Introduction to Lie Groups and Lie Algebras November 9, 2004

Gabriel C. Drummond-Cole (based on the notes of Jaimie Thind) November 30, 2004

Recall: 𝔤 is a semisimple complex Lie algebra, and 𝔥 ∈ 𝔤 is a Cartan subalgebra so that 𝔤 = 𝔥 ⊕ ⨁_{α∈R} 𝔤_{α}. Then R ∈ 𝔥^{∗} has a number of properties: Definition 1 Let E be a finite dimensional real Euclidean space. A finite subset R ⊂ E\{0} is called a root system if it satisfies

1. R spans E.

2. If α, β ∈ R then S_{α}(β) = β − \frac{2(β,α)}{(α,α)}α = β − ⟨α^{∨}, β⟩α ∈ R. This is reflection in the hyperplane perpendicular to α. Here, also, α^{∨} ∈ E^{∗} is defined by the above by ⟨α^{∨}, λ⟩ = \frac{2(α,λ)}{(λ,λ)}.

3. If α, β ∈ R then ⟨α^{∨}, β⟩ = \frac{2(α,β)}{(α,α)} ∈ ℤ. R is called reduced if α ∈ R implies 2α ∉ R. Some remarks:

1. If R_{1} ⊂ E_{1}, R_{2} ⊂ E_{2} are root systems then R_{1} ⊔ R_{2} ⊂ E_{1} ⊕ E_{2} is a root system.

2. If we rescale a root system α ↦ cα for c ∈ ℝ_{∗} it will again be a root system.

3. The third part of the definition means that the projection of β on α is an integer multiple of \frac{α}{2}.

Example 1 [figure] (Pictures)

Assume we are working with reduced root systems. Back to Lie algebras, let 𝔤 be as above. So 𝔥^{∗} = 𝔥_{ℝ}^{∗} ⊕ i𝔥_{ℝ}^{∗}. Then ( , )|_{𝔥_{ℝ}^{∗}} is positive definite and R ⊂ 𝔥_{ℝ}^{∗} is a reduced root system. Example 2 Take 𝔤 = 𝔰𝔩(n + 1, ℂ). Then take 𝔥 to be traceless diagonal matrices

\{( \begin{matrix}λ_{1} \\ & . & . & . \\ & & & & λ_{n+1} \end{matrix} ) | ∑ λ_{i} = 0\} ⊂ ℂ^{n+1}.

We’ll index the roots by α_{ij}, i ≠ j. Let 𝔤_{α_{ij}} = ℂE_{ij}. Then [h, E_{ij}] = (λ_{i} − λ_{j})E_{ij} = (ϵ_{i} −

ϵ_{j})(h)E_{ij}, where ϵ_{i}(( \begin{matrix}λ_{1} \\ & . & . & . \\ & & & & λ_{n+1} \end{matrix} )) = λ_{i}.

Then 𝔥^{∗} = ℂϵ_{1} ⊕ ⋯ ⊕ ℂϵ_{n+1}/ ∑ ϵ_{i} = 0. So the root system is {ϵ_{i} − ϵ_{j}, i ≠ j}.

𝔥_{ℝ} = \{( \begin{matrix}λ_{1} \\ & . & . & . \\ & & & & λ_{n+1} \end{matrix} ) |λ_{i} ∈ ℝ, ∑ λ_{i} = 0\}.

So 𝔥_{ℝ}^{∗} = ℝϵ_{1} ⊕ ⋯ ⊕ ℝϵ_{n+1}/ ∑ ϵ_{i} = 0. Theorem 1 Let α, β ∈ R and α ≠ cβ. Then, up to interchanging α, β, we must have one of the following.

1. α ⊥ β (like A_{1} × A_{1}).

2. (a) |α| = |β|, ϕ = π/3. (b) |α| = |β|, ϕ = 2π/3. These cases correspond to A_{2}.

3. (a) |α| = \sqrt{2}|β|, ϕ = π/4. (b) |α| = \sqrt{2}|β|, ϕ = 3π/4. These cases correspond to B_{2}.

4. (a) |α| = \sqrt{3}|β|, ϕ = π/6. (b) |α| = \sqrt{3}|β|, ϕ = 5π/6. These cases correspond to G_{2}.

Look at n_{αβ} = ⟨b^{∨}, α⟩ = \frac{2(β,α)}{(α,α)} ∈ ℤ. This is \frac{2|α||β|}{|α|^{2}} cos ϕ = \frac{2|β|}{|α|} cos ϕ. Then n_{αβ}n_{βα} = 4 cos^{2} ϕ. The options are • n_{αβ} = nβα = 0. This is case one.

• n_{αβ} = ±1, nβα = ±1 so that cos^{2} ϕ = \frac{1}{4} and cos ϕ = ±\frac{1}{2}. This is case two above.

• n_{αβ} = ±1, nβα = ±2 so that cos^{2} ϕ = \frac{1}{2} and cos ϕ = ±\frac{1}{\sqrt{2}}. This is case three above.

• n_{αβ} = ±1, nβα = ±3 so that cos^{2} ϕ = \frac{3}{4} and cos ϕ = ±\frac{\sqrt{3}}{2}. This is case four above. Theorem 2 The only rank two root systems are A_{1} × A_{1}, A_{2}, B_{2}, G_{2}. Let R be a rank two root system. Take α, β ∈ R such that the angle between them is the smallest possible. Then we must be in one of the cases 1, 2a, 3a, 4a (to eliminate the b cases reflect to get a root with smaller angle). If we are in the case 2a, by reflections it must contain all of the six roots in A_{2}; if it contained anything else we’d get a pair of roots with smaller angle. Similarly for 3a, 4a. Now let t ∈ E such that (t, α) ≠ 0 for all α ∈ R. Then R = R_{+} ⊔ R_{−}, where R_{+} = {α ∈ R|(α, t) > 0} and R_{−} = {α ∈ R|(α, t) < 0}. Example 3 Look at the root system A_{n} of 𝔰𝔩(n + 1, ℂ). Take t ∈ 𝔥_{ℝ}^{∗} as t = (t_{1}, ⋯, t_{n+1}) with t_{1} > ⋯ > t_{n+1}. Then (t, α_{ij}) = (t, ϵ_{i} − ϵ_{j}) = t_{i} − t_{j}, so this is positive when i < j, negative when i > j. Then R_{+} = {α_{ij}|i < j}; R_{−} = {α_{ij}|i > j}. So ⊕_{α∈R_{+}}𝔤_{α} is the set of strictly upper triangular matrices, and ⊕_{α∈R_{−}}𝔤_{α} is the set of strictly lower triangular matrices. Definition 2 A root α ∈ R_{+} is simple if it cannot be written as a sum of positive roots. Let Π = {α_{1}, ⋯, α_{k}} be the set of simple roots. Lemma 1 Any positive root can be written as α = ∑ n_{i}α_{i} for n_{i} ∈ ℤ_{+}. Lemma 2 (α_{i}, α_{j}) ≤ 0. Lemma 3 {α_{i}, …, α_{k}} are linearly independent The proof is an exercise. In 𝔰𝔩(n + 1), the simple roots are ϵ_{1} − ϵ_{2}, ⋯, ϵ_{n} − ϵ_{n+1}.