Jacobi identity - Citizendia

In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Carl Gustav Jacob Jacobi ( December 10, 1804 - February 18, 1851) was a Prussian Mathematician, widely considered to be Unlike for associative operations, order of evaluation is significant for operations satisfying Jacobi identity. In Mathematics, associativity is a property that a Binary operation can have

1 Definition
2 Interpretation
3 Examples
4 See also

Definition

A binary operation $*$ on a set $S$ possessing a commutative binary operation $+$ , satisfies the Jacobi identity if

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

Interpretation

In a Lie algebra, the objects that obey the Jacobi identity are infinitesimal motions. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie When acting on an operator with an infinitesimal motion, the change in the operator is the commutator.

The Jacobi Identity can then be translated into words:

$[ [A , B] , C ] = [A , [B , C]] - [ B , [A , C]] \,$

meaning "the infinitesimal motion of B followed by the infinitesimal motion of A ( $[A,[B,\cdot]]$ ), minus the infinitesimal motion of A followed by the infinitesimal motion of B ( $[B,[A,\cdot]]$ ), is the infinitesimal motion of [A,B] ( $[[A,B],\cdot]$ ), when acting on any arbitrary infinitesimal motion C (thus, these are equal)".

Examples

The Jacobi identity is satisfied by the multiplication (bracket) operation on Lie algebras and Lie rings and these provide the majority of examples of operations satisfying the Jacobi identity in common use. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In Mathematics a Lie ring is a structure related to Lie algebras that can arise as a generalisation of Lie algebras or through the study of the Lower central Because of this the Jacobi identity is often expressed using Lie bracket notation:

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

If the multiplication is antisymmetric, the Jacobi identity admits two equivalent reformulations. In mathematics anticommutativity refers to the property of an operation being anticommutative, i Defining the adjoint map

$\operatorname{ad}_x: y \mapsto [x,y],$

after a rearrangement, the identity becomes

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

Thus, the Jacobi identity for Lie algebras simply becomes the assertion that the action of any element on the algebra is a derivation. In Mathematics, the adjoint representation (or adjoint action) of a Lie group G is the natural representation of G on its In Abstract algebra, a derivation is a function on an algebra which generalizes certain features of the Derivative operator This form of the Jacobi identity is also used to define the notion of Leibniz algebra. In Mathematics, a (left Leibniz algebra (sometimes called a Loday algebra) is a module A over a commutative ring or field R with a bilinear

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. In Mathematics, the adjoint representation (or adjoint action) of a Lie group G is the natural representation of G on its In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie

A similar identity, called the Hall-Witt identity, exists for the commutators in groups. In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative. In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element

In analytical mechanics, Jacobi identity is satisfied by Poisson brackets, while in quantum mechanics it is satisfied by operator commutators. Analytical mechanics is a term used for a refined highly mathematical form of Classical mechanics, constructed from the Eighteenth century onwards as a formulation In Mathematics and Classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative.

Contents

Definition

Interpretation

Examples

See also