for allx1, . . . , xn∈Rand equality holds if and only if allxi = 0.
4.6.1 Lemma Subdiagrams of admissible diagrams are admissible.
We call a pair of nodes alinkif they are connected by at least one line.
4.6.2 Lemma An admissible diagram withnnodes contains at mostn−1links.
Proof.
0<
Xn i=1
ei
2
=n+ 2X
i<j
hei, eji.
and note thathei, eji ≤ −12 for alli,j.
It follows from Lemmas 4.6.1 and 4.6.2 that
4.6.3 Lemma An admissible diagram contains no cycles (closed paths).
4.6.4 Lemma Not more than three lines can be joined to a given node.
Proof. Suppose e1, . . . , em are joined to em+1. Owing to Lemma 4.6.3, e1, . . . , emis an orthonormal set. Sincee1does not belog to their span,
1 =|e1|2>
Xm i=1
he1, eii2.
Now just recall that the number of lines joined toe1isPm
i=14he1, eii2. 4.6.5 Lemma Suppose an admissible diagram contains a subdiagram of type
such that only the end nodes are connected to other nodes. Then the diagram obtained by contracting this subdiagram to a single node is admissible.
Proof. Suppose that e1, . . . , em correspond to the nodes of the subdia-gram. Thene1+· · ·+emis a unit vector and(e1+· · ·+em, em+1, . . . , en)is a configuration corresponding to the contracted diagram.
4.6.6 Lemma The only admissible diagram with a triple link is
>
Moreover, the following arrangements are forbidden for an admissible diagram:
two double links; two triple nodes; a double link and a triple node.
4.6. CLASSIFICATION OF ROOT SYSTEMS 71 Proof. In view of Lemma 4.6.5, the diagrams
contract to a diagram with a quadruple node and hence are forbidden by
Lemma 4.6.4.
It only remains to determine the admissible diagrams with one double link and diagrams with one triple node.
4.6.7 Lemma Suppose
e1 e2 ep fq f2 f1
is an admissible diagram with a double link, where we indicate the unit vectors corresponding to the nodes andp≥q≥1. Thenq = 1orp=q = 2.
Proof. Lete = Pp
i=1iei andf = Pq
j=1jfj. Then the Cauchy-Schwarz inequalityhe, fi2≤ ||e||2· ||f||2yields(p−1)(q−1)<2, which implies the
desired result.
4.6.8 Lemma Suppose
b1 b2 bp−1
a
cq−1
c2 c1 dr−1
d2 d1
is an admissible diagram with one triple node, where we indicate the unit vectors corresponding to the nodes andp≥q ≥r ≥2. Thenq =r = 2orr = 2,q = 3 andp≤5.
Proof. Letb=Pp−1
i=1ibi,c=Pq−1
j=1jcj andd=Pr−1
k=1kdk. Thena,b,c,d
are linearly independent vectors, so 1 =||a||2
> ha, bi2
||b||2 +ha, ci2
||c||2 +ha, di2
||d||2
= 1 2
1− 1
p
+ 1 2
1−1
q
+1 2
1−1
r
,
which gives1p +1q +1r >1. This implies the desired result.
4.6.9 Theorem The following is a complete list of connected Dynkin diagrams:
Cartan type Diagram Condition
An −
Bn > n≥2
Cn < n≥3
Dn n≥4
G2 > −
F4 > −
E6 −
E7 −
E8 −
Proof. The list of admissible Coxeter graphs is obtained from Lem-mas 4.6.4, 4.6.6, 4.6.7 and 4.6.8. Regarding the lengths of roots, the diagrams
> > >
are symmetric, so it does not matter which root is shorter. Therefore the only essential difference is in the Coxeter graph
>
forn≥3, which gives rise to two different Dynkin diagrams, namely,
> and < .
4.7. PROBLEMS 73
This yields the table.
The Cartan types An, Bn, Cn, Dn correspond to the classical series of compact connected simple Lie groupsSU(n),SO(2n+ 1),Sp(n),SO(2n), respectively. In fact, in the next chapter we will compute explicitly the root systems of the classical groups. The remaining types are calledexceptional, and the construction of the corresponding Lie algebras is more involved.
4.6.10 Example We will determine the root system of typeAnfrom the sim-ple rootsΠ = {α1, . . . , αn}. We need to determine which nonnegative in-tegral linear combinationsPn
i=1miαi are roots. The roots of level one are the simple roots, by definition. For level two, there are no roots of the form 2αj, due to (4.5.3), and we need to find out which sumsαi+αj withi6=j are roots. By (xix)αi−αj 6∈∆, so in view of (4.5.4)αi+αj is a root if and onlyhαi, αji<0if and only if|i−j|<1by the form ofAn. Next, suppose inductively that a root of levelmhas the formβ =αi+αi+1+· · ·+αi+m−1. No root is a linear combination of simple roots with coefficients of mixed sign, soβ−αjis never a root. In view of (4.5.4),β+αjis a root if and only hβ, αji<0if and only ifj=i−1orj =i+mby the form ofAn. We have shown that
∆+={αi+αi+1+· · ·+αj |1≤i≤j≤n}. In particular, there aren2+nroots inAn.
4.7 Problems
1 Use root systems to check the following special isomorphisms in low dimensions:
sl(2,C)∼=so(3,C)∼=sp(1,C) so(5,C)∼=sp(2,C) sl(4,C)∼=so(6,C) so(4,C)∼=sl(2,C)×sl(2,C)
The universal covering Lie group pofSO(n)forn≥3is denoted bySpin(n).
Deduce from the above that2
Spin(3) =SU(2)
Spin(4) =SU(2)×SU(2) Spin(5) =Sp(2)
Spin(6) =SU(4)
2It is know thatSU(n)andSp(n)are connected and simply-connected.
2 Let∆be a root system and assumeα,β ∈∆withα+β ∈∆. Prove that (Zα+Zβ)∩∆
is a root system of typeA2,B2orG2.
3 Consider a root system∆with an ordering and denote byα1, . . . , αnthe simple roots, whereα1 >· · ·> αn. Prove that, in the following cases for the type of∆, the indicated expression is a root and it belongs to the positive Weyl chamberC:
Bn:α1+· · ·+αn
Cn:α1+ 2α2+· · ·+ 2αn−1+αn
F4 :α1+ 2α2+ 3α3+ 2αn
G2 : 2α1+α2
Can you see that it is the unique root of∆inC? 4 Use the root space decomposition to:
a. Classify compact connected Lie groups of dimension3.
b. Prove that there exist no compact semisimple Lie groups in dimen-soions4,5or7.
5 Let g be a complex simple Lie algebra with root system ∆ and Weyl groupW.
a. Prove that two simple roots whose nodes in the Dinkin diagram are connected by a single edge are in the same orbit underW.
b. Deduce that all roots in ∆of a particular length form a single orbit underW.
6 Consider a root system ∆with an ordering. Letα andβ be two roots whose nodes in the Dynkin diagram are joined bynedges, where0≤k≤ 3. Letsαandsβ be the associated reflections in the Weyl group. Show that
(sαsβ)m = 1, wherem=
2 ifk= 0;
3 ifk= 1;
4 ifk= 2;
6 ifk= 3.
7 Prove that any element of order2in the Weyl group of a root system is the product of two commuting reflections.
4.7. PROBLEMS 75 8 Let∆be a root system of typeB3, namely:
∆ ={±(θ1±θ2),±(θ2±θ3),±(θ1±θ3)} ∪ {±θ1,±θ2,±θ3}.
Prove that the orthogonal projection of∆on the hyperplane orthogonal to θ1+θ2+θ3is congruent to a root system of typeG2.
9 Lete1,e2,e3,e4 be the canonical basis of Euclidean spaceR4. Consider the configuration∆of48vectors
±ei(1≤i≤4), ±ei±ej(1≤i < j ≤4), 1
2(±e1±e2±e3±e4).
a. Let α, β ∈ ∆. Show that aα,β := 2hβ,αi
||α||2 is an integer and sα(β) :=
β−aα,βα∈∆.
b. Check that∆is congruent to the root system of typeF4.
c. Deduce that the dimension ofF4 is52 and compute the order of its Weyl group to be1152.