12. Exceptional Lie Algebras and Automorphisms We Have Constructed Four Inﬁnite Families of Semisimple Algebras

MATH 223A NOTES 2011 LIE ALGEBRAS 51

12. Exceptional Lie algebras and automorphisms We have constructed four inﬁnite families of semisimple algebras:

(1) sl(n + 1, F ) has type An (2) so(2n, F ) has type Dn (3) so(2n + 1, F ) has type Bn (4) sp(2n, F ) has type Cn. Theorem 12.0.2. If L is a simple Lie algebra over an algebraically closed field of char- acteristic zero then L either belongs to one of the four infinite families listed above or it is isomorphic to one of the five exceptional Lie algebras of type E6, E7, E8, F4, G2.

We will construct a Lie algebra of type G2 later. For now we will just construct the root systems of type E8 and F4.

12.1. E6, E7, E8. The root systems E6, E7 can be constructed from the root system (E, Φ) of type E8 as follows. Let ∆! be the subset of the base ∆ corresponding to the subdiagram of E corresponding to E or E . Let E! be the span of ∆! in E. Let Φ! = Φ E!. Then 8 6 7 ∩ (E/Φ!) will be a root system of type E6 or E7. Therefore, it suﬃces to construct a root system of type E8. Let E = R8. The root system is: 1 8 Φ = ! ! ! with even number of signs {± i ± j} ∪ 2 ± i − ! i=1 # " Note that the ﬁrst part !i !j is the root system of type D8. Also, all of these roots have the same length. A{±base±for}this root system is

1 7 ! ! , , ! ! , ! + ! , ! + ! ! 2 − 3 · · · 7 − 8 7 8 2 1 8 − i $ i=2 % " We show that this is a base: Since the ﬁrst 7 roots form a base for D7 we know that any root of the form !i !j for 2 i < j n is a positive linear combination of the ﬁrst 7 roots. By adding twice± the last≤ simple≤root α , we can get ! ! : 8 1 − 2 ! ! = 2α + (! + ! ) + (! + ! ) + (! ! ) 1 − 2 8 3 4 5 6 7 − 8 with a single α plus roots of the form ! ! we get any sum 1 ! with the sign of ! 8 i ± j 2 ± i 1 being positive (with an even number of - signs). Note that α8 is perpendicular to every other simple root except for ! ! = α . So, the Dynkin diagram& is: 7 − 8 6 ◦ ◦ ◦ ◦ ◦ ◦ ◦ α1 α2 α3 α4 α6 α8 ◦ α7 52 MATH 223A NOTES 2011 LIE ALGEBRAS

4 12.2. F4. This root system is given by E = R and Φ = ! ! ! 1 ( ! ! ! ! ) {± i} ∪ {± i ± j} ∪ { 2 ± 1 ± 2 ± 3 ± 4 } where these roots have lengths 1, √2, 1 resp. The base is given by: ! ! , ! ! , ! , 1 ( ! ! ! + ! ) 1 − 2 2 − 3 3 2 − 1 − 2 − 3 4 The last two roots are smaller so the Dynkin diagram is: = = ◦ ◦ ⇒ ◦ ◦ α1 α2 α3 α4 12.3. Automorphisms of Φ. Definition 12.3.1. Aut(Φ) := f O(E) f(Φ) = Φ { ∈ | } The orthogonal group O(E) is the group of all linear isometries of E, i.e., f preserves the inner product, equivalently, f preserves lengths and angles. Lemma 12.3.2. The Weyl group W is a normal subgroup of Aut(Φ). Proof. Clearly, W is a subgroup of Aut(Φ) since it sends roots to roots and is generated by reflections. It remains to show that W is a normal subgroup. W is generated by the reflections σ where β Φ. For any τ Aut(Φ) we have: β ∈ ∈ 1 τσ τ − = σ W β τ(β) ∈ 1 So, W is normal in Aut(Φ). The equation follows from the fact that τσβτ − sends τ(β) to τ(β) and also fixes every x τ(β). − ⊥ ! Theorem 12.3.3. Aut(Φ) ∼= W ! Γ where Γ is the group of all τ Aut(Φ) so that τ(∆) = ∆. ∈ Proof. To prove that Aut(Φ) is this semi-direct product it suffices to show that W Γ = 1 and that W Γ = Aut(Φ). ∩ { }To prove the first statement we recall that W acts simply transitively on the set of all bases. So, the only element of W which fixes ∆ is the identity. To prove the second statement, take any τ Aut(Φ). Then τ(∆) = ∆! is another base for Φ. So, there ∈ 1 1 is some w W so that w(∆) = ∆!. Then w− τ Γ and τ = w(w− τ) W Γ. So, ∈ ∈ ∈ Aut(Φ) = W ! Γ. ! Theorem 12.3.4. Γ is the group of automorphisms of the Dynkin diagram. Before proving this, I pointed out that the automorphisms of the diagram are very easy to compute.

(1) For An, n 2, Γ = Z/2. ≥ (2) For Dn, n 5, Γ = Z/2. ≥ (3) For D4, Γ = S3, the symmetric group on 3 letters. (4) For E6, Γ = Z/2. (5) For all other Dynkin diagrams, Γ is trivial. MATH 223A NOTES 2011 LIE ALGEBRAS 53

Proof. Take any τ Γ. Then τ(∆) = ∆ means that τ permutes the elements of ∆ and preserves lengths and∈ angles. Therefore, τ gives an automorphism of the Dynkin diagram. Conversely, suppose that τ is an automorphism of the Dynkin diagram. Then τ sends basic roots to basic roots. Since the basic roots form a basis for E, τ gives a linear isometry of E. The Dynkin diagram contains the information of which simple roots are long and which are short. So, τ sends long/short simple roots to long/short simple roots and also preserves the angles. So, τ is an isometry of E. It remains to show that τ sends roots to roots. (I will ﬁnish this later.) !