Identities". MathWorld.

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[1] Hall 2015 Example 3.3

[2] Alekseev, Ilya; Ivanov, Sergei O. (18 April 2016). "Higher Jacobi Identities". arXiv:1604.05281.