Lie Groups and Representation Theory

Structure of finite-dimensional semisimple Lie algebras #

Let \(\mathbb{F}\) be a field. A vector space \(\mathfrak{g}\) over \(\mathbb{F}\) is called a Lie algebra if there exists an \(\mathbb{F}\)-bilinear map \(\mathfrak{g\times g\to g}\) \((x,y)\mapsto [x,y]\) (called the commutator/Lie bracket) such that

• \([x,x]=0\) (alternating)
• \([x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0\) (Jacobi identity)

for all \(x,y,z\in\mathfrak{g}.\) Lie algebras are a class of non-associative algebras - the Jacobi identity controls how non-associative the Lie bracket is.

Note: if \(\text{char}(\mathbb F)\neq2\) then \([x,x]=0\;\forall x\in\mathfrak g\) is equivalent to \([x,y]=-[y,x]\;\forall x,y\in\mathfrak g.\) This is because

$$[x,y]=[x,y]+0=[x,y]-[y+x,y+x]$$

$$=[x,y]-[y,x]-[y,y]-[x,x]-[x,y]=-[y,x].$$

On the other hand, \([x,y]=-[y,x]\implies[x,x]=-[x,x]\implies 2[x,x]=0\implies\text{char}(\mathbb F)=2\) or \([x,x]=0.\)

Definition. A vector subspace \(s\) of \(\mathfrak{g}\) is called a

• subalgebra if \([x,y]\in s\) for all \(x,y\in s,\) and
• ideal if \(x\in\mathfrak{g},\) \(y\in s\implies [x,y]\in s.\)

A Lie algebra \(\mathfrak{g}\) is called

• abelian if \([x,y]=0\) for all \(x,y\in s,\)
• simple if \(\{0\}\) and \(\mathfrak{g}\) are the only ideals of \(\mathfrak{g},\) and
• semisimple if \(\mathfrak{g}\) is a direct sum of its simple ideals.

Example.

1. Any vector space \(V\) is an abelian Lie algebra.
2. \(\mathbb{R}^3\) is a Lie algebra with \([x,y]=\vec{x}\times \vec{y}.\)
3. Any associative algebra \(A\) can be made into a Lie algebra by defining \([a,b]=ab-ba.\)

The affine Lie algebras #

Proof. cf. Biswas-Habib-Venkatesh¹.