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单词 LieAlgebra
释义

Lie algebra


A Lie algebraMathworldPlanetmath over a field k is a vector spaceMathworldPlanetmath 𝔤 with a bilinear map [,]:𝔤×𝔤𝔤, called the Lie bracket and denoted (x,y)[x,y]. It is required to satisfy:

  1. 1.

    [x,x]=0 for all x𝔤.

  2. 2.

    The Jacobi identityMathworldPlanetmath: [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0 for all x,y,z𝔤.

1 Subalgebras & Ideals

A vector subspace 𝔥 of the Lie algebra 𝔤 is a subalgebraMathworldPlanetmathPlanetmathPlanetmath if 𝔥 is closed underPlanetmathPlanetmath the Lie bracket operation, or, equivalently, if 𝔥 itself is a Lie algebra under the same bracket operation as 𝔤. An ideal of 𝔤 is a subspace 𝔥 for which [x,y]𝔥 whenever either x𝔥 or y𝔥. Note that every ideal is also a subalgebra.

Some general examples of subalgebras:

  • The center of 𝔤, defined by Z(𝔤):={x𝔤[x,y]=0for all y𝔤}. It is an ideal of 𝔤.

  • The normalizerMathworldPlanetmath of a subalgebra 𝔥 is the set N(𝔥):={x𝔤[x,𝔥]𝔥}. The Jacobi identity guarantees that N(𝔥) is always a subalgebra of 𝔤.

  • The centralizerMathworldPlanetmathPlanetmath of a subset X𝔤 is the set C(X):={x𝔤[x,X]=0}. Again, the Jacobi identity implies that C(X) is a subalgebra of 𝔤.

2 Homomorphisms

Given two Lie algebras 𝔤 and 𝔤 over the field k, a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath from 𝔤 to 𝔤 is a linear transformation ϕ:𝔤𝔤 such that ϕ([x,y])=[ϕ(x),ϕ(y)] for all x,y𝔤. An injectivePlanetmathPlanetmath homomorphism is called a monomorphismMathworldPlanetmathPlanetmathPlanetmath, and a surjectivePlanetmathPlanetmath homomorphism is called an epimorphismMathworldPlanetmath.

The kernel of a homomorphism ϕ:𝔤𝔤 (considered as a linear transformation) is denoted ker(ϕ). It is always an ideal in 𝔤.

3 Examples

  • Any vector space can be made into a Lie algebra simply by setting [x,y]=0 for all vectors x,y. The resulting Lie algebra is called an abelianMathworldPlanetmath Lie algebra.

  • If G is a Lie group, then the tangent space at the identityPlanetmathPlanetmathPlanetmathPlanetmath forms a Lie algebra over the real numbers.

  • 3 with the cross productMathworldPlanetmath operation is a nonabelianPlanetmathPlanetmathPlanetmath three dimensional Lie algebra over .

4 Historical Note

Lie algebras are so-named in honour of Sophus Lie, a Norwegianmathematician who pioneered the study of these mathematical objects.Lie’s discovery was tied to his investigation of continuoustransformation groups and symmetriesPlanetmathPlanetmathPlanetmath. One joint project with FelixKlein called for the classification of all finite-dimensionalPlanetmathPlanetmath groupsacting on the plane. The task seemed hopeless owing to the generallynon-linear nature of such group actionsMathworldPlanetmath. However, Lie was able tosolve the problem by remarking that a transformation group can belocally reconstructed from its corresponding “infinitesimalgenerators”, that is to say vector fields corresponding to various1-parameter subgroups. In terms of this geometric correspondence, thegroup composition operation manifests itself as the bracket of vectorfields, and this is very much a linear operation. Thus the task ofclassifying group actions in the plane became the task of classifyingall finite-dimensional Lie algebras of planar vector field; a projectthat Lie brought to a successful conclusionMathworldPlanetmath.

This “linearization trick” proved to be incredibly fruitful and ledto great advances in geometry and differential equations. Suchadvances are based, however, on various results from the theory of Liealgebras. Lie was the first to make significant contributions to thispurely algebraic theory, but he was surely not the last.

TitleLie algebra
Canonical nameLieAlgebra
Date of creation2013-03-22 12:03:36
Last modified on2013-03-22 12:03:36
Ownerdjao (24)
Last modified bydjao (24)
Numerical id18
Authordjao (24)
Entry typeDefinition
Classificationmsc 17B99
Related topicCommutatorBracket
Related topicLieGroup
Related topicUniversalEnvelopingAlgebra
Related topicRootSystem
Related topicSimpleAndSemiSimpleLieAlgebras2
DefinesJacobi identity
Definessubalgebra
Definesideal
Definesnormalizer
Definescentralizer
Defineskernel
Defineshomomorphism
Definescenter
Definescentre
Definesabelian Lie algebra
Definesabelian
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