Lie algebras

It cannot be stressed enough how ubiquitous Lie algebras (specifically their representations) are in theoretical physics. One reason they arise is because for every Lie group (for example, GL(n,C), the group of all invertible nxn matrices) one obtains a Lie algebra by looking at the tangent space at the identity element. In this way one can reduce problems about representations of Lie groups (which turn up naturally when we consider the symmetry of physical systems) to problems about representations of Lie algebras, for which there is powerful algebraic machinery available.

Definition Let k be a field (for example the real numbers or complex numbers). A Lie algebra (over k) is a k-vector space g equipped with a binary operation (x,y) |--> [x,y] called the bracket satisfying:

1. [ax+by,z]=a[x,z] + b[y,z] and [z,ax+by]=a[z,x] + b[z,y], for a,b in k and x,y,z in g (bilinearity)
2. [x,x]=0
3. [x,[y,z]] + [y,[z,x]] + [z,[x,y]]=0 (Jacobi identity)

A Lie algebra is called abelian if [x,y]=0 for all x,y in g. Obviously the most interesting ones are non-abelian. A one-dimensional Lie algebra is forced to be abelian but there is a non-abelian two dimensional Lie algebra. It has a basis {x,y} such that [x,y]=y.

The most important example of a Lie algebra is gl(n,k) which is the collection of all nxn matrices over k. What is the bracket? For matrices A,B define their bracket by [A,B]=AB-BA. It's not difficult to check that all the axioms are satisfied and this bracket makes gl(n,k) into a Lie algebra.

As usual in algebra we think about substructures, homomorphisms and ideals. A subspace h of a Lie algebra of g is a Lie subalgebra if it is closed under the bracket, that is [x,y] is in h whenever x,y are in h. A Lie subalgebra is a Lie algebra in its own right. A function f:g-->h between Lie algebras is a homomorphism if it is a linear transformation and satisfies f([x,y])=[f(x),f(y)], for all x,y in g. An ideal h of a Lie algebra g is a subspace such that [x,y] is in h for each x in h and y in g. It's not hard to see that the quotient vector space g/h gets the structure of a Lie algebra in this case.

The most important examples of Lie algebras occur as Lie subalgebras of gl(n,k) (in fact a theorem of Ado tells us that if the field has characteristic zero every finite-dimensional Lie algebra turns up this way). Here are some examples:

• sl(n,k) consists of all nxn matrices which have trace zero, that is the sum of the diagonal elements is zero. Because the trace of AB is the same as the trace of BA we see that sl(n,k) is a Lie subalgebra of gl(n,k).
• t(n,k) the collection of upper triangular matrices and n(n,k), the strictly upper triangular matrices, are also Lie subalgebras of gl(n,k).

All the examples I gave so far were finite-dimensional but there are also some very interesting infinite-dimensional ones. In general given two vector fields x,y their commutator xy-yx is another vector field and this operation defines a bracket. For example consider g the vector fields on C\{0}. This space is spanned by zⁿd/dz, for n in Z. Defining the bracket as above, [x,y]=xy-yx, one obtains an infinite-dimensional Lie algebra. In particular we have
[zⁿd/dz,z^md/dz]=(m-n)z^n+m-1d/dz

This Lie algebra is called the Virasoro Lie algebra.