Chevalley Groups

Elias Weber

January 07, 2010

Supervised by Dr Claude Marionand Prof Donna Testerman

AbstractThis project deals with the construction of the Chevalley group and the studies of its

structure and properties through the analysis of its relevant subgroups.

Introduction

Given a simple Lie algebra, one can study its automorphisms. The Chevalley group is aninteresting subgroup of this automorphism group. The generators of the Chevalley groupare constructed with the help of a basis of the Lie algebra called a Chevalley basis, thead-functor and the exponential map.In the first chapter of this project we construct the Chevalley group of a simple Lie algebradefined over the complex numbers and then generalize this construction to a simple Liealgebra defined over an arbitrary field.Once we constructed the Chevalley group, we study some of its subgroups in the secondchapter. First the subgroups presented in this chapter may seem to be chosen a bit arbitrary.But in the third chapter we use them to prove that a Chevalley group has a BN -pair. Therewe will understand the importance of those subgroups, since the existence of a BN -pairreveals a lot of information about the group. This is discussed in a more general setting inthe continuation of the third chapter.Finally, we state a theorem which uses the existence of a BN -pair and which implies thatmost Chevalley groups are simple.

The theory is based on the book ”Simple Groups of Lie Type” written by R. W. Carter.Most of the proofs are not given in this project. Instead we provide references whenever aproof is omitted. Also we illustrate the material discussed in this project through workedout examples.

Contents

1 Construction of Chevalley groups 31.1 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 The Chevalley Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 The Exponential Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Generalisation to an Arbitrary Field . . . . . . . . . . . . . . . . . . . . . . 7

2 Important Subgroups of Chevalley groups 92.1 The Nilpotent Subgroups U and V . . . . . . . . . . . . . . . . . . . . . . . 92.2 The Subgroups 〈Xr, X−r〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 The Diagonal and Monomial Subgroups H and N . . . . . . . . . . . . . . . 152.4 The Borel subgroup B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Properties of Chevalley groups 203.1 The BN -pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Simplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2

1 Construction of Chevalley groups

1.1 Lie Algebras

It is assumed that the reader is familiar with the basic theory of simple Lie algebras. Onecan find a good introduction to Lie algebras in [Hu, Chapters 1 - 4].

Recall that a simple Lie algebra can be decomposed as follows:

L = H ⊕∑r∈Φ

Lr

where H is a Cartan subalgebra of L, Φ is a root system of (L,H) and for r ∈ Φ ⊆ H∗,Lr := x ∈ L| [x, h] = r(h)x ∀ h ∈ H.

For r ∈ Φ, hr := 2r(r,r)

is called the co-root of r. The scalar product (·, ·) : R× R −→ R isinduced by the Killing form and gives R the structure of a Euclidean space. Therefore thefollowing holds: (r, s) := ‖r‖ · ‖s‖ · cos θ, where θ is the angle between the roots r and s.

Since Φ is a Euclidean space, one can define the reflection in the hyperplane orthogonal tor, wr(x) := x− 2(r,x)

(r,r)r. The group generated by the reflections wr, r ∈ Φ, is called the Weyl

group of Φ.

Recall also that a Lie algebra is called simple if it has no non-trivial ideals and is non-zero.

1.2 The Chevalley Basis

Let L be a simple Lie algebra defined over C.In order to construct the Chevalley group over an arbitrary field K, we need to pass by abasis of L, called Chevalley basis. We will find that the so called structure constants areintegers. This fact will allow us to generalize the results to algebras defined over an arbitraryfield K.

To construct this basis one chooses arbitrary non-zero elements er ∈ Lr for r ∈ Φ+. Thesegenerate Lr, since Lr are 1-dimensional over C. Furthermore, there exist unique elementse−r ∈ L−r for r ∈ Φ+, such that [er, e−r] = hr as defined above.

Theorem 1.1. Let Π be a fundamental basis of Φ. Then the set

hr, r ∈ Π; er, r ∈ Φ

forms a basis of L, called a Chevalley basis of L.Moreover, the elements of this basis multiply together in the following way:

[hr, hs] = 0

[hr, es] = Ar,ses

[er, e−r] = hr

[er, es] = Nr,ser+s, if r + s ∈ Φ

[er, es] = 0, if r + s /∈ Φ

where Ar,s := 2(r,s)(r,r)

are integers and Nr,s depend on the choice of the elements er, r ∈ Φ+.

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Proof. A proof can be found in [Ca, Sections 4.1 and 4.2].

Remark 1.2. It is possible to choose the elements er in such a way that Nr,s = ±(p + 1),where p is defined by the r-chain through s. This is the chain

−pr + s, ..., s, ..., qr + s

where p and q are such that −pr+ s, ..., s, ..., qr+ s ∈ Φ but −(p+ 1)r+ s, (q+ 1)r+ s /∈ Φ.

Proof. This is discussed at the beginning of [Ca, Sections 4.2].

From now on we assume that the elements er are chosen in that way, if we talk aboutChevalley bases. We see that in this case the multiplication constants Ar,s and Nr,s areintegers.

The signs of the so-called structure constants Nr,s are not fully determined by the Liealgebra structure. However, these constants fulfill some (further) relations:

Proposition 1.3. Let r1, r2, r3, r4 be roots. Then the following hold.

(i) Nr2,r1 = −Nr1,r2

(ii) N−r1,−r2 = −Nr1,r2

(iii) If r1 + r2 + r3 = 0, thenNr1,r2(r3,r3)

=Nr2,r3(r1,r1)

=Nr3,r1(r2,r2)

(iv) If r1 + r2 + r3 + r4 = 0, thenNr1,r2 ·Nr3,r4(r1+r2,r1+r2)

+Nr2,r3 ·Nr1,r4(r2+r3,r2+r3)

+Nr3,r1 ·Nr2,r4(r3+r1,r3+r1)

= 0

Proof. The corresponding calculations can be found in [Ca, Theorem 4.1.2 and Section 4.2].

One can therefore fix the sign of a number of structure constants which will determine theothers.

Example 1. Let L be a Lie algebra of type B2 over C. Recall that a B2-root system consistsof the roots α (long), β (short), α+ β (short), α+ 2β (long) and the corresponding negativeroots. We set the lenght of the short roots to be 1. The long roots then have length

√2.

We choose Nα,β and Nβ,α+β to be respectively negative and positive. Then all structureconstants are determined.

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Proof. First we want to see which structure constants are zero. This is the case for Nr,s,(r, s ∈ Φ) with r + s /∈ Φ. We illustrate this in Table 1 below, where r is the root chosenin the leftmost column and s the root in the first row. Note that we only have to calculatethe upper right triangle of Table 1 since we have Ns,r = −Nr,s. This holds by Proposition1.3 (i). It follows that the lower left triangle has the same entries but with opposite signs.The entries in the diagonal are all 0, since 2r /∈ Φ whenever r ∈ Φ.

α β α + β α + 2β −α −β −α− β −α− 2βα 0 0 0 0 0 0β 0 0 0 0α + β 0 0 0α + 2β 0 0 0−α 0 0 0−β 0 0−α− β 0 0−α− 2β 0

Table 1: Structure constants being zero

So it is the aim now to fill in the gaps in the upper right triangle of Table 1 above.

(a) Nα,β = −(0 + 1) = −1 and Nβ,α+β = +(1 + 1) = 2 due to Theorem 1.1 and theassumptions, since −1 · α+ β /∈ B2 and −1 · β + (α+ β) = α ∈ B2,−2 · β + (α+ β) =α− β /∈ B2.

(b) N−α,−β = −Nα,β = 1 by (a) and Proposition 1.3 (ii).Analogously, N−β,−α−β = −Nβ,α+β = −2.

(c) We now use Proposition 1.3 (iii), with r1 = α, r2 = β, r3 = −α − β. (Note that wedo have r1 + r2 + r3 = 0). This yields

Nα,β(−α−β,−α−β)

=Nβ,−α−β

(α,α)=

N−α−β,α(β,β)

⇒ −11

=Nβ,−α−β

2=

N−α−β,α1

⇒ Nβ,−α−β = −2 and Nα,−α−β = −N−α−β,α = −(−1) = 1.

(d) In the same way as in part (b) we get Nα+β,−α = Nα,−α−β = 1 and Nα+β,−β =Nβ,−α−β = −2.

(e) We again use Proposition 1.3 (iii). Set r1 = −β, r2 = −α − β, r3 = α + 2β. (Notethat we do have r1 + r2 + r3 = 0). We get

N−β,−α−β(α+2β,α+2β)

=Nα+2β,−β

(−α−β,−α−β)=

N−α−β,α+2β

(−β,−β)

⇒ −22

=Nα+2β,−β

1=

N−α−β,α+2β

1

⇒ Nα+2β,−β = −1 and Nα+2β,−α−β = −N−α−β,α+2β = 1.

(f) Finally, we apply Proposition 1.3 (ii) once more. This yields Nβ,−α−2β = −1 andNα+β,−α−2β = 1

Let us summarise the calculated constants in Table 2 below.

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α β α + β α + 2β −α −β −α− β −α− 2βα 0 -1 0 0 0 0 1 0β 1 0 2 0 0 0 -2 -1α + β 0 -2 0 0 1 -2 0 1α + 2β 0 0 0 0 0 -1 1 0−α 0 0 -1 0 0 1 0 0−β 0 0 2 1 -1 0 -2 0−α− β -1 2 0 -1 0 2 0 0−α− 2β 0 1 -1 0 0 0 0 0

Table 2: Structure constants for L of type B2

Remark 1.4. We did not use Proposition 1.3 (iv) here, since the root system B2 is verysmall. However, if there are more roots, this condition is important as well.

1.3 The Exponential Map

The next step in constructing the Chevalley group is to define the exponential map. Weapply it to the functions ad er (r ∈ Φ) which yield the generators of the Chevalley group.

Definition 1.5. Let L be a Lie algebra defined over a field of characteristic 0 (e.g. C).Then define the exponential map by:

exp : Nilpotent derivations of L −→ Automorphisms of L,

δ 7→ exp δ := 1 + δ +δ2

2!+ ...+

δn−1

(n− 1)!, if δn = 0.

Remark 1.6. The map exp δ is in fact an automorphism of L.

Proof. A proof can be found in [Ca].

Remark 1.7. Since ad er are nilpotent derivations, the elements xr(ξ) := exp (ξ · ad er) aretherefore automorphisms of L for ξ ∈ C, r ∈ Φ.

Proposition 1.8. These automorphisms act on the Chevalley basis as follows, if r and sare linearly independent:

xr(ξ).er = er

xr(ξ).e−r = e−r + ξ · hr − ξ2 · er

xr(ξ).hr = hr − 2ξ · er

xr(ξ).hs = hs − As,r · ξ · er

xr(ξ).es =∑q

i=0Mr,s,i · ξi · eir+s, with Mr,s,i := 1i!·Nr,s ·Nr,r+s · · ·Nr,(i−1)r+s = ±

(p+ii

),

where p is defined by the r-chain through s, like in Remark 1.2.

Proof. The calculation can be found in [Ca, Section 4.3].

Corollary 1.9. The automorphisms xr(ξ), r ∈ Φ, ξ ∈ C transform the basis elementsinto linear combinations of basis elements. The coefficients are integral multiples of positivepowers of ξ.

We are now ready to define the Chevalley group of type L over C.

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Definition 1.10. The group

L(C) := 〈xr(ξ)| r ∈ Φ, ξ ∈ C〉

is called Chevalley group of type L over C.

Thanks to Corollary 1.9 we can generalize this to a simple Lie algebra L defined over anarbitrary field K.

1.4 Generalisation to an Arbitrary Field

Let L be a simple Lie algebra defined over C and let hr, r ∈ Π; er, r ∈ Φ be a Chevalleybasis of L. Furthermore, let LZ denote the subset of L consisting of all integral linearcombinations of the basis elements. The pair (LZ,+) is an abelian group. Because ofTheorem 1.1, the Lie bracket of two elements of LZ lies again in LZ and thus LZ is a Liealgebra defined over Z.

Let K be an arbitrary field and define LK := K ⊗LZ. Since LZ and K are additive abeliangroups, (LK ,+) is also an abelian group. Every element a of LK can be written as follows:

a =∑r∈Π

λr(1K ⊗ hr) +∑r∈Φ

µr(1K ⊗ er),

where 1K is the unit element of K and λr, µr ∈ K.Thinking of the elements of LK as above, the operation

“ ∗ ” : K × LK → LKk ∗ a 7→

∑r∈Π (k · λr)(1K ⊗ hr) +

∑r∈Φ (k · µr)(1K ⊗ er),

is well defined and gives the abelian group (LK ,+) the structure of a K-vector space. If wewrite h′r := 1K ⊗ hr and e′r := 1K ⊗ er then

h′r, r ∈ Π; e′r, r ∈ Φ

is a basis of the vector space LK , called a Chevalley basis of LK .

We want to define a Lie multiplication on LK that gives LK the structure of a Lie algebra:

Proposition 1.11. Let x, y be elements of the Chevalley basis of the simple Lie algebra Lover C. Then

[1K ⊗ x, 1K ⊗ y] := 1K ⊗ [x, y]

defines a Lie bracket on LK.Moreover the multiplication constants of LK with respect to h′r, r ∈ Π; e′r, r ∈ Φ are thesame as the multiplication constants of L with respect to the basis hr, r ∈ Π; er, r ∈ Φ.

Proof. A proof can be found in [Ca, Proposition 4.4.1].

We want also to generalize the maps xr(ξ).Let Ar (ξ) denote the matrix corresponding to xr (ξ) with respect to the Chevalley basishr, r ∈ Π; er, r ∈ Φ of L. The entries of Ar(ξ) have the form a · ξi, where a and i areintegers, i ≥ 0. By changing a ∈ K to a′ := 1K ⊗ a ∈ K ×Z, and by taking an element t ofK, one gets the matrix with the entries a′ · ti. We denote it by A′r(t).

Definition 1.12. Let x′r(t) : LK −→ LK be the linear map corresponding to the matrixA′r(t) with respect to the basis h′r, r ∈ Π; e′r, r ∈ Φ.

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Proposition 1.13. x′r(t) is an automorphism of LK , ∀ r ∈ Φ, t ∈ K.

Proof. See [Ca, Proposition 4.4.2].

The elements x′r(t) act on the Chevalley basis of LK in the same way modulo K as theelements xr(ξ) on the Chevalley basis of L.

For matter of conciseness, we simplify our notation in the following way: We write hr forh′r, er for e′r, xr(t) for x′r(t), and Ar(t) for A′r(t). Note that this does not lead to confusion,since the elements we already called like that are just a special case of the more generalsetting that we have now (namely for K = C).

Definition 1.14. The Chevalley group of type L over the field K is the following groupof Lie algebra automorphisms of LK :

L(K) := 〈xr(t)| r ∈ Φ, t ∈ K〉

Proposition 1.15. The Chevalley group L(K) is determined up to isomorphism by thesimple Lie algebra L over C and the field K.

Proof. A proof can be found in [Ca, Proposition 4.4.3].

To summarise this chapter, we started with a simple Lie algebra defined over C and con-structed its Chevalley group. To do so, we passed by the Chevalley basis and used thenotion of the exponential map to define the Chevalley group of a simple Lie algebra definedover C. Finally we generalized this construction to simple Lie algebras over arbitrary fieldsK.

In the following chapters we will study this group of automorphisms of simple Lie algebras.

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2 Important Subgroups of Chevalley groups

In this chapter we study some important subgroups of the Chevalley group that revealinteresting properties of the Chevalley group itself. Maybe the biggest aim of this section isto define the so-called Borel subgroup B, which forms together with the monomial subgroupN a BN-pair of the Chevalley group. The fact that a certain group has a BN-pair gives alot of information about that group. The notion of the BN-pair is introduced and discussedin Section 3.1.

In this chapter L denotes a Lie algebra defined over an arbitrary field K if not otherwisestated. Furthermore we set G := L(K) for shorter writing.

2.1 The Nilpotent Subgroups U and V

We start with the study of two subgroups of G that are isomorphic. One of them, U , willbe used in Section 2.4 below to define the Borel subgroup of G. To understand the structureof these groups, we will also study the commutator of the generators of the Chevalley groupG.

Definition 2.1. For r ∈ Φ, the subgroups Xr := 〈xr(t)| t ∈ K〉 are called root subgroupsof G.Fix now a basis Π of the root system Φ.

Definition 2.2. Let U be the subgroup of G generated by the root subgroups Xr, r ∈ Φ+

and let V the subgroup of G generated by the root subgroups Xr, r ∈ Φ−.

To investigate the structure of these two groups, we need a tool, namely the Chevalleycommutator formula.

Theorem 2.3. Let r, s be linearly independent roots of L and t, u be elements of K. Thenthe commutator

[xs(u), xr(t)] := xs(u)−1 · xr(t)−1 · xs(u) · xr(t)

is equal to ∏i,j>0

xir+js ·(Cijrs · (−t)i · uj

), (1)

where the product is taken in increasing order of the sum i+ j over all i,j for which ir+ jsis a root. Furthermore, the constants Cijrs are equal to ±1,±2 or ±3 and are defined by:

Ci1rs = Mr,s,i

C1jrs = (−1)j ·Ms,r,j

C32rs = 13·Mr+s,r,2

C23rs = −23Ms+r,s,2

Proof. For a complete proof, see [Ca, Section 5.2].

Remark 2.4. In the theorem given a fixed value for i+j the order of the factors of the formxir+js · (Cijrs · (−t)i · uj) in the product (1) is not determined. However, it can be shownthat these factors do commute.

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Corollary 2.5. If r 6= s ∈ Φ+ then

[xs(u), xr(t)] =∏i,j>0

xir+js ·(Cijrs · (−t)i · uj

),

where the product is taken over all positive roots of the form is + js in increasing order ofthe sum i+ j.

We do now an example to illustrate the use of the commutator formula.

Example 2. Let L be a simple Lie algebra defined over C, Φ a root system of L and B aChevalley basis of L. Let α, β ∈ Φ be such that α and β generate a root system of type A2

and Nα,β = 1.Let K be a field of characteristic p > 2 and let LK := K ⊗ LZ be the Lie algebra definedover K which is induced by L.Prove that

F :=

xα(t).xβ(t).xα+β(−1

2· t2)

∣∣∣∣ t ∈ Kis a subgroup of the Chevalley group L(K) that is isomorphic to (K,+).

Proof. Note first, that in a root system of type A2, the only roots are α, β, α + β,−α,−βand −α− β.

To check that F is a subgroup of H we need the following lemma.

Lemma 2.6. Let L be a simple Lie algebra defined over K and α ∈ Φ.Then we have xα(t).xα(s) = xα(t+ s). And in particular, xα(t)−1 = xα(−t).

Proof of Lemma 2.6. We have

xα(t).xα(s) = exp(t · ad eα) · exp(s · ad eα)

=n∑i=0

ti · (ad eα)i

i!·m∑j=0

sj · (ad eα)j

j!

=

(n,m)∑(i,j)=(0,0)

ti · sj · (ad eα)i+j

i! · j!

=n+m∑k=0

(k∑i=0

ti · sk−i · (ad eα)k

i! · (k − i)!

)

=n+m∑k=0

(k∑i=0

k!

i! · (k − i)!· t

i · sk−i · (ad eα)k

k!

)

=n+m∑k=0

(k∑i=0

(k

i

)· ti · sk−i

)· (ad eα)k

k!

10

=∑n+m

k=0(t+s)k·(ad eα)k

k!

= exp((t+ s) · ad eα)

= xα(t+ s).

This implies xα(t).xα(−t) = xα(t+ (−t)︸ ︷︷ ︸=0

) = exp(0) = id and hence xα(t)−1 = xα(−t).

We are now ready to prove that F is a subgroup of L(K).For this, let a := xα(t).xβ(t).xα+β(−1

2· t2) and b := xα(s).xβ(s).xα+β(−1

2· s2) be arbitrarily

chosen elements of F . We show that a−1 and a · b are again in F .

a−1 = xα+β(−1

2· t2)−1.xβ(t)−1.xα(t)−1

Lemma 2.6= xα+β(

1

2· t2).xβ(−t).xα(−t)

2 x Commutator= xβ(−t).xα(−t).xα+β(

1

2· t2)

Commutator= xα(−t).xβ(−t).xα+β(−t2).xα+β(

1

2· t2)

Lemma 2.6= xα(−t).xβ(−t).xα+β(−1

2(−t)2).

This is in F since −t ∈ K. We used the commutator formula three times. We prove the threecorresponding equalities for general arguments since we will reuse them in the remainingpart of this exercise. First, xα+β(u) and xβ(t) commute:

xα+β(u).xβ(t) = xβ(t).xα+β(u). [xα+β(u), xβ(t)]

= xβ(t).xα+β(u).∏i,j>0

xiβ+j(α+β)(Cijβ(α+β) · (−t)i · uj)

= xβ(t).xα+β(u).

since there exist no roots of the form iβ + j(α + β) for i, j ≥ 1.Analogously one can show that xα+β(u) and xα(t) commute since there exist no roots of theform iα + j · (α + β), i, j ≥ 1 in an A2 system.Finally,

xβ(u).xα(t) = xα(t).xβ(u). [xβ(u), xα(t)]

= xα(t).xβ(u).∏i,j>0

xiα+jβ(Cijαβ(−t)i · uj)

= xα(t).xβ(u).xα+β( C11αβ︸ ︷︷ ︸=Mα,β,1= 1

1!·Nα,β

(−t) · u)

= xα(t).xβ(u).xα+β(1 · (−t) · u)

= xα(t).xβ(u).xα+β(−t · u).

Note for the next paragraph that this equation is equivalent toxα(t).xβ(u) = xβ(u).xα(t).xα+β(t · u). We have shown that the inverses are again in F .

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Moreover, F is closed under multiplication:

a · b = xα(t).xβ(t).xα+β(−1

2· t2).xα(s).xβ(s).xα+β(−1

2· s2)

= xα(t).xβ(t).xα(s).xβ(s).xα+β(−1

2· t2).xα+β(−1

2· s2)

= xα(t).xα(s).xβ(t).xα+β(−s · t).xβ(s).xα+β(−1

2· (t2 + s2)).

= xα(t+ s).xβ(t).xβ(s).xα+β(−s · t).xα+β(−1

2· (t2 + s2)).

= xα(t+ s).xβ(t+ s).xα+β(−1

2· (t2 + 2s · t+ s2︸ ︷︷ ︸

=(t+s)2

)).

This is in F since t+ s ∈ K.Therefore F is a subgroup of L(K) and it remains to show that the group (F, .) is isomorphicto (K,+). Define ϕ : K −→ F, t 7→ xα(t).xβ(t).xα+β(−1

2t2). Then ϕ is an isomorphism.

(a) ϕ is a homomorphism. This follows directly from the first part:

ϕ(s).ϕ(t) = xα(t).xβ(t).xα+β(−1

2· t2).xα(s).xβ(s).xα+β(−1

2· s2)

= xα(t+ s).xβ(t+ s).xα+β(−1

2· ((t+ s)2))

= ϕ(t+ s).

(b) For injectivity of ϕ we use the calculation of the inverse, i.e. that ϕ(s)−1 = ϕ(−s):

ϕ(t) = ϕ(s)

⇒ 1 = ϕ(t).ϕ(s)−1 = ϕ(t).ϕ(−s) = ϕ(t− s)⇒ 1 = xα(t− s).xβ(t− s).xα+β(−1

2· (t− s)2).

Applying this function to eβ yields the equation

eβ = xα(t− s).xβ(t− s).xα+β(−1

2· (t− s)2).eβ

= xα(t− s).xβ(t− s).

(q∑i=0

Mα+β,β,i · (−1

2(t− s)2)i · ei(α+β)+β

)= xα(t− s).xβ(t− s).Mα+β,β,0︸ ︷︷ ︸

=1

·1 · eβ

= xα(t− s).eβ

=

q∑i=0

Mα,β,i · (t− s)i · eiα+β

= Mα,β,0︸ ︷︷ ︸=1

·(t− s)0 · eβ + Mα,β,1︸ ︷︷ ︸= 1

1!·Nα,β

·(t− s)1eα+β

= eβ + (t− s) · eα+β

⇒ t− s = 0 ⇒ t = s.

We used the relations stated in Proposition 1.8, the fact that eα+β 6= 0 since α + β isa root, and that 2α + β is not a root in A2.

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(c) ϕ is surjective by the definition of F .

So (F, . ) is isomorphic to (K,+) and we are done.

We now apply the Chevalley commutator formula to the subgroups U and V of G anddetermine their structure.

To do so we need another notion:

Definition 2.7. The height of a root h(r) is defined as the sum of its coefficients withrespect to the fixed base Π, i.e. if r =

∑s∈Π λs · s, then h(r) :=

∑s∈Π λs.

Moreover, an ordering of Φ+ is a reflexive, transitive and complete relation ” ≺ ” of Φ suchthat h(r) ≤ h(s) =⇒ r ≺ s.

Finally we can state a theorem about the structure of U .

Theorem 2.8. Let U ⊇ Um := 〈Xr| r ∈ Φ+, h(r) ≥ m〉

(i) U is nilpotent and U = U1 ⊇ U2 ⊇ ... ⊇ Uh ⊇ 1

(ii) Each element x ∈ U can uniquely be written as∏ri∈Φ+

xri(ti),

where the product is taken over all positive roots in a fixed increasing order.

Proof. The proof uses the commutator formula discussed above (see Theorem 2.3) and canbe found in [Ca, Theorem 5.3.3].

Naturally, V is also nilpotent and its elements can be uniquely expressed in a similar wayas in part (ii) of the above theorem, where the product is taken over all negative roots indecreasing order.

These two subgroups are used in Section 3.1 to explore the Borel subgroup. We study nowthe subgroups 〈Xr, X−r〉.

2.2 The Subgroups 〈Xr, X−r〉The subgroups 〈Xr, X−r〉 ≤ G are like U and V generated by root subgroups.As shown in Theorem 2.9 below, the subgroups 〈Xr, X−r〉 are quotients of the classical groupSL2(K). This is helpful since SL2(K) has a nice and simple structure.

Recall that SL2(K) is generated by the elements

(1 t0 1

),

(1 0t 1

), (t ∈ K).

Theorem 2.9. For all r ∈ Φ, there is a surjective homomorphism ϕr : SL2(K) −→

〈Xr, X−r〉, such that ϕr

((1 t0 1

))= xr(t) and ϕr

((1 0t 1

))= x−r(t).

Proof. For a proof see [Ca, Theorem 6.3.1].

We are interested in how the images of diagonal matrices and anti-diagonal matrices ofSL2(K) under this homomorphism ϕr act on the Chevalley basis of LK . These elements areimportant for the next subgroups of G we will consider.

Given r, s ∈ Φ recall the definition of the integer Ar,s given in Theorem 1.1.

13

Proposition 2.10. Let hr(λ) denote the image of

(λ 00 λ−1

)under ϕr.

Then hr(λ) acts on the Chevalley basis of LK as follows:

hr(λ).hs = hs, s ∈ Π

hr(λ).es = λAr,s · es, s ∈ Φ

Proof. A proof can be found in [Ca, Proposition 6.4.1].

In the following statement, wr denotes the element of the Weyl group of LK which is thesimple reflection in the hyperplane orthogonal to r.

Proposition 2.11. Let nr := ϕr

((0 1−1 0

)). Then nr acts on the Chevalley basis of LK

as follows:

nr.hs = hwr(s), s ∈ Π

nr.es = ηr,s · ewr(s), s ∈ Φ, where ηr,s = ±1.

Moreover the numbers ηr,s satisfy the following relations:

ηr,r = −1

ηr,−r = −1

ηr,s · ηr,wr(s) = (−1)Ar,s

ηr,s · ηr,−s = 1.

Proof. For a proof, see [Ca, Proposition 6.4.2].

More generally we can take a look at the matrices nr(t) := ϕ

((0 t−t−1 0

)). In this setting

nr = nr(1).

Example 3. Let L(K) be a Chevalley group and let α ∈ Φ. Show that

nr = xr(1).x−r(−1).xr(1)

Proof. Using Theorem 2.9 we can calculate

nr = ϕr

((0 1−1 0

))= ϕr

((1 10 1

)·(

1 0−1 1

)·(

1 10 1

))ϕr Homomorphism

= ϕr

((1 10 1

)).ϕr

((1 0−1 1

)).ϕr

((1 10 1

))Theorem 2.9

= xr(1).x−r(−1).xr(1).

One can get more. Thanks to the properties of ϕr stated in Theorem 2.9 and Proposition2.11 one derives the following equalities.

14

Lemma 2.12. We have the following:

(i) nr(−1) = n−1r

(ii) nr(t) = xr(t) · x−r(−t−1) · xr(t)

(iii) hr(t) = nr(t)nr(−1).

Proof. See [Ca, Lemma 6.4.4].

So, we have found some elements of the groups 〈Xr, X−r〉 that act in a nice way on theChevalley basis of LK . We now use the elements hr(λ), nr to define two further importantsubgroups both needed to find a BN-pair of G.

2.3 The Diagonal and Monomial Subgroups H and N

The two subgroups H and N are generated by the elements hr(λ) and nr. So they aredefined by the homomorphisms in Theorem 2.9 above. Let us consider the subgroup H firstand once we are done with that we will take a look at N .

Definition 2.13. The diagonal subgroup H is generated by

hr(λ)| r ∈ Φ, 0 6= λ ∈ K .

Note that each element hr(λ) acts trivially on HK , the Cartan subalgebra of LK , since wehave hr(λ).hs = hs, s ∈ Π. Thus, each element of H is an automorphism of LK that fixesHK pointwise. Moreover, it sends each element es of the Chevalley basis to a multiple ofitself, since we have hr(λ).es = λArs · es, s ∈ Φ.

This yields the possibility to express H in terms of characters which are defined below. Forthis purpose let P be the set of all linear combinations of roots with integer coefficients, sothat P := ZΦ. Componentwise addition gives P the structure of an abelian group.

Definition 2.14. The homomorphisms χ : P −→ K\0 are called K-characters of P .For simplicity, we refer to them as characters.

Every character χ is a homomorphism and hence for all a, b ∈ P we have

χ(a+ b) = χ(a) · χ(b) and χ(−a) = χ(a)−1.

The characters form a additive group with the operation χ2 + χ1(s) := χ2(s) · χ1(s).

Now, every character χ gives rise to a map h(χ) : LK → LK , which is in fact a automorphismof LK . It is defined as follows:

For r ∈ Π, h(χ).hr = hr and

for r ∈ Φ, h(χ).er = χ(r) · er.

For λ ∈ K, r ∈ Φ, let χr,λ be the homomorphism from P to K\0 defined by χr,λ(a) :=

λ2·(r,a)(r,r) . We see that h(χr,λ) = hr(λ). So every element of H can be identified with a K-

character of P .

15

We can say more. As usual let Π := p1, ..., pl be a basis of Φ. Then there is a basis of H,B = q1, ..., ql, such that

pi =l∑

j=1

Apj ,pi · qi. (2)

A proof of the existence of such a basis can be found in [Ca, Section 7.1].

Recall that the constants Apj ,pi =2(pj ,pj)

(pi,pi)are integers. Hence we define Q := ZB, the set

of fundamental weights of L, and we get P ⊆ Q since every pi, (i ∈ 1, ..., l) is alinear combination of the qi, (i ∈ 1, ..., l), with integer coefficients due to (2). Now, everycharacter of Q can be restricted to a character of P . But not every character of P can beextended to a character of Q.

Since the characters are important to understand the structure of the diagonal subgroup,we give some of their properties.

Example 4. Let P = ZΦ as usual and let Hom(P,K\0) be the group of K-characters ofP. Then

(a) H := h(χ)|χ ∈ Hom(P,K\0) is a subgroup of the group of automorphisms of LK,

(b) the map χ 7→ h(χ) is an isomorphism from Hom(P,K\0) onto H and

(c) the left group action of the Weyl group W on Φ induces an action of W on the group

of K-characters of P and on H.

Proof. We prove the three parts consecutively.

(a) We have to show that the subset H of LK is closed under multiplication and inversion.

We haveh(χ1).h(χ2).hs = h(χ1).hs = hs = h(χ2 · χ1).hs

by definition of an element of H.Moreover,

h(χ1).h(χ2).es = h(χ1) · χ2(s).es = χ2(s) · h(χ1).es (since χ2(s) ∈ K)

= χ2(s) · χ1(s) · es = (χ2 + χ1)(s) · es= h(χ2 + χ1)(s).es,

where we again used the definition of an element of H.Since χ2 +χ1 ∈ Hom(P,K\0), we have shown that H is closed under multiplication.

The set H is also closed under inversion. In fact one can show

h(χ)−1 = h(−χ),

which is in H. To do so, we check the action of h(χ).h(−χ) on the Chevalley basis ofLK .

h(χ).h(χ−1).hs = h(χ).hs = hs

h(χ).h(χ−1).es = h(χ) · χ−1(s).es = χ−1(s) · χ(s) · es= (−χ+ χ)(s) · es = 1 · es.

Therefore h(χ).h(χ−1) = id and we are done.

16

(b) We check that χ 7→ h(χ) is a group isomorphism by looking at the action on theChevalley basis of LK .

The map χ 7→ h(χ) is a homomorphism since we have

h(χ2 + χ1).hs = hs = h(χ2).hs = h(χ2).(h(χ1).hs)

and

h(χ2 + χ1).es = ((χ2 + χ1)(s)) · es= χ2(s).χ1(s) · es= h(χ2).(χ1(s) · es)= h(χ2).h(χ1).es,

by the definition of an element of H.

Furthermore, χ 7→ h(χ) is injective since we have

h(χ1) = h(χ2) =⇒ χ1(r).er = χ2(r)er ∀ r ∈ Φ

=⇒ χ1(r) = χ2(r)∀ r ∈ Φ

=⇒ χ1 = χ2.

and χ 7→ h(χ) is surjective by definition of H.

(c) Set w.χ(s) := χ(w−1(s)). This defines an action of W on the K-characters of P , sincethe two following conditions hold.

(w2 w1).χ(s) = χ((w2 w1)−1(s))

= χ(w−11 (w−1

2 (s)))

= w1.χ(w−12 (s))

= w2.w1.χ(s)

andid.χ(s) = χ(id(s)) = χ(s),

since W acts on the left on Φ.

Set now w.h(χ) := h(w−1.χ). This defines an action of W on H.

(w2 w1).h(χ) = h((w2 w1)−1.(χ))

= h(w−11 .(w−1

2 .χ))

= w1.h(w−12 .χ)

= w2.w1.h(χ)

andid.h(χ) = h(id.χ) = h(χ),

since W acts on the K-characters of P as we showed before.

We now state a theorem that describes which characters give rise to elements in H.

17

Theorem 2.15. The group H consists of all automorphisms h(χ) of LK for which χ is aK-character of P that can be extended to a K-character of Q.

Proof. See [Ca, Theorem 7.1.1] for a proof.

Through calculation one can derive the following property of the diagonal subgroup H.

Proposition 2.16. The group H normalizes all root subgroups Xr, r ∈ Φ.

Proof. See [Ca, Section 7.1] for a proof.

Since U and V are generated by root subgroups, we get the following important corollary.

Corollary 2.17. The group H normalizes U and V . In particular, UH and V H are sub-groups of G.

We get the following intersection rules for these groups.

Lemma 2.18. We have UH ∩ V = 1 and V H ∩ U = 1

Proof. See [Ca, Lemma 7.1.2].

Corollary 2.19. We have UH ∩ V H = H

We now have a good understanding of the diagonal subgroup. To summarise, we saw thatthe elements of H can be identified with characters which we discussed in a more generalcontext. Moreover, we found that UH is a subgroup of G.

Now we will study the monomial subgroup N which forms together with UH a BN -pair ofG as we will see in Section 3.1.

Definition 2.20. The subgroup N of G is defined as follows:

N := 〈H, nr| r ∈ Φ〉

and is called the monomial subgroup of G.

We now state a remarkable theorem. For this purpose, recall that W denotes the Weylgroup associated to (LK ,Φ).

Theorem 2.21. There is a homomorphism ϕ : N → W with kernel H such that ϕ(nr) = wr,for all r ∈ Φ. Hence H is a normal subgroup of N and the quotient N/H ∼= W .If n ∈ N and h(χ) ∈ H, we have n · h(χ) · n−1 = h(χ′), with χ′(r) = χ(w−1(r)), wherew = ϕ(n).

Proof. A proof of this theorem can be found in [Ca, Theorem 7.2.2].

We now consider the action of the elements nr on the root subgroups Xr.

Lemma 2.22. Let r, s ∈ Φ. Then the following hold.

(i) nr.xs(t).n−1r = xwr(s)(ηr,s · t)

(ii) nr ·Xs · n−1r = Xwr(s),

Proof. For a proof see [Ca, Lemma 7.2.1].

18

Corollary 2.23. We have U ∩N = 1 and UH ∩N = H.

Proof. See [Ca, Corollary 7.2.4] for a proof.

We saw that H is normal in N and that the quotient N/H is isomorphic to the Weyl groupW . Moreover we found the property that UH ∩ N = H. These two properties will beimportant to prove that G has a BN -pair.To do so, we introduce the Borel subgroup.

2.4 The Borel subgroup B

As mentioned earlier (see the introduction to Chapter 2), we are interested in a subgroupof G called the Borel subgroup. We already know it. Here is its definition.

Definition 2.24. Let B := UH. We call B the Borel subgroup of G.

As already stated, B and N form a BN -pair of G in the sense of Definition 3.1. We nowgive a technical proposition concerning double cosets of N with respect to B. As we willsee in the next chapter, it corresponds to axiom four of the definition of a BN -pair (seeDefinition 3.1).

Proposition 2.25. Let r ∈ Π and n ∈ N . Then

BnB.BnrB ⊆ BnnrB ∪BnB.

Proof. For a proof of this proposition see [Ca, Proposition 8.1.5].

To find the Borel subgroup and the monomial subgroup, we studied the subgroups U, V,〈Xr, X−r〉 and H in this chapter. Using these subgroups we now derive the structure of G.

19

3 Properties of Chevalley groups

Discussing the preceding subgroups already gives us an understanding of the Chevalleygroup. However, there are some further interesting properties which we point out in thissection.

We are now ready to define the notion of a BN-pair. We will then state a theorem aboutthe simplicity of the Chevalley groups G = L(K). Sketching its proof will illustrate theimportance of the existence of such a BN -pair.

3.1 The BN-pair

The concept of a BN-pair is much more general and is used in a broad spectrum in algebra.The fact that a certain group has a BN-pair reveals many of its properties.Let us define it in a general context.

Definition 3.1. Let G be any group. A pair of subgroups (B,N) of G is called a BN-pairof G if the following axioms are satisfied:

(1) G = 〈B,N〉 .

(2) B ∩N is a normal subgroup of N .

(3) The group W := N/(B ∩N) is generated by a set of involutions wi (i ∈ I).

(4) For any preimage ni of wi under the natural homomorphism ϕ : N → W and anyn ∈ N , we have

BniB.BnB ⊆ BninB ∪BnB.

(5) For ni as in (4), niBni 6= B.

Let us give an example.

Example 5. Let G := PSL2(K) := SL2(K)/ ±id. Let B be the subgroup consisting ofthe classes of matrices of the form (

a b0 a−1

)with a, b ∈ K and let T be the subgroup of B that consists of the classes of the diagonalmatrices (

a 00 a−1

)with a ∈ K. Then B, and N := NG(T ) form a BN-pair of G = PSL2(K).

Proof. Note first that a class of a matrix in PSL2(K) is just the matrix itself and its additiveinverse, i.e.

Class of

(a bc d

)= ±id ·

(a bc d

)=

(a bc d

),

(−a −b−c −d

)

From now on we just write

(a bc d

)and mean the class of

(a bc d

).

20

(a) We calculate the normalizer of T in G. For this, set g :=

(c de f

)and note that

g−1 =

(f −d−e c

), since cf − de = 1.

NG(T ) =g ∈ G

∣∣gTg−1 = T

=

g ∈ G

∣∣∣∣∀ a ∈ K, ∃ a′ ∈ K, g · (a 00 a−1

)· g−1 =

(a′ 00 a′−1

)= T ∪

(0 a−a−1 0

)∣∣∣∣ a ∈ KThe last equation holds because of the following.If ∀ a ∈ K, ∃ a′ ∈ K such that,(

a′ 00 a′−1

)=

(c de f

)·(a 00 a−1

)·(f −d−e c

)=

(acf − a−1d2 −ace+ cda−1

(a− a−1)ef −ae2 + a−1cf

),

i.e. (a′ 00 a′−1

)=

(acf − a−1d2 −ace+ cda−1

(a− a−1)ef −ae2 + a−1cf

),

then due to the lower left component, e = 0 or f = 0. So, we have two cases.

(1) Case e = 0: Then 1 = cf−de = cf , hence f = c−1 and in particular c, f 6= 0. We

get

(a′ 00 a′−1

)=

(a− a−1d2 cda−1

0 a−1

). Now the upper right component implies

that d = 0. So there are no matrices with e = 0 in NG(T ) but those in T .

(2) Case f = 0: In this case, ed = −1 hence e = −d−1 and e, d 6= 0. We get(a′ 00 a′−1

)=

(−a−1d2 acd−1 + cda−1

0 −ad−2

)and the equation holds if and only if

acd−1 + cda−1 = 0 for all a ∈ K. This is the case if and only if c = 0. So, NG(T )

contains also the matrices of the form

(0 d−d−1 0

).

(b) First we prove the third axiom. Consider the bijection ψ : T → N\T,(a 00 a−1

)7→(

0 a−a−1 0

). Hence there are only two cosets, idW = T and w1 := N\T . Since(

0 a−a−1 0

)·(

0 b−b−1 0

)=

(−ab−1 0

0 −a−1b

)∈ T , we have that w2

1 = idW and

therefore w1 is an involution that generates W .

(c) Now we prove that the first axiom of the definition of a BN -pair, G = 〈B,N〉 , holds.

Let g :=

(a bc d

)be an arbitrary element in G.

· If c = 0, then g ∈ B.

· If a = 0, then g =

(0 b−b−1 d

)=

(0 1−1 0

)·(b−1 −d0 b

)∈ NB.

· If d = 0, then g =

(a b−b−1 0

)=

(b −a0 b−1

)·(

0 1−1 0

)∈ BN.

21

· Else, we can assume that a, b, c are all non-zero. In this case since c−1(ad−1) = b,

g =

(a bc d

)=

(−ad−1 −c−1d

0 −a−1d

)·(

0 1−1 0

)·(cad−1 a

0 c−1a−1d

)∈ BNB.

Therefore, every element of G is generated by some elements in B and N .

(d) The second axiom, namely that B∩N E N is not hard to prove. B∩N = T ⊆ NG(T )by part (a) and NG(T ) normalizes T by definition.

(e) To prove the technical fourth axiom let s be a preimage of w1, i.e. an element of N\T .We distinguish two cases.

(1) Case n ∈ T . Hence n ∈ B and BsB.BnB = BsB = BsnB.

(2) Case n :=

(0 a−a−1 0

)∈ N\T . In this case, we have

(0 a−a−1 0

)=

(0 1−1 0

)·(a−1 00 a

)and thus

BnB = B

(0 1−1 0

)B.

But if we take a close look at part (c), every matrix in G\B can be written as

b1

(0 1−1 0

)b2 for some b1, b2 ∈ B. Since sn ∈ (N\T )2 = T ⊆ B, BsnB = B.

Therefore BsnB ∪BnB = G.

We see that in both cases BsB.BnB ⊆ BsnB ∪BnB and the axiom holds.

(f) The last axiom is just a calculation. Let s :=

(0 a−a−1 0

)be an arbitrary element of

N\T . Then

sBs 3(

0 a−a−1 0

)·(

1 10 1

)·(

0 a−a−1 0

)=

(−1 0a−2 −1

)/∈ B

and therefore sBs 6= B for all s ∈ N\T which is the preimage of w1.

One could show that PSL2(K) corresponds to a Chevalley group, but this is not subject ofthis project. However, not only PSL2(K) has a BN -pair but all Chevalley groups.

Proposition 3.2. The Chevalley group G := L(K) has a BN-pair.

As an illustration of the importance of the subgroups discussed in Chapter 2, we give aproof of the above proposition.

Proof. We want to show that the Borel subgroup B and the monomial subgroup N form aBN -pair of G. Let us proof the axioms:

(2) By Corollary 2.23 we know that B ∩ N = UH ∩ N=H, the diagonal subgroup of G.Since H is normal in N by Theorem 2.21, the second axiom follows.

22

(3) By the same theorem we have N/(B ∩N) = N/H ∼= W , where W denotes the Weylgroup of (LK ,Φ). But the Weyl group is generated by the simple reflections in thehyperplane orthogonal to r, for r ∈ Π. Denote them by wr. We have that w2

r = id,since wr is a reflection. The elements wr are therefore involutions. This proves thethird axiom.

(4) The fourth axiom holds by Proposition 2.25.

(5) Fix an arbitrary element r ∈ Π. Then Xr ⊆ B since B contains U and U containsXr. By using Lemma 2.22, we get nr ·Xr · n−1

r = Xwr(r) = X−r. But X−r * B sinceV contains X−r and B ∩ V = 1 by Lemma 2.18. Hence the fifth axiom follows.

(1) First recall that G is generated by the root subgroups Xr. Therefore it is sufficientto show that every root subgroup Xr is contained in 〈B,N〉. Thus arbitrarily fixan r ∈ Φ. It is a general fact that every root r ∈ Φ can be written as w(ri) forsome w ∈ W and ri ∈ Π, where Π denotes a fundamental basis of Φ as usual [Ca,Proposition 2.1.8]. Choose an element n ∈ N with ϕ(n) = w. Note that such anelement n exists since ϕ is surjective by Theorem 2.21. Lemma 2.22 now yields thatXr = Xw(ri) = n ·Xri · n−1. Therefore Xr is generated by N and B and so is G. Thefirst axiom follows.

Since all five axioms are fulfilled, we have shown that the Borel subgroupB and the monomialsubgroup N form a BN -pair of G.

The fact that a Chevalley group has a BN -pair motivates the investigation of further prop-erties of groups having a BN -pair.

Proposition 3.3. Let G be a group with a BN-pair. Then the following two propertieshold:

(i) G = BNB

(ii) For each subset J ⊆ I, let WJ be the subgroup of W generated by the elements wiwhere i ∈ J . Furthermore, let NJ denote the subgroup of N mapping to WJ under ϕ.Then PJ := BNJB is a subgroup of G.

Proof. A proof can be found in [Ca, Proposition 8.2.2].

Proposition 3.4. Let G be a group with a BN-pair. Let n and n′ be two elements of N .Then

BnB = Bn′B ⇐⇒ ϕ(n) = ϕ(n′)

Thus there is a one-to-one correspondence between double cosets of B in G and elements ofW .

Proof. A proof can be found in [Ca, Proposition 8.2.3].

Let us translate this property to the case where G = L(K).We have a one-to-one correspondence between double cosets of B in G and the Weyl groupof (LK ,Φ). Since the Weyl group is finite, this in particular yields that the number of doublecosets of B in G is finite, which is not the case in general.

For the next proposition, we need the notion of the length of an element in W .Let G be a group with a BN -pair. Then the group W is generated by involutions wi, (i ∈ I).Because of that, every element w ∈ W can be written as a product of the involutions wi, i.e.

23

w = wi1 ·wi2 · · ·wik . Define the length of w to be the minimal length of such an expression,i.e

l(w) := min k ∈ N| ∃ i1, ..., ik ∈ I such that w = wi1 · · · wik .

We are now ready to state a technical proposition which is needed to prove the simplicityof the Chevalley groups.

Proposition 3.5. Let G be a group with a BN-pair. Let wi be one of the generators of Wand w any element of W such that l(wi · w) ≥ l(w). Let ni and n be elements of N withϕ(ni) = wi and ϕ(n) = w. Then BniB.BnB ⊆ BninB.

Proof. See [Ca, Proposition 8.2.4] for a proof.

Corollary 3.6. If like in the preceding proposition, ϕ(ni) = wi and ϕ(n) = w, then we havethe following:

l(wi · w) < l(w)⇒ ni ∈ BnBn−1B.

Proof. See [Ca, Corollary 8.2.6] for a proof.

Finally, we take a look at the so called parabolic subgroups of a group G with a BN -pair.

Definition 3.7. Let G be a group with a BN -pair. We call the groups which contain aconjugate gBg−1 of B parabolic subgroups of G.

For example, the groups PJ = BNJB are parabolic subgroups. In fact their conjugates arethe only ones.

Theorem 3.8. Let G be a group with a BN-pair. Then the conjugates of the subgroups PJare the only parabolic subgroups of G.

Proof. For a proof see [Ca, Theorem 8.3.2].

To prove the above theorem, one has to take a look at the subgroup generated by a singleelement of the monomial subgroup N and the Borel subgroup B.

Proposition 3.9. Let G be a group with a BN-pair and let n be an element of N . Letw ∈ W be the image of n under the natural homomorphism and let

w = wi1 · wi2 · · · wik , i1, ..., ik ∈ I

where l(w) = k. Let J be i1, ..., ik ⊆ I. Then

〈B, n〉 =⟨B, nBn−1

⟩= PJ .

Proof. A proof of this proposition can be found in [Ca, Proposition 8.3.1].

Parabolic subgroups have some further nice properties which are summarised in the twofollowing theorems.

Theorem 3.10. Let G be a group with a BN-pair. Then each PJ ⊆ G is equal to itsnormalizer. Moreover, if PJ and PK are distinct, then PJ and PK are not conjugate to oneanother.

Proof. For a proof see [Ca, Theorem 8.3.3].

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Theorem 3.11. Let G be a group with a BN-pair. Then the parabolic subgroups PJ fordistinct subsets J ⊆ I are all distinct. Furthermore, PJ ∩ PK = PJ∩K.

Proof. For a proof see [Ca, Theorem 8.3.4].

Thus the parabolic subgroups form a lattice in bijective correspondence to the lattice ofsubsets of I.

With this statement we close the section about BN -pairs.We discussed the notion of the BN -pair, gave an example of a group, with a BN -pair, andproved that for every Chevalley group its Borel and monomial subgroups form a BN -pair.Because of that we were motivated to state some additional properties of groups having aBN -pair. Finally, we talked about parabolic subgroups.We focus now on the simplicity of the Chevalley group.

3.2 Simplicity

To prove the simplicity of the Chevalley group, we use - as already stated - a more generalresult for groups with a BN -pair. For this, let G′ := ghg−1h−1| g, h ∈ G denote thecommutator subgroup of any group G. Moreover, let I denote the same set as in thedefinition of the notion of the BN -pair, i.e. I is the index set of the involutions thatgenerate the group W = N/(B ∩N).

Theorem 3.12. Let G be any group with a BN-pair. If moreover, G satisfies the followingconditions:

(a) G = G′,

(b) B is solvable,

(c)⋂g∈G gBg

−1 = 1,

(d) the set I cannot be decomposed into two non-empty subsets J and K such that J∪K =I and that all wj commute with all wk, j ∈ J , k ∈ K,

then G is simple.

Proof. For a proof see [Ca, Theorem 11.1.1].

We want to show that the BN -pair we constructed for the Chevalley group satisfies the fourconditions of Theorem 3.12 in most of the cases.To do so, we need an auxiliary lemma.

Lemma 3.13. Fix two elements r ∈ Φ and t ∈ K\0. Moreover let Q denote the additivegroup of fundamental weights as in Section 2.3.

(i) There is a K-character of Q such that χ(r) = t2.

(ii) There is a K-character of Q such that χ(r) = t, unless L = A1 or L = Cl and r is along root.

Proof. See [Ca, Lemma 11.1.3] for a proof.

Now we can state the important theorem about the simplicity of the Chevalley group.

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Theorem 3.14. Let L be a simple Lie algebra defined over C and let K be an arbitrary field.Then the Chevalley group G = L(K) is simple except for A1(2), A1(3), B2(2) and G2(2).

Remark 3.15. A classification of the simple Lie algebras and the definition ofA1, B2 and G2 can be found in [Hu, Section 11.4].

Proof. (sketch) The idea is to use Theorem 3.12 and Lemma 3.13. Therefore, we haveto show that the Borel subgroup B and the monomial subgroup N satisfy the conditions(a)− (d). We give the idea of the proof for each condition. A complete proof can be foundin [Ca, Theorem 11.1.2].

(b) The Borel subgroup B = UH is solvable since U is nilpotent and H abelian.

(d) Since L is simple, the generating involutions of the Weyl group W , which are thesimple reflections, cannot be decomposed into two complementary, non-empty subsetsthat commute.

(c) One can show that G has no non-trivial normal subgroup contained in B. This impliesthe condition

⋂g∈G gBg

−1 = 1.

(a) First one can show that G has trivial center. Then Lemma 3.13 (i) can be used toprove that G is perfect, i.e. the condition G = G′ for each field K having at least fourdistinct elements. Note that in such a field there exists a non-zero element t, suchthat t2 6= 1. Finally, one can show for all smaller fields that G is perfect by usingLemma 3.13 (ii). Note that this part of the lemma is applicable in all cases except forA1(2), A1(3), B2(2) and G2(2).

The above theorem leaves us four cases to consider separately.

Remark 3.16. The Chevalley groups A1(2), A1(3), B2(2) and G2(2) are in fact not simple.

Proof. We give some references, some ideas and some complete proofs.

(1) The Chevalley group A1(2) is non-abelian and has order 6. Since S3 is the only non-abelian group of order 6, A1(2) ∼= S3. But we know that S3 is solvable.

(2) The Chevalley group A1(3) is isomorphic to A4, the alternating group, which is solv-able. This can be seen as follows. Let G be PSL2(3) which is the Chevalley group ofA1(3) and let V be a two dimensional vector space over F3 with basis 〈e1, e2〉. DefineΩ := 〈e1〉 , 〈λ · e1 + e2〉|λ ∈ F3, so |Ω| = 4. Now G acts on Ω by left multiplication.Hence it exists an F ≤ S4 such that G ∼= F . Since |G| = 12 and since A4 is the onlysubgroup of order 12 of S4, we get G ∼= A4.

(3) See [Di] for a proof concerning the non simplicity of B2(2).

(4) Finally consider the Chevalley group G := G2(2). The commutator G′ is a propersubgroup of G by [Co]. Hence G has a proper normal subgroup and is therefore notsimple.

To summarise we used the subgroups defined and studied in the second chapter to find aBN -pair of G. With the help of this BN -pair we showed that all Chevalley groups aresimple except A1(2), A1(3), B2(2) and G2(2) which in fact are not.

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Conclusion

To conclude, one can say that the Chevalley group has a very interesting structure.

When we constructed the Chevalley group, it looked like a group with quite strange gener-ators. We considered the Chevalley group of a simple Lie algebra defined over the complexnumbers and could generalise the concept thanks to the properties of the Chevalley basis tosimple Lie algebras over arbitrary fields.We may have asked ourself why this group, which is constructed in this quite complicatedway, should have interesting properties. By studying subgroups of the Chevalley group, thisfeeling did not vanish very quickly and we were probably still skeptical about the use of theChevalley group.However, once we saw the variety of these subgroups and encountered big theorems con-cerning them, we began to understand the importance of this theory. In the moment wherewe found a BN -pair of the Chevalley group, we certainly understood why we had studiedall those subgroups before.Finally, we could even see that the Chevalley group is simple in most of the cases, which isreally an interesting property.

I hope that the reader is pleased about the theory presented in this project and enjoyed thereading.

Acknowledgment

Doctor Claude Marion invested many hours to supervise this project. He always tookhis time to help me, had good advises which improved my project and knew the rightformulations to make it also under a formal aspect more appropriate. Thank you!Also thanks to Professor Donna Testerman who helped me finding this interesting subject,provided helpful material and set guidelines which made a successful studying and writingpossible.

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References

[Ca] R. W. Carter. Simple Groups of Lie Type,John Wiley & Sons Ltd, London, 1972.

[Co] S. H. Conway, R. T. Curties, S. P. Norton, R. A. Parker, R. A. Wilson.Atlas of finite groups, Oxford University Press, Eynsham ,1985.

[Di] J. Dieudonne. Les isomorphismes exeptionnels entre les groupes classiques finis,Canadian Journal of Mathematics 6, pp 305-315, 1954.

[Hu] J. E. Humphreys. Introduction to Lie Algebras and Representation TheorySpringer-Verlag, New York, 1972.

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