Lie algebra

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

  1. 1.

    [x,x]=0 for all xβˆˆπ”€.

  2. 2.

    The Jacobi identityMathworldPlanetmath: [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0 for all x,y,zβˆˆπ”€.

1 Subalgebras & Ideals

A vector subspace π”₯ of the Lie algebra 𝔀 is a subalgebraMathworldPlanetmathPlanetmathPlanetmath if π”₯ is closed underPlanetmathPlanetmath the Lie bracket operation, or, equivalently, if π”₯ itself is a Lie algebra under the same bracket operation as 𝔀. An ideal of 𝔀 is a subspace π”₯ for which [x,y]∈π”₯ whenever either x∈π”₯ or y∈π”₯. Note that every ideal is also a subalgebra.

Some general examples of subalgebras:

  • β€’

    The center of 𝔀, defined by Z⁒(𝔀):={xβˆˆπ”€βˆ£[x,y]=0⁒for all ⁒yβˆˆπ”€}. It is an ideal of 𝔀.

  • β€’

    The normalizerMathworldPlanetmath of a subalgebra π”₯ is the set N⁒(π”₯):={xβˆˆπ”€βˆ£[x,π”₯]βŠ‚π”₯}. The Jacobi identity guarantees that N⁒(π”₯) is always a subalgebra of 𝔀.

  • β€’

    The centralizerMathworldPlanetmathPlanetmath of a subset XβŠ‚π”€ is the set C⁒(X):={xβˆˆπ”€βˆ£[x,X]=0}. Again, the Jacobi identity implies that C⁒(X) is a subalgebra of 𝔀.

2 Homomorphisms

Given two Lie algebras 𝔀 and 𝔀′ over the field k, a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath from 𝔀 to 𝔀′ is a linear transformation Ο•:𝔀→𝔀′ such that ϕ⁒([x,y])=[ϕ⁒(x),ϕ⁒(y)] for all x,yβˆˆπ”€. An injectivePlanetmathPlanetmath homomorphism is called a monomorphismMathworldPlanetmathPlanetmathPlanetmath, and a surjectivePlanetmathPlanetmath homomorphism is called an epimorphismMathworldPlanetmath.

The kernel of a homomorphism Ο•:𝔀→𝔀′ (considered as a linear transformation) is denoted ker⁑(Ο•). It is always an ideal in 𝔀.

3 Examples

  • β€’

    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.

  • β€’

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

  • β€’

    ℝ3 with the cross productMathworldPlanetmath operation is a nonabelianPlanetmathPlanetmathPlanetmath three dimensional Lie algebra over ℝ.

4 Historical Note

Lie algebras are so-named in honour of Sophus Lie, a Norwegian mathematician who pioneered the study of these mathematical objects. Lie’s discovery was tied to his investigation of continuous transformation groups and symmetriesPlanetmathPlanetmathPlanetmath. One joint project with Felix Klein called for the classification of all finite-dimensionalPlanetmathPlanetmath groups acting on the plane. The task seemed hopeless owing to the generally non-linear nature of such group actionsMathworldPlanetmath. However, Lie was able to solve the problem by remarking that a transformation group can be locally reconstructed from its corresponding β€œinfinitesimal generators”, that is to say vector fields corresponding to various 1-parameter subgroups. In terms of this geometric correspondence, the group composition operation manifests itself as the bracket of vector fields, and this is very much a linear operation. Thus the task of classifying group actions in the plane became the task of classifying all finite-dimensional Lie algebras of planar vector field; a project that Lie brought to a successful conclusionMathworldPlanetmath.

This β€œlinearization trick” proved to be incredibly fruitful and led to great advances in geometry and differential equations. Such advances are based, however, on various results from the theory of Lie algebras. Lie was the first to make significant contributions to this purely algebraic theory, but he was surely not the last.

Title Lie algebra
Canonical name LieAlgebra
Date of creation 2013-03-22 12:03:36
Last modified on 2013-03-22 12:03:36
Owner djao (24)
Last modified by djao (24)
Numerical id 18
Author djao (24)
Entry type Definition
Classification msc 17B99
Related topic CommutatorBracket
Related topic LieGroup
Related topic UniversalEnvelopingAlgebra
Related topic RootSystem
Related topic SimpleAndSemiSimpleLieAlgebras2
Defines Jacobi identity
Defines subalgebra
Defines ideal
Defines normalizer
Defines centralizer
Defines kernel
Defines homomorphism
Defines center
Defines centre
Defines abelian Lie algebra
Defines abelian