Lie Algebras

4 \(\msl _2k\)

The structure theory of semisimple Lie algebras is centered around the behavior of \(\msl _2k\). Before moving on let’s study it more carefully.

Let’s consider a root system for \(\msl _2\).

It has a basis \(e = \begin {bmatrix} 0 & 1\\0 & 0 \end {bmatrix},f = \begin {bmatrix} 0 & 0 \\ 1 & 0 \end {bmatrix},h = \begin {bmatrix} 1 & 0\\ 0 & -1 \end {bmatrix}\), and can be presented via the relations \([e,f]=h, [h,e]=2e,[h,f]=-2f\). \(kh\) is a Cartan subalgebra, and \(e,f\) generate the root spaces.

We can try to classify finite-dimensional irreducible \(\msl _2\)-modules. There is a standard one, given on a vector space \(V\) of dimension \(2\). \(\Sym ^nV\) is a representation of dimension \(n+1\).

Proposition 4.1. \(\Sym ^nV\) is a complete list of f.d irreducible \(\msl _2\)-representations.

Proof. Let \(v\) be a nonzero element of \(V\) in the kernel of \(e\). \(v,fv\) form a basis of \(V\). Note that \(e\) acts nilpotently with \(1\)-dimensional kernel \(v^{\otimes n}\). Thus any subrepresentation contains that vector, and since it generates the rest of the space by powers of \(f\), \(\Sym ^nV\) is irreducible.

Now let \(W\) be an any f.d irreducible representation. It decomposes into eigenspaces \(W_\lambda \) for \(h\), and \(eW_{\lambda } \subset W_{\lambda +2}, fW_{\lambda } = W_{\lambda -2}\). We see then by irreducibility \(\oplus _n W_{\lambda +2n} = W\), where the nonzero terms occur in a segment, and are one-dimensional as \([e,f] = h\). Since \(\tr h\) is \(0\), we must have the highest and lowest weights be opposite in signs, so \(\lambda \) is an integer \(n\). This is enough to see that \(\lambda \) uniquely characterizes the irrep, so it is \(\Sym ^n V\). □

By this theorem, we can understand the isomorphism class of any finite-dimensional \(\msl _2\)-module by looking at how \(h\) acts on the kernel of \(e\). Namely, every eigenvalue of \(n\) corresponds to a copy of \(\Sym ^n V\).

Corollary 4.2. Let \(W\) be a finite dimensional representation of \(\msl _2\), and \(v\) a vector such that \(ev = 0\). Then \(hv = \lambda v\), \(\lambda \in \ZZ _{\geq 0}\). \(\lambda \) is the largest \(k\) such that \(f^k v \neq 0\). Moreover, \(v,fv,\dots f^\lambda v\) are linearly independent, \(hf^kv=(\lambda -2k)f^kv\) and \(ef^kv = n(\lambda -n+1)f^{k-1}v\).

Corollary 4.3. A f.d representation of \(\msl _2\) is irreducible iff it satisfies any of the following conditions:
- • The kernel of \(e\) is \(1\)-dimensional.
- • The kernel of \(f\) is \(1\)-dimensional.
- • The eigenspaces of \(h\) are \(1\)-dimensional.

Given a finite-dimensional \(\msl _2\)-module \(V\), let \(\tau = \exp (e)\exp (f)\exp (-e)\).

Lemma 4.4. \(\tau \) swaps the \(-n\) and \(n\) \(h\)-eigenspaces of any f.d. \(\msl _2\)-module \(W\).

Proof. Since \(W\) is a sum of summands of tensor products of the standard representation, it suffices to verify for the standard representation. Here \(\tau \) is \(\begin {bmatrix} 0 & 1 \\ -1 & 0 \end {bmatrix}\). □

Definition 4.5. An \(\msl _2\)-triple is a homomorphism \(\msl _2 \to \mg \).

For any semisimple Lie algebra, we will get an \(\msl _2\)-triple for each root, so we will use the representation theory of \(\msl _2\) to show that the roots form a root system.