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# 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. [1]

117 relations: Abelian group, Adjoint representation, Ado's theorem, Alternating multilinear map, Angular momentum, Angular momentum operator, Anticommutativity, Anyonic Lie algebra, Associative algebra, Associative property, Élie Cartan, Baker–Campbell–Hausdorff formula, Bilinear map, Binary operation, Cartan's criterion, Center (algebra), Central series, Centralizer and normalizer, Characteristic (algebra), Chiral Lie algebra, Commutative ring, Commutator, Commutator subgroup, Connected space, Covering space, Cross product, Derivation (differential algebra), Differentiable manifold, Differential graded Lie algebra, Differential topology, Direct sum of modules, Endomorphism, Euclidean space, Euclidean vector, Exponential map (Lie theory), Field (mathematics), Flexible algebra, Fraktur, General Leibniz rule, General linear group, Generator (mathematics), Group (mathematics), Heisenberg group, Hermann Weyl, Homeomorphism, Hydrogen-like atom, Index of a Lie algebra, Infinitesimal transformation, Isomorphism theorems, Jacobi identity, ... Expand index (67 more) »

## Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

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.

In abstract algebra, Ado's theorem is a theorem characterizing finite-dimensional Lie algebras.

## Alternating multilinear map

In mathematics, more specifically in multilinear algebra, an alternating multilinear map is a multilinear map with all arguments belonging to the same space (e.g., a bilinear form or a multilinear form) that is zero whenever any two adjacent arguments are equal.

## Angular momentum

In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of linear momentum.

## Angular momentum operator

In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum.

## Anticommutativity

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

## Anyonic Lie algebra

In mathematics, an anyonic Lie algebra is a U(1) graded vector space L over \mathbb equipped with a bilinear operator and linear maps \varepsilon\colon L\to\mathbb and \Delta\colon L \to L\otimes L satisfying for pure graded elements X, Y, and Z.

## Associative algebra

In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field.

## Associative property

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

## Élie Cartan

Élie Joseph Cartan, ForMemRS (9 April 1869 &ndash; 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups and their geometric applications.

## Baker–Campbell–Hausdorff formula

In mathematics, the Baker–Campbell–Hausdorff formula is the solution to the equation for possibly noncommutative and in the Lie algebra of a Lie group.

## Bilinear map

In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments.

## 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.

## Cartan's criterion

In mathematics, Cartan's criterion gives conditions for a Lie algebra in characteristic 0 to be solvable, which implies a related criterion for the Lie algebra to be semisimple.

## Center (algebra)

The term center or centre is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements.

## Central series

In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial.

## Centralizer and normalizer

In mathematics, especially group theory, the centralizer (also called commutant) of a subset S of a group G is the set of elements of G that commute with each element of S, and the normalizer of S are elements that satisfy a weaker condition.

## Characteristic (algebra)

In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0) if the sum does indeed eventually attain 0.

## Chiral Lie algebra

In algebra, a chiral Lie algebra is a D-module on a curve with a certain structure of Lie algebra.

## Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.

## Commutator

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

## Commutator subgroup

In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.

## Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets.

## Covering space

In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below.

## 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.

## Differentiable manifold

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.

In mathematics, in particular abstract algebra and topology, a differential graded Lie algebra (or dg Lie algebra, or dgla) is a graded vector space with added Lie algebra and chain complex structures that are compatible.

## Differential topology

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds.

## Direct sum of modules

In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module.

## Endomorphism

In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself.

## Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

## Euclidean vector

In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction.

## Exponential map (Lie theory)

In the theory of Lie groups, the exponential map is a map from the Lie algebra \mathfrak g of a Lie group G to the group, which allows one to recapture the local group structure from the Lie algebra.

## Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

## Flexible algebra

In mathematics, particularly abstract algebra, a binary operation • on a set is flexible if it satisfies the flexible identity: for any two elements a and b of the set.

## Fraktur

Fraktur is a calligraphic hand of the Latin alphabet and any of several blackletter typefaces derived from this hand.

## General Leibniz rule

In calculus, the general Leibniz rule, named after Gottfried Wilhelm Leibniz, generalizes the product rule (which is also known as "Leibniz's rule").

## General linear group

In mathematics, the general linear group of degree n is the set of invertible matrices, together with the operation of ordinary matrix multiplication.

## Generator (mathematics)

In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts.

## 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.

## Heisenberg group

In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3&times;3 upper triangular matrices of the form \end under the operation of matrix multiplication.

## Hermann Weyl

Hermann Klaus Hugo Weyl, (9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher.

## Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.

## Hydrogen-like atom

A hydrogen-like ion is any atomic nucleus which has one electron and thus is isoelectronic with hydrogen.

## Index of a Lie algebra

Let g be a Lie algebra over a field K. Let further \xi\in\mathfrak^* be a one-form on g. The stabilizer gξ of ξ is the Lie subalgebra of elements of g that annihilate ξ in the coadjoint representation.

## Infinitesimal transformation

In mathematics, an infinitesimal transformation is a limiting form of small transformation.

## Isomorphism theorems

In mathematics, specifically abstract algebra, the isomorphism theorems are three theorems that describe the relationship between quotients, homomorphisms, and subobjects.

## 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.

## Kac–Moody algebra

In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently discovered them) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a generalized Cartan matrix.

## Karin Erdmann

Karin Erdmann (born 1948) is a German mathematician specializing in the areas of algebra known as representation theory (especially modular representation theory) and homological algebra (especially Hochschild cohomology).

## Killing form

In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras.

## Levi decomposition

In Lie theory and representation theory, the Levi decomposition, conjectured by Killing and Cartan and proved by, states that any finite-dimensional real Lie algebra g is the semidirect product of a solvable ideal and a semisimple subalgebra.

## Lie algebra cohomology

In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras.

## Lie algebra extension

In the theory of Lie groups, Lie algebras and their representation theory, a Lie algebra extension is an enlargement of a given Lie algebra by another Lie algebra.

## Lie algebra representation

In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator.

## Lie bialgebra

In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it's a set with a Lie algebra and a Lie coalgebra structure which are compatible.

## Lie bracket of vector fields

In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi&ndash;Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields X and Y on a smooth manifold M a third vector field denoted.

## Lie coalgebra

In mathematics a Lie coalgebra is the dual structure to a 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 function, vector field and one-form), along the flow defined by another vector field.

## Lie group

In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

## Lie group–Lie algebra correspondence

In mathematics, Lie group–Lie algebra correspondence allows one to study Lie groups, which are geometric objects, in terms of Lie algebras, which are linear objects.

## Lie superalgebra

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

## Lie's third theorem

In mathematics, Lie's third theorem states that every finite-dimensional Lie algebra g over the real numbers is associated to a Lie group G. Historically, the third theorem referred to a different but related result.

## Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

## List of simple Lie groups

In mathematics, the simple Lie groups were first classified by Wilhelm Killing and later perfected by Élie Cartan.

## Mathematics

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

## Matrix exponential

In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function.

## Matrix group

In mathematics, a matrix group is a group G consisting of invertible matrices over a specified field K, with the operation of matrix multiplication.

## Module (mathematics)

In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra.

## Moyal bracket

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

## Nilpotent Lie algebra

In mathematics, a Lie algebra is nilpotent if its lower central series eventually becomes zero.

## Non-associative algebra

A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative.

## Nondegenerate form

In linear algebra, a nondegenerate form or nonsingular form is a bilinear form that is not degenerate, meaning that v \mapsto (x \mapsto f(x,v)) is an isomorphism, or equivalently in finite dimensions, if and only if.

## Orthogonal group

In mathematics, the orthogonal group in dimension, denoted, is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations.

## Orthogonal symmetric Lie algebra

In mathematics, an orthogonal symmetric Lie algebra is a pair (\mathfrak, s) consisting of a real Lie algebra \mathfrak and an automorphism s of \mathfrak of order 2 such that the eigenspace \mathfrak of s corrsponding to 1 (i.e., the set \mathfrak of fixed points) is a compact subalgebra.

In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems.

## P-group

In mathematical group theory, given a prime number p, a p-group is a group in which each element has a power of p as its order.

## Particle physics and representation theory

There is a natural connection between particle physics and representation theory, as first noted in the 1930s by Eugene Wigner.

## Poisson algebra

In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation.

## Pro-p group

In mathematics, a pro-p group (for some prime number p) is a profinite group G such that for any open normal subgroup N\triangleleft G the quotient group G/N is a ''p''-group.

## Quantum group

In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebras with additional structure.

## 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.

## Quasi-Frobenius Lie algebra

In mathematics, a quasi-Frobenius Lie algebra over a field k is a Lie algebra equipped with a nondegenerate skew-symmetric bilinear form for all X, Y, Z in \mathfrak.

## Quasi-Lie algebra

In mathematics, a quasi-Lie algebra in abstract algebra is just like a Lie algebra, but with the usual axiom replaced by In characteristic other than 2, these are equivalent (in the presence of bilinearity), so this distinction doesn't arise when considering real or complex Lie algebras.

## Radical of a Lie algebra

In the mathematical field of Lie theory, the radical of a Lie algebra \mathfrak is the largest solvable ideal of \mathfrak.

## Reductive Lie algebra

In mathematics, a Lie algebra is reductive if its adjoint representation is completely reducible, whence the name.

## Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.

## Restricted Lie algebra

In mathematics, a restricted Lie algebra is a Lie algebra together with an additional "p operation.".

## Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

## Root system

In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties.

## Rotation group SO(3)

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition.

## Scalar (mathematics)

A scalar is an element of a field which is used to define a vector space.

## Semidirect product

In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product.

## Semisimple Lie algebra

In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., non-abelian Lie algebras \mathfrak g whose only ideals are and \mathfrak g itself.

## Simple Lie group

In group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.

## Simplicial Lie algebra

In algebra, a simplicial Lie algebra is a simplicial object in the category of Lie algebras.

## Skew-Hermitian matrix

In linear algebra, a square matrix with complex entries is said to be skew-Hermitian or antihermitian if its conjugate transpose is equal to the original matrix, with all the entries being of opposite sign.

## Smoothness

In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous.

## Solvable Lie algebra

In mathematics, a Lie algebra \mathfrak is solvable if its derived series terminates in the zero subalgebra.

## Sophus Lie

Marius Sophus Lie (17 December 1842 – 18 February 1899) was a Norwegian mathematician.

## Special linear group

In mathematics, the special linear group of degree n over a field F is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion.

## Special linear Lie algebra

In mathematics, the special linear Lie algebra of order n (denoted \mathfrak_n(F) or \mathfrak(n, F)) is the Lie algebra of n \times n matrices with trace zero and with the Lie bracket.

## Special unitary group

In mathematics, the special unitary group of degree, denoted, is the Lie group of unitary matrices with determinant 1.

## String theory

In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings.

## Symmetric bilinear form

A symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map.

## Torus

In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.

## Trace (linear algebra)

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.e., where aii denotes the entry on the ith row and ith column of A. The trace of a matrix is the sum of the (complex) eigenvalues, and it is invariant with respect to a change of basis.

## Unitarian trick

In mathematics, the unitarian trick is a device in the representation theory of Lie groups, introduced by for the special linear group and by Hermann Weyl for general semisimple groups.

## Unitary group

In mathematics, the unitary group of degree n, denoted U(n), is the group of unitary matrices, with the group operation of matrix multiplication.

## Universal enveloping algebra

In mathematics, a universal enveloping algebra is the most general (unital, associative) algebra that contains all representations of a Lie algebra.

## Vector field

In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space.

## Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

## Virasoro algebra

In mathematics, the Virasoro algebra (named after the physicist Miguel Angel Virasoro) is a complex Lie algebra, the unique central extension of the Witt algebra.

## Weyl's theorem on complete reducibility

In algebra, Weyl's theorem on complete reducibility is a fundamental result in the theory of Lie algebra representations.

## Wilhelm Killing

Wilhelm Karl Joseph Killing (10 May 1847 &ndash; 11 February 1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry.

## References

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