I am currently learning Lie algebras as they will be really important for my master project. Here are some notes that I have taken about basic notions. I will probably post more stuff about what I am doing later. I am using a program that convert LaTex to WordPress, so I hope it will be readable. Most of the following notions about Lie Algebras were taken in Humphreys’ book. The book has a good basic knowledge in classification of Lie Algebras.
Definition (Lie Algebra)
A vector space over a field , with an operation , denoted and called the bracket or commutator of and , is called a Lie Algebra over if the following axioms are satisfied:
Bilinearity: The bracket operation is bilinear, i.-e. ,
Antisymmetriy: for all .
Jacobi identity: .
Notice that we can deduced from these properties that .
Definition (Lie Isomorphisms)
We say that two Lie algebras over are isomorphic if there exists a vector space isomorphism satisfying for all .
Definition (Lie Subalgebras)
A Lie Subalgebras is a subspace of such that whenever .
Lie algebras first arose with the study of Lie groups, which are groups that are also smooth manifolds. Lee gives a good introduction in his book to Lie algebras from that point of view.
Let and be smooth vector fields on a smooth manifold . Then the operator given by:
is a Lie bracket. It is a smooth vector fields.
Definition (Left-Invariant Vector Fields)
Suppose is a Lie group. Any defines maps , called Left translation and Right translation, respectively, by
A vector field on is said to be left-invariant if
The space of all smooth vector fields on a smooth manifold is a Lie algebra under the Lie bracket of vector fields. Given is a Lie group, the set of all smooth left-invariant vector fields on is a Lie subalgebra of and is therefore a Lie algebra. It is called the Lie algebra of G and is denoted by . The reason why this algebra is of particular importance is given by the following theorem:
Let be a Lie group. The evaluation map , given by , is a vector space isomorphism. Thus, dim = dim.
Let be a finite dimensional vector space over , denote by End the set of linear transformations . Define the bracket as . With this operation End is called the general linear algebra, denoted by . One interesting theorem links to .
is isomorphic to .
Any subalgebra of a Lie algebra is called a linear Lie algebra.
Some Lie algebras of linear transformations arise most naturally as derivations of algebras. By an F-algebra, we simply mean a vector space over endowed with a bilinear operation . By a derivation of , we mean a linear map satisfying the Liebnitz’s rule: . Let Der denote the set of all derivations of . Then, Der is a subspace of End . Since the commutator of two derivations is again a derivation, Der is a subalgebra of . Certain derivations arise quite naturally. If is an endomorphism of , which is denoted ad . In fact, ad Der. Derivations of this form are called inner and all other are called outer. The map sending to ad is called the adjoint representation of .
It is know (cf. Jacobson Chapter VI) that every finite dimensional Lie algebra is isomorphic to some linear Lie algebra.
Definition (Abelian Lie algebras)
A Lie algebra is called abelian if for all .
1.1. Ideals and homomorphism
A subspace of a Lie algebra is called an ideal of if , together imply .
Example 1 The centre is an ideal. Another important example of ideal is the derived algebra of , denoted , which is analogous to the commutator subgroup of a group. It consists of all linear combinations of commutators . It is clear that if are two ideals, then is an ideal. Similarly, is an ideal.
If has no ideals except itself and , and if moreover , then is called a simple Lie algebra.
In the case where is not simple, and is a proper ideal of , then the construction of a quotient algebra is formally the same as the construction of a quotient ring: A quotient space with Lie multiplication defined by .
The normalizer of a subalgebra (or just subspace) of is defined by . By the Jacobi identity, is a subalgebra of . It is the largest subalgebra of which includes as an ideal. If , we call self-normalizing. The centralizer of a subset of is . It is a subalgebra of .
A linear transformation is called an homomorphism if .
Note that is an ideal of . Also, the standard isomorphism theorems hold.
Let be a vector space over . A Representation of a Lie algebra is a homomorphism .
An important example to bear in mind is the adjoint representation . The kernel of ad is . If is simple, then , so is a monomorphism. Then, any simple Lie algebra is isomorphic to a linear Lie algebra.
1.2. Solvable and nilpotent Lie algebras
The derived series of is defined by is called solvable if for some .
Let be a Lie algebra.
If L is solvable, then so are all subalgebras and homomorphic images of L.
If I is a solvable ideal of such that is solvable, then itself is solvable.
If are solvable ideals of , then so is .
We are now about to define really important Lie algebras. Let be a maximal solvable ideal of . Such an ideal exists. Call this ideal the radical of and denoted Rad . If and Rad , then is called semisimple. If is not solvale, i.e., Rad , then is semisimple.
Define a sequence of ideals of (the lower central series) by is called nilpotent if for some .
Let L be a Lie algebra.
If is nilpotent, then so are all subalgebras and homomorphic images of .
If is nilpotent, then so is .
If is nilpotent, then .
Now if is any Lie algebra, and , we call x ad-nilpotent if ad is a nilpotent endomorphism.
All elements of are ad-nilpotent if and only if is nilpotent.
James E. Humphreys, Introduction to Lie Algebras and Representation theory, Second Edition, Springer-Verlag, New-York, 1972.
John M. Lee, Introduction to Smooth Manifolds, Springer-Verlag, New-York, 2003.