Lie algebra
A Lie algebra over a field is a vector space
with a bilinear map , called the Lie bracket and denoted . It is required to satisfy:
- 1.
for all .
- 2.
The Jacobi identity
: for all .
1 Subalgebras & Ideals
A vector subspace of the Lie algebra is a subalgebra if is closed under
the Lie bracket operation, or, equivalently, if itself is a Lie algebra under the same bracket operation as . An ideal of is a subspace for which whenever either or . Note that every ideal is also a subalgebra.
Some general examples of subalgebras:
- •
The center of , defined by . It is an ideal of .
- •
The normalizer
of a subalgebra is the set . The Jacobi identity guarantees that is always a subalgebra of .
- •
The centralizer
of a subset is the set . Again, the Jacobi identity implies that is a subalgebra of .
2 Homomorphisms
Given two Lie algebras and over the field , a homomorphism from to is a linear transformation such that for all . An injective
homomorphism is called a monomorphism
, and a surjective
homomorphism is called an epimorphism
.
The kernel of a homomorphism (considered as a linear transformation) is denoted . It is always an ideal in .
3 Examples
- •
Any vector space can be made into a Lie algebra simply by setting for all vectors . The resulting Lie algebra is called an abelian
Lie algebra.
- •
If is a Lie group, then the tangent space at the identity
forms a Lie algebra over the real numbers.
- •
with the cross product
operation is a nonabelian
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 symmetries. One joint project with FelixKlein called for the classification of all finite-dimensional
groupsacting on the plane. The task seemed hopeless owing to the generallynon-linear nature of such group actions
. 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 conclusion
.
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.
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 |