Lie algebra
A Lie algebra^{} over a field $k$ is a vector space^{} $\mathrm{\pi \x9d\x94\u20ac}$ with a bilinear map $[,]:\mathrm{\pi \x9d\x94\u20ac}\Gamma \x97\mathrm{\pi \x9d\x94\u20ac}\beta \x86\x92\mathrm{\pi \x9d\x94\u20ac}$, called the Lie bracket and denoted $(x,y)\beta \x86\xa6[x,y]$. It is required to satisfy:

1.
$[x,x]=0$ for all $x\beta \x88\x88\mathrm{\pi \x9d\x94\u20ac}$.

2.
The Jacobi identity^{}: $[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0$ for all $x,y,z\beta \x88\x88\mathrm{\pi \x9d\x94\u20ac}$.
1 Subalgebras & Ideals
A vector subspace $\mathrm{\pi \x9d\x94\u20af}$ of the Lie algebra $\mathrm{\pi \x9d\x94\u20ac}$ is a subalgebra^{} if $\mathrm{\pi \x9d\x94\u20af}$ is closed under^{} the Lie bracket operation, or, equivalently, if $\mathrm{\pi \x9d\x94\u20af}$ itself is a Lie algebra under the same bracket operation as $\mathrm{\pi \x9d\x94\u20ac}$. An ideal of $\mathrm{\pi \x9d\x94\u20ac}$ is a subspace $\mathrm{\pi \x9d\x94\u20af}$ for which $[x,y]\beta \x88\x88\mathrm{\pi \x9d\x94\u20af}$ whenever either $x\beta \x88\x88\mathrm{\pi \x9d\x94\u20af}$ or $y\beta \x88\x88\mathrm{\pi \x9d\x94\u20af}$. Note that every ideal is also a subalgebra.
Some general examples of subalgebras:

β’
The center of $\mathrm{\pi \x9d\x94\u20ac}$, defined by $Z\beta \x81\u2019(\mathrm{\pi \x9d\x94\u20ac}):=\{x\beta \x88\x88\mathrm{\pi \x9d\x94\u20ac}\beta \x88\pounds [x,y]=0\beta \x81\u2019\text{for all\Beta}\beta \x81\u2019y\beta \x88\x88\mathrm{\pi \x9d\x94\u20ac}\}$. It is an ideal of $\mathrm{\pi \x9d\x94\u20ac}$.

β’
The normalizer^{} of a subalgebra $\mathrm{\pi \x9d\x94\u20af}$ is the set $N\beta \x81\u2019(\mathrm{\pi \x9d\x94\u20af}):=\{x\beta \x88\x88\mathrm{\pi \x9d\x94\u20ac}\beta \x88\pounds [x,\mathrm{\pi \x9d\x94\u20af}]\beta \x8a\x82\mathrm{\pi \x9d\x94\u20af}\}$. The Jacobi identity guarantees that $N\beta \x81\u2019(\mathrm{\pi \x9d\x94\u20af})$ is always a subalgebra of $\mathrm{\pi \x9d\x94\u20ac}$.

β’
The centralizer^{} of a subset $X\beta \x8a\x82\mathrm{\pi \x9d\x94\u20ac}$ is the set $C\beta \x81\u2019(X):=\{x\beta \x88\x88\mathrm{\pi \x9d\x94\u20ac}\beta \x88\pounds [x,X]=0\}$. Again, the Jacobi identity implies that $C\beta \x81\u2019(X)$ is a subalgebra of $\mathrm{\pi \x9d\x94\u20ac}$.
2 Homomorphisms
Given two Lie algebras $\mathrm{\pi \x9d\x94\u20ac}$ and ${\mathrm{\pi \x9d\x94\u20ac}}^{\beta \x80\xb2}$ over the field $k$, a homomorphism^{} from $\mathrm{\pi \x9d\x94\u20ac}$ to ${\mathrm{\pi \x9d\x94\u20ac}}^{\beta \x80\xb2}$ is a linear transformation $\mathrm{{\rm O}\x95}:\mathrm{\pi \x9d\x94\u20ac}\beta \x86\x92{\mathrm{\pi \x9d\x94\u20ac}}^{\beta \x80\xb2}$ such that $\mathrm{{\rm O}\x95}\beta \x81\u2019([x,y])=[\mathrm{{\rm O}\x95}\beta \x81\u2019(x),\mathrm{{\rm O}\x95}\beta \x81\u2019(y)]$ for all $x,y\beta \x88\x88\mathrm{\pi \x9d\x94\u20ac}$. An injective^{} homomorphism is called a monomorphism^{}, and a surjective^{} homomorphism is called an epimorphism^{}.
The kernel of a homomorphism $\mathrm{{\rm O}\x95}:\mathrm{\pi \x9d\x94\u20ac}\beta \x86\x92{\mathrm{\pi \x9d\x94\u20ac}}^{\beta \x80\xb2}$ (considered as a linear transformation) is denoted $\mathrm{ker}\beta \x81\u2018(\mathrm{{\rm O}\x95})$. It is always an ideal in $\mathrm{\pi \x9d\x94\u20ac}$.
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 abelian^{} Lie algebra.

β’
If $G$ is a Lie group, then the tangent space at the identity^{} forms a Lie algebra over the real numbers.

β’
${\mathrm{\beta \x84\x9d}}^{3}$ with the cross product^{} operation is a nonabelian^{} three dimensional Lie algebra over $\mathrm{\beta \x84\x9d}$.
4 Historical Note
Lie algebras are sonamed in honour of Sophus Lie, a Norwegian mathematician who pioneered the study of these mathematical objects. Lieβs discovery was tied to his investigation of continuous transformation groups and symmetries^{}. One joint project with Felix Klein called for the classification of all finitedimensional^{} groups acting on the plane. The task seemed hopeless owing to the generally nonlinear nature of such group actions^{}. However, Lie was able to solve the problem by remarking that a transformation group can be locally reconstructed from its corresponding βinfinitesimal generatorsβ, that is to say vector fields corresponding to various 1parameter 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 finitedimensional Lie algebras of planar vector field; a project that Lie brought to a successful conclusion^{}.
This βlinearization trickβ 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  20130322 12:03:36 
Last modified on  20130322 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 