In linear algebra, we have learned that an - is simply a vector space over equipped with an additional product structure that is , i.e. a binary operation where such that The Lie algebra is a special kind of algebra with additional requirements, and it can be seen as a generalization of the following important operation .
The notion of Lie algebra
We have the following immediate consequence from the definition
Proof. Applying (L1) to the bracket , we have: Using (L2), , so we have . ◻
In fact, if , then anti-commutativity implies (L2) because then Here we need the assumption that in the final implication since we need to conclude that .
As with all abstract structures in mathematics, it is very important to study the structure of a newly-defined algebraic object. There are mainly two ways we study the structure
We study "" maps in between them to understand whether two objects are essentially the same, relating complicated-seeming objects to simpler ones.
We study the to understand the internal structure and hierarchy within the object.
These two ideas leads to the following notions for Lie algebras:
We shall begin with the following simplest exmaple of Lie algebra.
Linear Lie Algebras
In this subsection, we introduce a family of important, examples of Lie algebras – the . The moral of the construction is the following: while it is difficult to define a bracket operation directly in the vector spaces themselves, it is much easier to define a Lie algebra structure on the linear maps from a vector space to itself. Therefore, we consider the following vector space:
We may verify that is itself a vector space with addition and scalar multiplication defined component-wise. With a choice of basis of , we may represent every element of as an matrix, assuming that . We may treat each matrix as a vector of length , then a basis for would simply be the matrices with only a in a particular entry, and zero in all others, just like the standard orthonormal basis of . More precisely, define then is a basis for , so .
We notice that matrix multiplication / composition of linear maps gives a natural product structure on , thereby making an -algebra. Moreover, this product structure also gives rise to a natural bracket operation on , as we have hinted at the beginning of the chapter:
Proof. It suffices to verify axioms (L1), (L2), (L3).
(L1) For any , : and similarly for the component.
(L2) This is trivial as .
(L3) We verify the Jacobi Identity. We first compute Similarly we have: and adding the three together, we see that all terms cancel, and we are left with , thus verifying the Jacobi identity.
We thereby have the notion of linear Lie algebras ◻
Next, we introduce four families of examples of linear Lie algebras: for , called the , that will be highly important for the development of the theory.
Proof. Firstly, we notice that is always a subalgebra of , as contains endomorphism with non-zero trace. Hence is at most . Next, it suffices to find a linearly independent set consisting of elements. For this purpose, we choose where and . This is clearly linearly independent, and its length is: so we must have that , and the above chosen basis will always be regarded as the standard basis for . ◻
To illustrate the basis more concretely, we write down the basis for : The reader is encouraged to write down the standard basis for given in the proof above. The above proof hints at the general method in finding a standard basis for linear Lie algebras:
Split matrices into blocks in a suitable fashion.
Compute to find restrictions on each block.
Find a basis for each block under the restriction, then put them together to form a basis of the entire algebra.
Proof. We adopt the same methodology as above. For any , we split into blocks in the same way as , i.e. where . We then notice that: and using the blocks Comparing each entry, we then see that are restriction on each block matrix. We find a basis for each block separately. For , notice that the diagonal entries are free of choice, and for the off-diagonal entries, as long as the entry in the lower-left triangular region ( where ) is chosen, then the reflection across the diagonal is determined. This is because must be a symmetric matrix. Hence, a basis for would be and for . Similarly for . For , we see that as long as the entries of are determined, will be determined by the relation , so together, the - diagonal blocks is generated by where . Putting these together gives a basis of , and counting gives  ◻
It is noticeable that based on the construction above, has the same standard basis as , given above by . This implies that is isomorphic to .
Finally, we include some other subalgebras of which are also important but do not belong to the classical algebras.
: If are subalgebras of a , we use or to denote the subspace of spanned by the commutators where .
Proof. We first show that . It is trivial to see that the LHS is contained in the RHS. For any , Let be the matrix whose th row is the same as the th row of , and the rest are all . We then have that and consequently: which shows that the RHS is contained in the LHS. We then show that . Again, LHS is contained in the RHS, and moreover, we see that which proves the equality. Using both identities, we then see that which completes the proof. (cf. Exercise 5). ◻
Lie Algebras of Derivations and the Adjoint Representation
Linear Lie algebras are subalgebras of , where is an arbitrary finite dimensional vector space. We get another important example of Lie algebra if we further require to be an -algebra itself:
The important fact about is that it is itself a Lie algebra with the bracket operation inherited from :
Proof. It is easy to verify that is a linear subspace of . We shall only verify that it is closed under the bracket operation: for any , we want to show that . We have: which completes the proof. ◻
As a Lie algebra is itself an -algebra by definition, the Lie algebra of derivations is well-defined, and perhaps the most important elements in are following:
As mentioned before, the map is a derivation of .
Proof. We simply verify the definition. By the anti-commutativity of the bracket (Proposition ), we may rewrite the Jacobi identity as: Hence which completes the proof. ◻
When a basis of is chosen, expressing as a matrix makes computations more convenient:
Abstract Lie Algebras
In all of our previous investigations, we have taken a more "global" approach in defining Lie algebras, by using familiar operations on known vector spaces that are well-studied. However, we may also take a more abstract approach in the definition as follows.
The first key observation is that using bilinearity, the value of the Lie bracket of any arbitrary vectors is by the value of the bracket of the basis vectors. In other words, if is any Lie algebra with basis , and suppose , then so the entire multiplication table can be recovered simply from the Lie brackets , or even more specifically, suppose: then the multiplication table is completely determined by , called the of the Lie algebra . In the language of matrices, is the -entry in the matrix representation of .
In general, not any set of scalars can define a Lie algebra structure. The specific relations they have to satisfy are given by the axioms (L1), (L2) and (L3) and are stated in Humphreys. The more significant ones are the ones given by (L2), which shows that , and anti-commutativity, which implies that . If we set all structure constants to , then clearly we get the abelian Lie algebra, and in fact, this is the only possible one-dimensional Lie algebra since the definition forces for any . Using this abstract point of view, we can actually do further and classify all two-dimensional Lie algebras
Proof. If , then we yield the abelian Lie algebra. If not, then we claim that there exists a basis such that . We first take to be a vector that spans the one-dimensional space of multiples of , and to be any vector independent to . Suppose , then: so it is a multiple of , and so must be some multiple of . Let , we then replace by , then we finally get . ◻
The natural follow-up question would then be: How can we move from the abstract to the concrete, and construct a more intuitive Lie algebra isomorphic to the ones defined abstract above? The key idea is to use the :