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(4.4.12) [gα,gα] =CHα.

We may suppose thatgadmits a compact real formuandhis the com-plexification of a Lie algebra t of u (either by assumption, or by Theo-rem 3.3.7). Since adH : u → u is skew-symmetric with respect to the real inner product−B : u×u → Rfor allH ∈ u, the rootα takes imaginary values ont. SinceBis negative definite ontandB(Hα, H) =α(H)∈iRfor allH∈t, we deduce thatiHα ∈t. SethR:=P

αRHα. Owing to (4.4.10),

(4.4.13) hR=it.

Moreover,

(4.4.14) The roots take real values onhR andB : hR×hR → Ris a real inner product.

Denote the restriction ofB tohR orhRbyh,iand writehα, αi = ||α||2 for α ∈ ∆; also, putH¯α = ||α2||2Hαand use (4.4.8) to findEα ∈ gα,Fα ∈ gα such thatB(Eα, Fα) = 1. Now, owing to (4.4.11),

[Eα, Fα] = ¯Hα, [ ¯Hα, Eα] = 2Eα, [ ¯Hα, Fα] =−2Fα, that is,

(4.4.15) g[α] := CH¯α +CEα +CFα is a subalgebra isomorphic to sl(2,C).

Also, iH¯α, Eα−Fα, i(Eα+Fα) span over Ra subalgebra isomorphic to su(2).

In the next section, we shall use our knowledge about representations ofg[α]to investigate the structure of root systems.

4.5 Root systems

Throughout this section, we fix a complex semisimple Lie algebra and a CSAh, with associated root system∆.

Fixα∈∆and putV :=h+P

cC\{0}g. In view of the Fundamental Calculation (4.4.4), the adjoint action ofgrestricted tog[α]∼=sl(2,C)defines a representationρ on V. Recall that the set of eigenvalues ofρ( ¯Hα), as a multiset, is a union of sets each of which has the form

(4.5.1) {−2m,−(2m−2), . . . ,2m} for some nonnegative integerm.

If0 6=X ∈ g, thenρ( ¯Hα)X = [ ¯Hα, X] = cα( ¯Hα)X = 2cX, implying that2c∈Z.

We also know some components ofV. In fact,ρis trivial onkerα⊂h(so ρ( ¯Hα)has eigenvalues0there) andρis irreducible ong[α]⊂CH¯α+gα+gα (soρ( ¯Hα)has eigenvalues0,2,−2there). In particular,W := kerα+g[α]

contains the zero eigenspace ofρ( ¯Hα)(that is, h). We claim thatW equals V. Indeed, if this is not the case, take an irreducible componentU in an invariant complement ofW inV. The subrepresentationU containsgfor somec, where the eigenvalue ofρ( ¯Hα)is the nonzero integern= 2c; in case nis even, U contains also a zero eigenvector, by the form (4.5.1), which is forbidden sinceW ⊃ h; in casenis odd,U contains an eigenvector with eigenvalue1, by the form (4.5.1), and this implies that β := 12α ∈ ∆; we now replaceα byβ in the definition ofV and deduce that2β = α ∈ ∆, which says thatρ( ¯Hβ)has an eigenvalue4, a contradiction to the above.

Now

(4.5.2) dimgα= 1.

and

(4.5.3) The only multiples ofα∈∆which are roots are±α.

Next, let α, β ∈ ∆, β 6= ±α, and considerρ = ad|g[α] acting onV = P

kZgβ+kα. ForX ∈gβ+kα, we have that

ρ( ¯Hα)X= [ ¯Hα, X] = (β+kα)( ¯Hα)X

=

2hβ, αi

||α||2 + 2k

X.

Sincedimgβ+kα = 1forβ+kα ∈ ∆, we see thatV is irreducible and the sequence of eigenvalues ofρ( ¯Hα)has the form2h||β,αα||2i+ 2k, where−p≤k≤ q, for integersp,q ≥0. Owing to (4.5.1), this set is symmetric with respect to0. In particular,−

2h||β,αα||2i−2p

= 2h||β,αα||2i + 2q. It follows that

(4.5.4) Forα,β ∈∆,β 6=±α, there exist integersp,q ≥0such that 2h||β,αα||2i =p−qwhereβ+kα∈∆if and only if−p≤k≤q.

{β+kα∈∆| −p≤k≤q}is called theα-string of roots throughβ.

Further,

(4.5.5) [gα,gβ] =gα+βifα,β,α+β ∈∆.

In factα+β∈∆impliesq≥1. If it were[gα,gβ] = 0, thenP

pk0gβ+kα would be a proper invariant subspace ofV =P

pkqgβ+kα, butV ad|g[α] -irreducible in view of the above discussion.

4.5. ROOT SYSTEMS 65 We further refine the information about the length of strings of roots.

(4.5.6) Ifα,β∈∆,β6=±α, then2hβ,αi

||α||2 ∈ {0,±1,±2,±3}.

Indeed we already know2hβ,αi

||α||2 is an integer; alsoα,βare linearly indepen-dent by (4.5.3), so the Cauchy-Schwarz inequality says that

0≤

2hβ, αi

||α||2 ·

2hα, βi

||β||2

= 4 hβ, αi2

||α||2||β||2 <4.

Further:

(4.5.7) Ifα,β∈∆,β6=±α, then theα-string throughβhas length≤4.

In fact, that string coincides with theα-string throughγ = β−pα, that is, {γ+kα |0≤k≤p+q}, and due to (4.5.4) and (4.5.6),

−2hγ, αi

||α||2 =p+q≤3.

Weyl group

Consider nexthR =P

αRα(resp. hR) equipped with the inner product h,iinduced from the Cartan-Killing form. Thereflectionin the hyperplane α(resp.kerα) is

(4.5.8) sα(λ) :=λ−2hλ, αi

||α||2α resp.sα(Hλ) :=Hλ−2hλ, αi

||α||2Hα

. TheWeyl groupW ofgrelative tohis the group generated by the orthogonal transformations{sα |α∈∆}.

(4.5.9) W permutes the roots; in particular,W is finite.

In fact, using (4.5.4), for eachβ ∈ ∆we have sα(β) = β + (q −p)α ∈ ∆, since−p≤q−p≤q.

For each α ∈ ∆, kerα ⊂ hR is called asingular hyperplane; ∪αkerα is called the set ofsingular elements ofhR, and its complement hRin hR is called the set ofregular elementsofhR. The connected components of ofhR

are calledWeyl chambers.

(4.5.10) W acts simply transitively on the set of Weyl chambers.

Indeed it follows from (4.5.9) thatW permutes the chambers. Given two chambersCandC, we chooseH ∈ C,H ∈ C and join them by an arc in hRthat at each time meets at most one singular hyperplane; the product in order of the reflections in the hyperplanes met by the arc is an element of W that sendsCtoC.1

1Missing argument for simple transitivity. Humphreys, use repr theory.

Positive roots and simple roots

Fix a Weyl chamberC. It is clear that each root takes a constant sign onC. We define

+:={α∈∆|α(C)>0}.

In view of (4.4.9), this determines a partition of the root systems into two halves:

(4.5.11) ∆ = ∆+∪˙(−∆+); andα,β ∈∆+,α+β∈∆impliesα+β ∈

+.

+is called apositive root systemand its elements are calledpositive roots. A positive root is calledsimpleif it cannot be written as a sum of two positive roots. Denote byΠthe set of simple roots.

(4.5.12) Ifα,β ∈Π,α6=βthenα−β 6∈∆andhα, βi ≤0.

Indeedα=β+ (α−β)andβ=α+ (β−α); ifα−βis a root, then either it is positive or its negative is positive; in either case we reach a contradiction to the fact thatαandβare simple. Nowp= 0in (4.5.4), so2h||β,αα||2i =−q≤0.

Since all simple roots lie in the same open half-space (as they are all positive), with all mutual angles non-acute by (4.5.12), an elementary argu-ment in linear algebra shows that they are linearly independent. Therefore:

(4.5.13) Every positive root can be written uniquely as a non-negative integral linear combination of simple roots; andΠ is a basis ofhRoverR.

The fact that every positive root is a positive sum of simple roots follows immediately from the definition of simple root; the uniqueness part fol-lows from the linear independence ofΠ; and the last statement now follows from (4.4.10).

Now we enumerateΠ ={α1, . . . , αr}, wherer = dimCh, and define aij = 2hαj, αii

||αi||2 .

The aij are called Cartan integers and the r ×r matrix(aij) is called the Cartan matrix.

A positive root is built up through roots by starting with a simple root and adding one simple root at a time. Indeed, define the level of β = Pr

i=1miαito bePr i=1mi.

(4.5.14) Every positive root of levelm+ 1can be written in at least one way as a the sum of a positive root of level m and a simple root.

4.5. ROOT SYSTEMS 67 To check this, supposeβ=Pr

i=1miαihas levelm+ 1. Then 0<hβ, βi=

Xr i1

mi0hβ, αii

shows thathβ, αi0iandmi0 >0for somei0. By (4.5.4),β−αi0 ∈∆is a root, as wished.

Now (4.5.11) and (4.5.14) imply thatΠand(aij)determine∆.

Isomorphism theorem

Consider a standard tripleHi,Ei,Fispanningg[αi]for eachi= 1, . . . , r, as in (4.4.15). Then we have the relations

[Hi, Hj] = 0, [Ei, Fi] =δijHi, [Hi, Ej] =aijEj, [Hi, Fj] =−aijFj, for alli,j. For eachα∈∆+, use (4.5.13) to write

(4.5.15) α=αi1 +· · ·+αis,

sum of simple roots, where each partial sum is a root; this decomposition is unique, up to reordering. Now define

Eα = [Eis,[Eis−1,· · · ,[Ei2, Ei1]· · ·]]

Fα = [Fis,[Fis−1,· · ·,[Fi2, Fi1]· · ·]].

(4.5.16)

In view of (4.5.5),

(4.5.17) {Hi}1ir∪ {Eα, Fα}α+

is a basis ofg. Now a delicate argument of induction onsshows that the multiplication table of this basis has rational entries determined by the Car-tan matrix. Here the difficulty lies in the fact that a different order in (4.5.15) entails different vectorsEα,Fα in (4.5.16), and the main step is to show that Eα (resp.Fα) is a rational multiple ofEα (resp.Fα), where the multiplica-tive constant is determined by the permutation and the Cartan matrix. This proves:

4.5.18 Theorem (Isomorphism Theorem) Letgandgbe complex semisimple Lie algebras with corresponding CSAhandh. Denote by∆and∆the associated root systems. IfΦ :h→ h is a linear isometry with respect to the Cartan-Killing forms such thatΦ= ∆, thenΦextends to an isomorphismg→g.

4.5.19 Corollary Every element of the Weyl groupW of a complex semisimple Lie algebragis induced by an automorphism ofg.

Proof. Each generator sα of W as in (4.5.8) is a linear isometry of the

CSA that preserves the root system.

In particular, a complex semisimple Lie algebra is determined, up to isomorphism, byΠand(aij). However, three arbitrary choices have been made in the construction of these invariants:

(a) A CSAhofg.

(b) A Weyl chamberCofgrelative toh.

(c) A labeling of the simple roots inΠ.

In case of (a), any two CSA’s ofg are conjugate under an inner automor-phism (Proposition 4.2.1). In case of (b), any two Weyl chambers are related by the action of an element of the Weyl group, and the latter is induced by an automorphism ofg(Corollary 4.5.19). We avoid the choice in (c) by in-troducing theDynkin diagramof g: taker nodes, one for each simple root inΠ; ifi6=j, joinαi toαj byaijaji = 4 cos2∠(αi, αj) ∈ {0,1,2,3}lines; if

||αi||>||αj||, we further orient the lines in the direction of the shorter root.

Note thataij 6= 0if and onlyaji6= 0, and in this case aaij

ji = ||||ααj||2

i||2. Root systems of rank two

The conditions on the angles and lengths of the simple roots strongly limit the possibilities for Dynkin diagrams. As a warm up, we list all the possi-bilities in case of rankr = 2.

α1 α2

a12=a21= 0 ∆+ ={α1, α2}

α1 α2

a12=a21=−1

+={α1, α2, α12}

α1 α2

a12=−2 a21=−1

+={α1, α2, α12,2α12}

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