Jacobi identity, the Glossary
In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation.[1]
Table of Contents
24 relations: Adjoint representation, Analytical mechanics, Anticommutative property, Associative property, Binary operation, Carl Gustav Jacob Jacobi, Commutator, Cross product, Derivation (differential algebra), Group (mathematics), Hilbert space, Identity element, Leibniz algebra, Lie algebra, Lie derivative, Lie superalgebra, Mathematics, Moyal bracket, Phase-space formulation, Poisson bracket, Product rule, Quantum mechanics, Structure constants, Three subgroups lemma.
- Non-associative algebra
- Properties of binary operations
Adjoint representation
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.
See Jacobi identity and Adjoint representation
Analytical mechanics
In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related formulations of classical mechanics.
See Jacobi identity and Analytical mechanics
Anticommutative property
In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. Jacobi identity and anticommutative property are Properties of binary operations.
See Jacobi identity and Anticommutative property
Associative property
In mathematics, the associative property is a property of some binary operations that means that rearranging the parentheses in an expression will not change the result. Jacobi identity and associative property are Properties of binary operations.
See Jacobi identity and Associative property
Binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element.
See Jacobi identity and Binary operation
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, determinants, and number theory.
See Jacobi identity and Carl Gustav Jacob Jacobi
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. Jacobi identity and commutator are mathematical identities.
See Jacobi identity and Commutator
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 is denoted by the symbol \times.
See Jacobi identity and Cross product
Derivation (differential algebra)
In mathematics, a derivation is a function on an algebra that generalizes certain features of the derivative operator.
See Jacobi identity and Derivation (differential algebra)
Group (mathematics)
In mathematics, a group is a set with an operation that associates an element of the set to every pair of elements of the set (as does every binary operation) and satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element.
See Jacobi identity and Group (mathematics)
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional.
See Jacobi identity and Hilbert space
Identity 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. Jacobi identity and identity element are Properties of binary operations.
See Jacobi identity and Identity element
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. Jacobi identity and Leibniz algebra are lie algebras.
See Jacobi identity and Leibniz algebra
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 identity. Jacobi identity and Lie algebra are lie algebras.
See Jacobi identity and Lie algebra
Lie derivative
In differential geometry, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector field.
See Jacobi identity and Lie derivative
Lie superalgebra
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a \Z/2\Zgrading. Jacobi identity and Lie superalgebra are lie algebras.
See Jacobi identity and Lie superalgebra
Mathematics
Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.
See Jacobi identity and Mathematics
Moyal bracket
In physics, the Moyal bracket is the suitably normalized antisymmetrization of the phase-space star product.
See Jacobi identity and Moyal bracket
Phase-space formulation
The phase-space formulation of quantum mechanics places the position and momentum variables on equal footing in phase space.
See Jacobi identity and Phase-space formulation
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.
See Jacobi identity and Poisson bracket
Product rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions.
See Jacobi identity and Product rule
Quantum mechanics
Quantum mechanics is a fundamental theory that describes the behavior of nature at and below the scale of atoms.
See Jacobi identity and Quantum mechanics
Structure constants
In mathematics, the structure constants or structure coefficients of an algebra over a field are the coefficients of the basis expansion (into linear combination of basis vectors) of the products of basis vectors. Jacobi identity and structure constants are lie algebras.
See Jacobi identity and Structure constants
Three subgroups lemma
In mathematics, more specifically group theory, the three subgroups lemma is a result concerning commutators.
See Jacobi identity and Three subgroups lemma
See also
Non-associative algebra
- Admissible algebra
- Associator
- Biracks and biquandles
- Bol loop
- CH-quasigroup
- Cancellation property
- Commutative magma
- Gyrovector space
- Heap (mathematics)
- Hyperbolic quaternion
- Isotopy of loops
- Jacobi identity
- Latin square
- Lie conformal algebra
- Lie-admissible algebra
- Magma (algebra)
- Malcev-admissible algebra
- Medial magma
- Monster vertex algebra
- Moufang loop
- Moufang polygon
- Non-associative algebras
- Noncommutative Jordan algebra
- Power associativity
- Pre-Lie algebra
- Quasifield
- Quasigroup
- Racks and quandles
- Small Latin squares and quasigroups
- Vertex operator algebra
Properties of binary operations
- Alternativity
- Anticommutative property
- Associative property
- Cancellation property
- Commutative property
- Distributive property
- Flexible algebra
- Idempotence
- Identity element
- Jacobi identity
- N-ary associativity
- Nilpotent algebra
- Power associativity
- Quasi-commutative property
- Symmetric function
References
[1] https://en.wikipedia.org/wiki/Jacobi_identity
Also known as Jacobi identities.