Logo
Unionpedia
Communication
Get it on Google Play
New! Download Unionpedia on your Android™ device!
Free
Faster access than browser!
 

Lie algebra

Index 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, ..., Kac–Moody algebra, Karin Erdmann, Killing form, Levi decomposition, Lie algebra cohomology, Lie algebra extension, Lie algebra representation, Lie bialgebra, Lie bracket of vector fields, Lie coalgebra, Lie derivative, Lie group, Lie group–Lie algebra correspondence, Lie superalgebra, Lie's third theorem, Linear map, List of simple Lie groups, Mathematics, Matrix exponential, Matrix group, Module (mathematics), Moyal bracket, Nilpotent Lie algebra, Non-associative algebra, Nondegenerate form, Orthogonal group, Orthogonal symmetric Lie algebra, P-adic number, P-group, Particle physics and representation theory, Poisson algebra, Pro-p group, Quantum group, Quantum mechanics, Quasi-Frobenius Lie algebra, Quasi-Lie algebra, Radical of a Lie algebra, Reductive Lie algebra, Representation theory, Restricted Lie algebra, Ring (mathematics), Root system, Rotation group SO(3), Scalar (mathematics), Semidirect product, Semisimple Lie algebra, Simple Lie group, Simplicial Lie algebra, Skew-Hermitian matrix, Smoothness, Solvable Lie algebra, Sophus Lie, Special linear group, Special linear Lie algebra, Special unitary group, String theory, Symmetric bilinear form, Torus, Trace (linear algebra), Unitarian trick, Unitary group, Universal enveloping algebra, Vector field, Vector space, Virasoro algebra, Weyl's theorem on complete reducibility, Wilhelm Killing. 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.

New!!: Lie algebra and Abelian group · See more »

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.

New!!: Lie algebra and Adjoint representation · See more »

Ado's theorem

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

New!!: Lie algebra and Ado's theorem · See more »

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.

New!!: Lie algebra and Alternating multilinear map · See more »

Angular momentum

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

New!!: Lie algebra and Angular momentum · See more »

Angular momentum operator

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

New!!: Lie algebra and Angular momentum operator · See more »

Anticommutativity

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

New!!: Lie algebra and Anticommutativity · See more »

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.

New!!: Lie algebra and Anyonic Lie algebra · See more »

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.

New!!: Lie algebra and Associative algebra · See more »

Associative property

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

New!!: Lie algebra and Associative property · See more »

Élie Cartan

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

New!!: Lie algebra and Élie Cartan · See more »

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.

New!!: Lie algebra and Baker–Campbell–Hausdorff formula · See more »

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.

New!!: Lie algebra and Bilinear map · See more »

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.

New!!: Lie algebra and Binary operation · See more »

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.

New!!: Lie algebra and Cartan's criterion · See more »

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.

New!!: Lie algebra and Center (algebra) · See more »

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.

New!!: Lie algebra and Central series · See more »

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.

New!!: Lie algebra and Centralizer and normalizer · See more »

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.

New!!: Lie algebra and Characteristic (algebra) · See more »

Chiral Lie algebra

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

New!!: Lie algebra and Chiral Lie algebra · See more »

Commutative ring

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

New!!: Lie algebra and Commutative ring · See more »

Commutator

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

New!!: Lie algebra and Commutator · See more »

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.

New!!: Lie algebra and Commutator subgroup · See more »

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.

New!!: Lie algebra and Connected space · See more »

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.

New!!: Lie algebra and Covering space · See more »

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.

New!!: Lie algebra and Cross product · See more »

Derivation (differential algebra)

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

New!!: Lie algebra and Derivation (differential algebra) · See more »

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.

New!!: Lie algebra and Differentiable manifold · See more »

Differential graded Lie algebra

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.

New!!: Lie algebra and Differential graded Lie algebra · See more »

Differential topology

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

New!!: Lie algebra and Differential topology · See more »

Direct sum of modules

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

New!!: Lie algebra and Direct sum of modules · See more »

Endomorphism

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

New!!: Lie algebra and Endomorphism · See more »

Euclidean space

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

New!!: Lie algebra and Euclidean space · See more »

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.

New!!: Lie algebra and Euclidean vector · See more »

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.

New!!: Lie algebra and Exponential map (Lie theory) · See more »

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.

New!!: Lie algebra and Field (mathematics) · See more »

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.

New!!: Lie algebra and Flexible algebra · See more »

Fraktur

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

New!!: Lie algebra and Fraktur · See more »

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").

New!!: Lie algebra and General Leibniz rule · See more »

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.

New!!: Lie algebra and General linear group · See more »

Generator (mathematics)

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

New!!: Lie algebra and Generator (mathematics) · See more »

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.

New!!: Lie algebra and Group (mathematics) · See more »

Heisenberg group

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

New!!: Lie algebra and Heisenberg group · See more »

Hermann Weyl

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

New!!: Lie algebra and Hermann Weyl · See more »

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.

New!!: Lie algebra and Homeomorphism · See more »

Hydrogen-like atom

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

New!!: Lie algebra and Hydrogen-like atom · See more »

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.

New!!: Lie algebra and Index of a Lie algebra · See more »

Infinitesimal transformation

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

New!!: Lie algebra and Infinitesimal transformation · See more »

Isomorphism theorems

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

New!!: Lie algebra and Isomorphism theorems · See more »

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.

New!!: Lie algebra and Jacobi identity · See more »

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.

New!!: Lie algebra and Kac–Moody algebra · See more »

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).

New!!: Lie algebra and Karin Erdmann · See more »

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.

New!!: Lie algebra and Killing form · See more »

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.

New!!: Lie algebra and Levi decomposition · See more »

Lie algebra cohomology

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

New!!: Lie algebra and Lie algebra cohomology · See more »

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.

New!!: Lie algebra and Lie algebra extension · See more »

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.

New!!: Lie algebra and Lie algebra representation · See more »

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.

New!!: Lie algebra and Lie bialgebra · See more »

Lie bracket of vector fields

In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–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.

New!!: Lie algebra and Lie bracket of vector fields · See more »

Lie coalgebra

In mathematics a Lie coalgebra is the dual structure to a Lie algebra.

New!!: Lie algebra and Lie coalgebra · See more »

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.

New!!: Lie algebra and Lie derivative · See more »

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.

New!!: Lie algebra and Lie group · See more »

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.

New!!: Lie algebra and Lie group–Lie algebra correspondence · See more »

Lie superalgebra

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

New!!: Lie algebra and Lie superalgebra · See more »

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.

New!!: Lie algebra and Lie's third theorem · See more »

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.

New!!: Lie algebra and Linear map · See more »

List of simple Lie groups

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

New!!: Lie algebra and List of simple Lie groups · See more »

Mathematics

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

New!!: Lie algebra and Mathematics · See more »

Matrix exponential

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

New!!: Lie algebra and Matrix exponential · See more »

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.

New!!: Lie algebra and Matrix group · See more »

Module (mathematics)

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

New!!: Lie algebra and Module (mathematics) · See more »

Moyal bracket

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

New!!: Lie algebra and Moyal bracket · See more »

Nilpotent Lie algebra

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

New!!: Lie algebra and Nilpotent Lie algebra · See more »

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.

New!!: Lie algebra and Non-associative algebra · See more »

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.

New!!: Lie algebra and Nondegenerate form · See more »

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.

New!!: Lie algebra and Orthogonal group · See more »

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.

New!!: Lie algebra and Orthogonal symmetric Lie algebra · See more »

P-adic number

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.

New!!: Lie algebra and P-adic number · See more »

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.

New!!: Lie algebra and P-group · See more »

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.

New!!: Lie algebra and Particle physics and representation theory · See more »

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.

New!!: Lie algebra and Poisson algebra · See more »

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.

New!!: Lie algebra and Pro-p group · See more »

Quantum group

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

New!!: Lie algebra and Quantum group · See more »

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.

New!!: Lie algebra and Quantum mechanics · See more »

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.

New!!: Lie algebra and Quasi-Frobenius Lie algebra · See more »

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.

New!!: Lie algebra and Quasi-Lie algebra · See more »

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.

New!!: Lie algebra and Radical of a Lie algebra · See more »

Reductive Lie algebra

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

New!!: Lie algebra and Reductive Lie algebra · See more »

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.

New!!: Lie algebra and Representation theory · See more »

Restricted Lie algebra

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

New!!: Lie algebra and Restricted Lie algebra · See more »

Ring (mathematics)

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

New!!: Lie algebra and Ring (mathematics) · See more »

Root system

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

New!!: Lie algebra and Root system · See more »

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.

New!!: Lie algebra and Rotation group SO(3) · See more »

Scalar (mathematics)

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

New!!: Lie algebra and Scalar (mathematics) · See more »

Semidirect product

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

New!!: Lie algebra and Semidirect product · See more »

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.

New!!: Lie algebra and Semisimple Lie algebra · See more »

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.

New!!: Lie algebra and Simple Lie group · See more »

Simplicial Lie algebra

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

New!!: Lie algebra and Simplicial Lie algebra · See more »

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.

New!!: Lie algebra and Skew-Hermitian matrix · See more »

Smoothness

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

New!!: Lie algebra and Smoothness · See more »

Solvable Lie algebra

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

New!!: Lie algebra and Solvable Lie algebra · See more »

Sophus Lie

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

New!!: Lie algebra and Sophus Lie · See more »

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.

New!!: Lie algebra and Special linear group · See more »

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.

New!!: Lie algebra and Special linear Lie algebra · See more »

Special unitary group

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

New!!: Lie algebra and Special unitary group · See more »

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.

New!!: Lie algebra and String theory · See more »

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.

New!!: Lie algebra and Symmetric bilinear form · See more »

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.

New!!: Lie algebra and Torus · See more »

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.

New!!: Lie algebra and Trace (linear algebra) · See more »

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.

New!!: Lie algebra and Unitarian trick · See more »

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.

New!!: Lie algebra and Unitary group · See more »

Universal enveloping algebra

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

New!!: Lie algebra and Universal enveloping algebra · See more »

Vector field

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

New!!: Lie algebra and Vector field · See more »

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.

New!!: Lie algebra and Vector space · See more »

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.

New!!: Lie algebra and Virasoro algebra · See more »

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.

New!!: Lie algebra and Weyl's theorem on complete reducibility · See more »

Wilhelm Killing

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

New!!: Lie algebra and Wilhelm Killing · See more »

Redirects here:

Abelian Lie algebra, Center of a Lie algebra, Ideal (Lie algebra), Ideal of a Lie algebra, Infinitesimal group, Lie Algebra, Lie algebra homomorphism, Lie algebras, Lie bracket, Lie brackets, Lie ring, Lie subalgebra, Topological Lie algebra.

References

[1] https://en.wikipedia.org/wiki/Lie_algebra

OutgoingIncoming
Hey! We are on Facebook now! »