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# 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. [1]

In mathematics, the adjoint representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space.

## Analytical mechanics

In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics.

## Anticommutativity

In mathematics, anticommutativity is a specific property of some non-commutative operations.

## Associative property

In mathematics, the associative property is a property of some binary operations.

## Binary operation

In mathematics, a binary operation on a set is a calculation that combines two elements of the set (called operands) to produce another element of the set.

## Carl Gustav Jacob Jacobi

Carl Gustav Jacob Jacobi (10 December 1804 – 18 February 1851) was a German mathematician, who made fundamental contributions to elliptic functions, dynamics, differential equations, and number theory.

## Commutator

In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative.

## Cross product

In mathematics and vector algebra, the cross product or vector product (occasionally directed area product to emphasize the geometric significance) is a binary operation on two vectors in three-dimensional space \left(\mathbb^3\right) and is denoted by the symbol \times.

## Derivation (differential algebra)

In mathematics, a derivation is a function on an algebra which generalizes certain features of the derivative operator.

## Germany

Germany (Deutschland), officially the Federal Republic of Germany (Bundesrepublik Deutschland), is a sovereign state in central-western Europe.

## Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

## Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.

## Leibniz algebra

In mathematics, a (right) Leibniz algebra, named after Gottfried Wilhelm Leibniz, sometimes called a Loday algebra, after Jean-Louis Loday, is a module L over a commutative ring R with a bilinear product satisfying the Leibniz identity In other words, right multiplication by any element c is a derivation.

## Lie algebra

In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.

## Lie superalgebra

In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2-grading.

## Mathematician

A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Moyal bracket

In physics, the Moyal bracket is the suitably normalized antisymmetrization of the phase-space star product.

## Phase space formulation

The phase space formulation of quantum mechanics places the position and momentum variables on equal footing, in phase space.

## 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.

## Quantum mechanics

Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.

## Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

## Structure constants

In mathematics, the structure constants or structure coefficients of an algebra over a field are used to explicitly specify the product of two basis vectors in the algebra as a linear combination.

## Three subgroups lemma

In mathematics, more specifically group theory, the three subgroups lemma is a result concerning commutators.

## References

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