Introduction to Lie Groups and Lie Algebras November 11, 2004

Gabriel C. Drummond-Cole

December 3, 2004

Recall: R = R_{+} ∪ R_{−} ⊂ E and r = dim E. Π = {α_{1}, ⋯, α_{r}} ⊂ R_{+} is the set of simple roots. For any α ∈ R_{+}, α = ∑ n_{i}α_{i} for n_{i} ≥ 0.

Example 1 R = {e_{i} − e_{j}, i ≠ j} ⊂ ℝ^{n+1}/(1, ⋯, 1). R_{+} = {e_{i} − e_{j}, i < j}. Π = {α_{1} = e_{1} − e_{2}, ⋯, α_{n} = e_{n} − e_{n+1}}.

Can one recover R from Π, and for different choices of polarization do we get “different” Π?

Note that R = R_{+} ∪ R_{−} depends only on the signs of (t, α). It changes when (t, α) = 0, i.e., t crosses the hyperplane H_{α} = {λ|(α, λ) = 0}.

Definition 1 A Weyl chamber is a connected component of E\ ∪ H_{α}.

Each Weyl chamber is the intersection of half-spaces so it is a convex polygonal cone. There are only finitely many.

Lemma 1 C → R = R_{+}^{C} ∪ R_{−}^{C}, where R_{+}^{C} = {α : α(C) > 0} is a bijection between Weyl chambers and polarizations.

The inverse map is R = R_{+} ∪ R_{−} to C_{+} = {t ∈ E|(α, t) > 0 for all α ∈ R_{+}} = {t ∈ E|(t, α_{i}) > 0 for all α_{i} ∈ Π}.

If (t, α_{i}) > 0 for all α_{i} ∈ Π then for all α ∈ R_{+} (t, α) = ∑ n_{i}(t, α_{i}) > 0.

Every Weyl chamber is bounded by exactly r hyperplanes, called the walls of the Weyl chamber.

Definition 2 The Weyl group W of R ⊂ GL(E) is the group generated by S_{α}, α ∈ R.

Example 2 Let R = A_{n}, the root system of 𝔰𝔩(n + 1). Then α_{ij} = (e_{i} − e_{j}) just as s_{ij} transposes i and j. So S_{α_{ij}} sends (t_{1}, ⋯, t_{i}, ⋯, t_{j}, ⋯, t_{n+1} to (t_{1}, ⋯, t_{j}, ⋯, t_{i}, ⋯, t_{n+1}. So W = S_{n+1}.

Some properties are that W ⊂ O(E) and |W | > ∞.

The first is obvious, the second is because W ⊂ S(R).

Theorem 1 1. W acts transitively on the set of Weyl chambers.

2. Fix R = R_{+} ∪ R_{−}. Then W is generated by S_{i} = S_{α_{i}} for α_{i} ∈ Π.

3. For every root α there exists w ∈ W such that w(α) ∈ Π so W Π = R.

Corollary 1 1. R can be recovered from Π.

2. Say R = R_{+} ∪ R_{−} = R′_{+} ∪ R′_{−}, with simple root sets Π, Π′. Then Π′ = wΠ for some w ∈ W.

Lemma 2 Fix R = R_{+}∪R_{−} and let W′ = ⟨S_{i}⟩, the group generated by the simple reflections. Then W′ acts transitively on the set of Weyl chambers.

It suffices to prove that for a Weyl chamber C, there exists w ∈ W′ such that w(C_{+}) = C. Let ℓ be the number of hyperplanes seperating C and C_{+}. Take a segment connecting a point in C with a point in C_{+}. It will intersect all these ℓ hyperplanes and only them at isolated points. Then C_{1} = S_{β_{1}}(C_{0}), C_{2} = S_{β_{2}}(C_{1}), and so on to C = C_{l} = S_{β_{ℓ}}(C_{ℓ−1}). ... Proof of theorem from lemma:

1. Obvious

2. Let β ∈ R and C be a Weyl chamber such that H_{β} is a wall of C. Then C = w(C_{+}); H_{β} = w(wall of C_{+}) = w(H_{α_{i}}) for some α_{i} ∈ Π and w ∈ W′. So β = ±wα_{i}. Then S_{β} = ws_{i}w^{−1} ∈ W′.

3. We already know β = ±w(α_{i}). If β = −wα_{i} then β = wS_{i}(α_{i}).