BOLETÍN DE SEGUIMIENTO A GRANDES PROYECTOS EN EL

Directly computing with the exceptional Lie algebras can be difficult due to their size and the complexity of their descriptions. Thus it is often easier to consider classical maximal subalgebras of the exceptional Lie algebras, and then try to extend results on the classical cases to the exceptional cases. Note that the maximal subalgebras are not always simple, but can at least be chosen to be direct sums of classical simple Lie al- gebras.

Forha Cartan subalgebra ofg, pick a maximal subalgebrak⊂gwith the same Cartan subalgebra such that the closure of the roots ofkunderWGis the set of roots ofg. We

can take a basis ofhand root vectors ofkandgrespectively as basis elements forkand g; denote byV the span of the root vectors ofgthat are not root vectors ofk. LetK⊂G

be the group corresponding tok.

We get a homomorphism r es : S(g) → S(k) by sending V to 0. We also get a map

M at(g)→M at(k) by sending A∈M at(g) to the submatrix where the coordinates are both ink. Combining these two maps gives a map

Res: (M at(g)⊗S(g))G→(M at(k)⊗S(k))K

In a matrix in (M at(g)⊗S(g))G, for a non-zero entry with one coordinate inkand one

inV, the entry must be in the kernel ofr es. Hence we get thatRes is an algebra ho- momorphism. ForA∈(M at(g)⊗S(g))G, denoteAk=Res(A).

Proof. Suppose that A_k=B_k. Then A_k−B_k=0k. So A−B has entries inS(V) on thek

submatrix. When restricting toh, this restricts to 0 on theksubmatrix. SinceWGallows

us to move roots ofgto roots ofk, we get thatA−B restricted tohmust also be 0 on theV submatrix. Hence, since the restriction of an element of (M at(g)⊗S(g))G toh

has non-zero entries only on the Cartan submatrix and the diagonal, we get thatA−B

restricted tohis 0 everywhere. But since

(M at(g)⊗S(g))G⊗F(g)=∼(M at(g)⊗S(h))WG_⊗F

G(h)

and since the base field doesn’t change if an element is 0 or not, we get that restriction tohis an injection. HenceA−B =0.

ThereforeResis an injection.

Using this injection, we can prove the following useful lemma:

Lemma 9.2.2(The Vector Restriction Lemma). For A∈Cg(g), if Res(A)vanishes on the

torus, then A is a multiple of M .

Proof. The exact descriptions of the family algebras for the classical cases has been

handled in the previous chapters. The observation to make is that the fake degrees of theW-representations thath⊗hdecomposes into match the degrees given by the el- ements of the family algebra that do not vanish on the torus, and hence anything that does vanish on the torus is a multiple ofM.

We handle the exceptional cases by reducing to the classical case. Forg6=G2,E6, we suppose that we have an element Awith vanishing torus part. Then the restriction of

Ato maximal subalgebrakalso has vanishing torus part. ForF4, we usek=B4, forE7 we use A7and for E8we useD8. We look at the part of the vector component of A that correspond to roots ofk. By the first part of the restriction lemma, this part is a multiple ofM restricted to this part and hence can be written asM|kP|k. P|k isW(k)-

invariant, and since both A|kandM|k are invariant under the subgroup ofW(g) that

fixesk,P|kis also invariant under that subgroup ofW(g) and hence can be extended to

the entire root system ofg; we denote the extension byPe. Since the extension ofM|_k

byW(g) is justM, we get thatA=MPe.

The cases ofG2andE6have to be handled separately, sinceE6has no maximal subal- gebras that are simple, whileG2hasA2but the image of the roots of A2underW(G2) misses some roots ofG2. So we consider the case of family algebras for semisimple Lie algebras.

For a semisimple algebrag⊕k, the adjoint group isG xKwhereGandK are the adjoint groups forgandkrespectively. For a representation ofg⊕kthat decomposes asU⊕V

wheregacts trivially onV andkacts trivially onU, we can write the corresponding family algebra by distributing End(U⊕V) and noting thatG leaves fixedV,V∨ and

S(k), and similarly forK:

(End(U⊕V)⊗S(g⊕k))G×K =(End(U)⊗S(g))G⊗I(k)⊕(U⊗S(g))G⊗(Vv⊗S(k))K

⊕(V⊗S(k))K⊗(U∨⊗S(g))G⊕(End(V)⊗S(k))K⊗I(g)

So we get that the family algebra ofU⊕V breaks into four blocks, depending on if the coordinates are inU orV. So it remains a free module overI(g⊕k)=I(g)⊗I(k).

ForG2, we use the maximal subalgebra k=k1⊕k2, wherek1∼=k2∼= A1, with one A1 containing a pair of long roots and the other a pair of short roots. The family alge- bra decomposes into four pieces, as above. For E6, we use the maximal subalgebra k=k1⊕k2⊕k3, wherek1∼=k2∼=k3∼=A2, and adjoint groupK1×K2×K3. We have that the family algebra of the adjoint representation ofkdecomposes into nine pieces, three of which are copies of the family algebra for A2tensored with two extra copies ofS(A2). Note that forX ∈ki andY ∈kj fori 6=j, the orbit ofX underKi spanski and thus the

orbit of (X,Y) underKi spans (ki,Y).

So suppose that we have an element ofCk(k) which vanishes on the torus. This ele-

ment vanishes on (h1,h2) whereh1 andh2 are ki andkj respectively. If i 6= j, then

by the above all entries in (ki⊗k∨_j) vanish. Ifi =j, then we’re in a copy of the family

algebra ofmki tensored withS(km)Km⊗S(kn)Kn fori,m,ndistinct, and thus since the

vector restriction lemma holds forki, we get that thekivector part of the family algebra

element thus of the formMiPi. BecauseW(g) intertwines the actions of theki, we get

that the element that the action ofW(g) sends thePi to each other, and henceMiPi

extends to aW(g)-invariant element on the vector part of the family algebra ofg, as was desired.

Chapter 10

The Exceptional Lie algebras

10.1 Invariants

The exceptional Lie algebras are not uniform in many senses, so we give a table listing the data for the reference representation and the exponents:

Table 10.1: Exponents for the Exceptional Lie algebras g dimV Exponents G2 7 1, 5 F4 26 1, 5, 7, 11 E6 27 1, 4, 5, 7, 8, 11 E7 56 1, 5, 7, 9, 11, 13, 17 E8 248 1, 7, 11, 13, 17, 19, 23, 29

tional Lie algebras carry invariants of degree higher than 2. ForG2, the 7 dimensional representation has a degree 2 symmetric invariant and a degree 3 antisymmetric in- variant. ForF4, the 26 dimensional representation has a degree 2 symmetric invariant and a degree 3 symmetric invariant. ForE6, the 27 dimensional representation has a degree 3 symmetric invariant. ForE7, the 56 dimensional representation has a degree 2 antisymmetric invariant and a degree 4 symmetric invariant. ForE8the 248 dimen- sional representation has a degree 2 symmetric invariant and a degree 3 antisymmetric invariant. The existence of these higher degree invariants gives us elements of (T(g))G

that are not obviously traces over the reference representation.