G2 : Aut Dyn(¦) = {1}, Λ/Λr = 0. c< c

§25. Split Semisimple Groups

In this section we give the classi¬cation of split semisimple groups over an

arbitrary ¬eld F . The classi¬cation does not depend on the base ¬eld and corre-

sponds to the classi¬cation over an algebraically closed ¬eld. The basic references

are Borel-Tits [?] and Tits [?]. An algebraic group G over F is said to be solvable

if the abstract group G(Falg ) is solvable, and semisimple if G = 1, G is connected,

and GFalg has no nontrivial solvable connected normal subgroups.

356 VI. ALGEBRAIC GROUPS

A subtorus T ‚ G is said to be maximal if it is not contained in a larger

subtorus. Maximal subtori remain maximal over arbitrary ¬eld extensions and are

conjugate over Falg by an element of G(Falg ). A semisimple group is split if it

contains a split maximal torus. Any semisimple group over a separably closed ¬eld

is split.

We will classify split semisimple groups over an arbitrary ¬eld. Let G be

split semisimple and let T ‚ G be a split maximal torus. Consider the adjoint

representation (see Example (??.??)):

ad : G ’ GL Lie(G) .

By the theory of representations of diagonalizable groups (see (??)) applied to the

restriction of ad to T , we get a decomposition

Lie(G) = V±

±

where the sum is taken over all weights ± ∈ T — of the representation ad |T . The

non-zero weights of the representation are called the roots of G (with respect to T ).

The multiplicity of a root is 1, i.e., dim V± = 1 if ± = 0 (we use additive notation

for T — ).

(25.1) Theorem. The set ¦(G) of all roots of G is a root system in T — —Z R.

The root system ¦(G) does not depend (up to isomorphism) on the choice of a

maximal split torus and is called the root system of G. We say that G is of type ¦

if ¦ ¦(G). The root lattice Λr clearly is contained in T — .

(25.2) Proposition. For any ± ∈ ¦(G) and χ ∈ T — one has ±— , χ ∈ Z. In

particular Λr ‚ T — ‚ Λ.

Consider pairs (¦, A) where ¦ is a root system in some R-vector space V and

∼

A ‚ V is an (additive) subgroup such that Λr ‚ A ‚ Λ. An isomorphism (¦, A) ’ ’

∼

(¦ , A ) of pairs is an R-linear isomorphism f : V ’ V such that f (¦) = ¦ and

’

f (A) = A . To each split semisimple group G with a split maximal torus T ‚ G

one associates the pair ¦(G), T — .

(25.3) Theorem. Let Gi be split semisimple groups with a split maximal torus Ti ,

—

i = 1, 2. Then G1 and G2 are isomorphic if and only if the pairs ¦(G1 ), T1 and

—

¦(G2 ), T2 are isomorphic.

When are two pairs (¦1 , A1 ) and (¦2 , A2 ) isomorphic? Clearly, a necessary

condition is that ¦1 ¦2 . Assume that ¦1 = ¦2 = ¦, then Λr ‚ Ai ‚ Λ for i = 1,

2.

(25.4) Proposition. (¦, A1 ) (¦, A2 ) if and only if A1 /Λr and A2 /Λr are con-

jugate under the action of Aut(V, ¦)/W (¦) Aut Dyn(¦) .

Thus, to every split semisimple group G one associates two invariants: a root

system ¦ = ¦(G) and a (¬nite) subgroup T — /Λr ‚ Λ/Λr modulo the action of

Aut Dyn(¦) .

(25.5) Theorem. For any root system ¦ and any additive group A with Λ r ‚ A ‚

Λ there exists a split semisimple group G such that ¦(G), T — (¦, A).

A split semisimple group G is called adjoint if T — = Λr and simply connected

simply connected if T — = Λ. These two types of groups are determined (up to

isomorphism) by their root systems.

§25. SPLIT SEMISIMPLE GROUPS 357

Central isogenies. Let π : G ’ G be a central isogeny of semisimple groups

and let T ‚ G be a split maximal torus. Then, T = π ’1 (T ) is a split maximal

torus in G and the natural homomorphism T — ’ T — induces an isomorphism of

∼

root systems ¦(G ) ’ ¦(G).

’

Let G be a split semisimple group with a split maximal torus T . The kernel

C = C(G) of the adjoint representation adG is a subgroup of T and hence is a diago-

nalizable group (not necessarily smooth!). The restriction map T — ’ C — induces an

isomorphism T — /Λr C — . Hence, C is a Cartier dual to the constant group T — /Λr .

One can show that C(G) is the center of G in the sense of Waterhouse [?]. The

image of the adjoint representation adG is the adjoint group G, so that G = G/C.

If G is simply connected then C — Λ/Λr and all other split semisimple groups

with root system isomorphic to ¦(G) are of the form G/N where N is an arbitrary

subgroup of C, Cartier dual to a subgroup in (Λ/Λr )const . Thus, for any split

semisimple G there are central isogenies

π π

(25.6) G’ G’ G

’ ’

with G simply connected and G adjoint.

(25.7) Remark. The central isogenies π and π are unique in the following sense:

If π and π is another pair of isogenies then there exist ± ∈ Aut(G) and β ∈ Aut(G)

such that π = π —¦ ± and π = β —¦ π.

25.A. Simple split groups of type A, B, C, D, F , and G. A split semi-

simple group G is said to be simple if Galg has no nontrivial connected normal

subgroups.

(25.8) Proposition. A split semisimple group G is simple if and only if ¦(G)

is an irreducible root system. A simply connected (resp. adjoint) split semisim-

ple group G is the direct product of uniquely determined simple subgroups G i and

¦(G) ¦(Gi ).

Type An , n ≥ 1. Let V be an F -vector space of dimension n + 1 and let

G = SL(V ). A choice of a basis in V identi¬es G with a subgroup in GL n+1 (F ).

The subgroup T ‚ G of diagonal matrices is a split maximal torus in G. Denote

by χi ∈ T — the character

χi diag(t1 , t2 , . . . , tn+1 ) = ti , i = 1, 2, . . . , n + 1.

The character group T — then is identi¬ed with Zn /(e1 +e2 +· · ·+en+1 )Z by ei ” χi .

¯

The Lie algebra of G consists of the trace zero matrices. The torus T acts

on Lie(G) by conjugation through the adjoint representation (see (??)). The weight

subspaces in Lie(G) are:

(a) The space of diagonal matrices (trivial weight),

(b) F · Eij for all 1 ¤ i = j ¤ n + 1 with weight χi · χ’1 .

j

We get therefore the root system { ei ’ ej | i = j } (in additive notation) in the

¯ ¯

—

space T —Z R, of type An . One can show that SL(V ) is a simple group and since

T — = Z · ei = Λ, it is simply connected. The kernel of the adjoint representation

¯

of G is µn+1 . Thus:

(25.9) Theorem. A split simply connected simple group of type An is isomorphic

to SL(V ) where V is an F -vector space of dimension n + 1. All other split semi-

simple groups of type An are isomorphic to SL(V )/µk where k divides n + 1. The

group SL(V )/µn+1 PGL(V ) is adjoint.

358 VI. ALGEBRAIC GROUPS

Type Bn , n ≥ 1. Let V be an F -vector space of dimension 2n+1 with a regular

quadratic form q and associated polar form bq . Assume that bq is of maximal Witt

index. Choose a basis (v0 , v1 , . . . , v2n ) of V such that bq (v0 , vi ) = 0 for all i ≥ 1

and

1 if i = j ± n, with i, j ≥ 1,

bq (vi , vj ) =

0 otherwise.

Consider the group G = O+ (V, q) ‚ GL2n+1 (F ). The subgroup T of diagonal

matrices t = diag(1, t1 , . . . , tn , t’1 , . . . , t’1 ) is a split maximal torus of G. Let χi

n

1

be the character χi (t) = ti , (1 ¤ i ¤ n), and identify T — with Zn via χi ” ei .

The Lie algebra of G consists of all x ∈ End(V ) = M2n+1 (F ) such that

bq (v, xv) = 0 for all v ∈ V and tr(x) = 0. The weight subspaces in Lie(G) with

respect to ad |T are:

(a) The space of diagonal matrices in Lie(G) (trivial weight),

(b) F · (Ei,n+j ’ Ej,n+i ) for all 1 ¤ i < j ¤ n with weight χi · χj ,

F · (Ei+n,j ’ Ej+n,i ) for all 1 ¤ i < j ¤ n with weight χ’1 · χ’1 ,

(c) i j

’1

(d) F · (Eij ’ En+j,n+i ) for all 1 ¤ i = j ¤ n with weight χi · χj ,

F · (E0i ’ 2aEn+i,0 ) where a = q(v0 ), for all 1 ¤ i ¤ n with weight χ’1 ,

(e) i

(f) F · (E0,n+i ’ 2aEi,0 ) for all 1 ¤ i ¤ n with weight χi .

We get the root system { ±ei , ±ei ±ej | i > j } in Rn of type Bn . One can show that

O+ (V, q) is a simple group, and since T — = Λr , it is adjoint. The corresponding

simply connected group is Spin(V, q). Thus:

(25.10) Theorem. A split simple group of type Bn is isomorphic to Spin(V, q)

(simply connected ) or to O+ (V, q) (adjoint) where (V, q) is a regular quadratic form

of dimension 2n + 1 with polar form bq which is hyperbolic on V / rad(bq ).

Type Cn , n ≥ 1. Let V be a F -vector space of dimension 2n with a nonde-

generate alternating form h. Choose a basis (v1 , v2 , . . . , v2n ) of V such that

±

1 if j = i + n,

h(vi , vj ) = ’1 if j = i ’ n,

0 otherwise.

Consider the group G = Sp(V, h) ‚ GL2n (F ). The subgroup T of diagonal

matrices t = diag(t1 , . . . , tn , t’1 , . . . t’1 ) is a split maximal torus in G. Let χi be

n

1

the character χi (t) = ti (1 ¤ i ¤ n) and identify T — with Zn via χi ” ei .

The Lie algebra of G consists of all x ∈ End(V ) = M2n (F ) such that

h(xv, u) + h(v, xu) = 0

for all v, u ∈ V . The weight subspaces in Lie(G) with respect to ad |T are:

(a) The space of diagonal matrices in Lie(G) (trivial weight),

(b) F · (Ei,n+j + Ej,n+i ) for all 1 ¤ i < j ¤ n with weight χi · χj ,

F · (Ei+n,j + Ej+n,i ) for all 1 ¤ i < j ¤ n with weight χ’1 · χ’1 ,

(c) i j

’1

(d) F · (Eij ’ En+j,n+i ) for all 1 ¤ i = j ¤ n with weight χi · χj ,

F · Ei,n+i for all 1 ¤ i ¤ n with weight χ2 ,

(e) i

F · En+i,i for all 1 ¤ i ¤ n with weight χ’2 .

(f) i

§25. SPLIT SEMISIMPLE GROUPS 359

We get the root system { ±2ei , ±ei ± ej | i > j } in Rn of type Cn . One can show

that Sp(V, h) is a simple group, and since T — = Λ, it is simply connected. The

corresponding adjoint group is PGSp(V, h). Thus

(25.11) Theorem. A split simple group of type Cn is isomorphic either to Sp(V, h)

(simply connected ) or to PGSp(V, h) (adjoint) where (V, h) is a non-degenerate al-

ternating form of dimension 2n.

Type Dn , n ≥ 2. Let (V, q) be a hyperbolic quadratic space of dimension 2n

over F . Choose a basis (v1 , v2 , . . . , v2n ) in V such that

1 if i = j ± n,

bq (vi , vj ) =

0 otherwise.

Consider the group G = O+ (V, q) ‚ GL2n (F ). The subgroup T of diagonal

matrices t = diag(t1 , . . . , tn , t’1 , . . . , t’1 ) is a split maximal torus in G. As in the

n

1

preceding case we identify T — with Zn .

The Lie algebra of G consists of all x ∈ End(V ) = M2n (F ), such that h(v, xv) =

0 for all v ∈ V .

The weight subspaces in Lie(G) with respect to ad |T are:

(a) The space of diagonal matrices in Lie(G) (trivial weight).

(b) F · (Ei,n+j ’ Ej,n+i ) for all 1 ¤ i < j ¤ n with the weight χi · χj ,

(c) F · (Ei+n,j ’ Ej+n,i ) for all 1 ¤ i < j ¤ n with weight χ’1 · χ’1 , i j

’1

(d) F · (Eij ’ Ej+n,i+n ) for all 1 ¤ i = j ¤ n with weight χi · χj .

We get the root system { ±ei ± ej | i > j } in Rn of type Dn . The group O+ (V, q)

T—

is a semisimple group (simple, if n ≥ 3) and Λr Λ. The corresponding

simply connected and adjoint groups are Spin(V, q) and PGO+ (V, q), respectively.

If n is odd, then Λ/Λr is cyclic and there are no other split groups of type Dn .

(Z/2Z)2 , one of which

If n is even, there are three proper subgroups in Λ/Λr

corresponds to O+ (V, q). The two other groups correspond to the images of the

compositions

Spin(V, q) ’ GL1 C0 (V, q) ’ GL1 C ± (V, q)

where C0 (V, q) = C + (V, q) • C ’ (V, q). We denote these groups by Spin± (V, q).

They are isomorphic under any automorphism of C0 (V, q) which interchanges its

two components.

(25.12) Theorem. A split simple group of type Dn is isomorphic to one of the

following groups: Spin(V, q) (simply connected ), O+ (V, q), PGO+ (V, q) (adjoint),

or (if n is even) to Spin± (V, q) where (V, q) is a hyperbolic quadratic space of

dimension 2n.

Type F4 and G2 . Split simple groups of type F4 and G2 are related to certain

types of nonassociative algebras:

(25.13) Theorem. A split simple group of type F4 is simply connected and adjoint

and is isomorphic to Autalg (J) where J is a split simple exceptional Jordan algebra

of dimension 27.

Reference: See Chevalley-Schafer [?], Freudenthal [?, Satz 4.11], Springer [?] or [?,

14.27, 14.28]. The proof given in [?] is over R, the proofs in [?] and [?] assume that

F is a ¬eld of characteristic di¬erent from 2 and 3. Springer™s proof [?] holds for

any ¬eld.

360 VI. ALGEBRAIC GROUPS

For a simple split group of type G2 we have

(25.14) Theorem. A split simple group of type G2 is simply connected and adjoint

and is isomorphic to Autalg (C) where C is a split Cayley algebra.

Reference: See Jacobson [?], Freudenthal [?] or Springer [?]. The proof in [?] is

over R, the one in [?] assumes that F is a ¬eld of characteristic zero and [?] gives

a proof for arbitrary ¬elds.

More on (??), resp. (??) can be found in the notes at the end of Chapter IX,

resp. VIII.

25.B. Automorphisms of split semisimple groups. Let G be a split semi-

simple group over F , let T ‚ G be a split maximal torus, and Π a system of simple

roots in ¦(G). For any • ∈ Aut(G) there is g ∈ G(F ) such that for ψ = Int(g) —¦ •,

one has ψ(T ) = T and ψ(Π) = Π, hence ψ induces an automorphism of Dyn(¦).

Thus, we obtain a homomorphism Aut(G) ’ Aut Dyn(¦) .

On the other hand, we have a homomorphism Int : G(F ) ’ Aut(G) taking

g ∈ G(F ) to the inner automorphism Int(gR ) of G(R) for any R ∈ Alg F where gR

is the image of g under G(F ) ’ G(R).

(25.15) Proposition. If G is a split semisimple adjoint group, the sequence

Int

1 ’ G(F ) ’’ Aut(G) ’ Aut Dyn(¦) ’ 1

’

is split exact.

Let G be a split semisimple group (not necessarily adjoint) and let C = C(G)

be the kernel of adG . Then G = G/C is an adjoint group with ¦(G) = ¦(G) and

we have a natural homomorphism Aut(G) ’ Aut(G). It turns out to be injective

and its image contains Int G(F ) .

(25.16) Theorem. Let G be a split semisimple group. Then there is an exact

sequence

1 ’ G(F ) ’ Aut(G) ’ Aut Dyn(¦)

where the last map is surjective and the sequence splits provided G is simply con-

nected or adjoint.

(25.17) Corollary. Let G be a split simply connected semisimple group. Then the

natural map Aut(G) ’ Aut(G) is an isomorphism.

§26. Semisimple Groups over an Arbitrary Field

In this section we give the classi¬cation of semisimple groups over an arbitrary

¬eld which do not contain simple components of types D4 , E6 , E7 or E8 . We recall

that a category A is a groupoid if all morphisms in A are isomorphisms. A groupoid

A is connected if all its objects are isomorphic.

Let “ be a pro¬nite group and let A be a groupoid. A “-embedding of A is a

functor i : A ’ A where A is a connected groupoid, such that for every X, Y in A,

there is a continuous “-action on the set HomA (i X, i Y ) with the discrete topology,

compatible with the composition law in A, and such that the functor i induces a

bijection

∼

HomA (X, Y ) ’ HomA (i X, i Y )“ .

’

It follows from the de¬nition that a “-embedding is a faithful functor.

§26. SEMISIMPLE GROUPS OVER AN ARBITRARY FIELD 361

(26.1) Examples. (1) Let 1 An = 1 An (F ) be the category of all central simple

algebras over F of degree n + 1 with morphisms being isomorphisms of F -algebras.

Then for the group “ = Gal(Fsep /F ) the natural functor

j : 1 An (F ) ’ 1 An (Fsep ), A ’ Asep = A —F Fsep

is a “-embedding.

(2) Let G be an algebraic group over a ¬eld F and let A = A(F ) be the groupoid

of all twisted forms of G (objects are algebraic groups G over F such that Gsep

Gsep and morphisms are algebraic group isomorphisms over F ). Then for “ =

Gal(Fsep /F ) the natural functor

j : A(F ) ’ A(Fsep ), G ’ Gsep

is a “-embedding.

Let i : A ’ A and let j : B ’ B be two “-embeddings and let S : A ’ B be a

functor. A “-extension of S (with respect to i and j) is a functor S : A ’ B such

that j —¦ S = S —¦ i and for all γ ∈ “, X, Y ∈ A, and f ∈ HomA (i X, i Y ) one has

γ S(f ) = S(γf ).

We call a continuous map γ ∈ “ ’ fγ ∈ AutA (i X) a 1-cocycle if

fγ —¦ γfρ = fγρ

for all γ, ρ ∈ “ and, we say that a “-embedding i: A ’ A satis¬es the descent

condition if for any object X ∈ A and for any 1-cocycle fγ ∈ AutA (i X) there exist

an object Y ∈ A and a morphism h : i Y ’ i X in A such that

fγ = h —¦ γh’1

for all γ ∈ “.

(26.2) Proposition. Let i : A ’ A and j : B ’ B be two “-embeddings and let

S : A ’ B be a functor having a “-extension S : A ’ B. Assume that the “-

embedding i satis¬es the descent condition. If S is an equivalence of categories,

then so is S.

Proof : Since i, j, and S are faithful functors, the functor S is also faithful. Let

g ∈ HomB (S X, S Y ) be any morphism for some X and Y in A. Since S is an

equivalence of categories, we can ¬nd f ∈ HomA (i X, i Y ) such that S(f ) = j(g).

For any γ ∈ “ one has

S(γf ) = γ S(f ) = γ j(g) = j(g) = S(f ),

hence γf = f . By the de¬nition of a “-embedding, there exists h ∈ HomA (X, Y )

such that i h = f . The equality j S(h) = S i(h) = S(f ) = j g shows that S(h) = g,

i.e., S is full as a functor. In view of Maclane [?, p. 91] it remains to check that any

object Z ∈ B is isomorphic to S(Y ) for some Y ∈ A. Take any object X ∈ A. Since

B is a connected groupoid, the objects j S(X) and j Z are isomorphic. Choose any

isomorphism g : j S(X) ’ j Z in B and set

gγ = g ’1 —¦ γg ∈ AutB j S(X)

for γ ∈ “. Clearly, gγ is a 1-cocycle. Since S is bijective on morphisms, there exists

a 1-cocycle

fγ ∈ AutA (i X)

362 VI. ALGEBRAIC GROUPS

such that S(fγ ) = gγ for any γ ∈ “. By the descent condition for the “-embedding i

one can ¬nd Y ∈ A and a morphism h : i Y ’ i X in A such that fγ = h —¦ γh’1 for

any γ ∈ “. Consider the composition

l = g —¦ S(h) : j S(Y ) = S i(Y ) ’ S i(X) = j S(X) ’ j Z

in B. For any γ ∈ “ one has

’1

γl = γg —¦ γ S(h) = g —¦ gγ —¦ S(γh) = g —¦ gγ —¦ S(fγ —¦ h) = g —¦ S(h) = l.

By the de¬nition of a “-embedding, l = j(m) for some isomorphism m : S(Y ) ’ Z

in B, i.e., Z is isomorphic to S(Y ).

(26.3) Remark. Since A and B are connected groupoids, in order to check that a

functor S : A ’ B is an equivalence of categories, it su¬ces to show that for some

object X ∈ A the map

AutA (X) ’ AutB S(X)

is an isomorphism, see Proposition (??).

We now introduce a class of “-embeddings satisfying the descent condition. All

“-embeddings occurring in the sequel will be in this class.

Let F be a ¬eld and “ = Gal(Fsep /F ). Consider a collection of F -vector spaces

V , V (2) , . . . , V (n) , and W (not necessarily of ¬nite dimension). The group “

(1)

(i)

acts on GL(Vsep ) and GL(Wsep ) in a natural way. Let

(1) (n)

ρ : H = GL(Vsep ) — · · · — GL(Vsep ) ’ GL(Wsep )

be a “-equivariant group homomorphism.

Fix an element w ∈ W ‚ Wsep and consider the category A = A(ρ, w) whose

objects are elements w ∈ Wsep such that there exists h ∈ H with ρ(h)(w) = w

(for example w is always an object of A). The set HomA (w , w ) consists of all

those h ∈ H such that ρ(h)(w ) = w . The composition law in A is induced by the

multiplication in H. Clearly, A is a connected groupoid.

Let A = A(ρ, w) be the subcategory of A consisting of all w ∈ W which are

objects in A. Clearly, for any w , w ∈ A the set HomA (w , w ) is “-invariant with

respect to the natural action of “ on H, and we set

HomA (w , w ) = HomA (w , w )“ ‚ H “ .

Clearly, the embedding functor i : A(ρ, w) ’ A(ρ, w) is a “-embedding.

(26.4) Proposition. The “-embedding i : A(ρ, w) ’ A(ρ, w) satis¬es the descent

condition.

Proof : Let w ∈ W be an object in A and let fγ ∈ AutA (w ) ‚ H be a 1-cocycle.

(i) (i)

Let fγ be the i-th component of fγ in GL(Vsep ). We introduce a new “-action

(i) (i) (i)

on each Vsep by the formula γ — v = fγ (γv) where γ ∈ “, v ∈ Vsep . Clearly,

(i)

γ — (xv) = γx · (γ — v) for any x ∈ Fsep . Let U (i) be the F -subspace in Vsep of

“-invariant elements with respect to the new action. In view of Lemma (??) the

natural maps

θ(i) : Fsep —F U (i) ’ Vsep ,

(i)

x — u ’ xu

§26. SEMISIMPLE GROUPS OVER AN ARBITRARY FIELD 363

are Fsep -isomorphisms of vector spaces. For any x ∈ Fsep and u ∈ U (i) one has

(i)’1

γu = fγ (u), hence (with respect to the usual “-action)

(fγ —¦ γ —¦ θ(i) )(x — u) = (fγ —¦ γ)(xu)

(i) (i)

’1

(i) (i) (i)

= fγ (γx · γu) = γx · fγ fγ (u) = γx · u

= θ(i) (γx — u) = θ(i) —¦ γ (x — u).

In other words, fγ = θ —¦ γθ’1 where γθ = γ —¦ θ —¦ γ ’1 .

The F -vector spaces U (i) and V (i) have the same dimension and are there-

fore F -isomorphic. Choose any F -isomorphism ±(i) : V (i) ’ U (i) and consider the

(i) (i)

composition β (i) = θ(i) —¦ ±sep ∈ GL(Vsep ). Clearly,

fγ = β —¦ γβ ’1 .

(26.5)

Consider the element w = ρ(β ’1 )(w ) ∈ Wsep . By de¬nition, w is an object

of A and β represents a morphism w ’ w in A. We show that w ∈ W , i.e.,

w ∈ A. Indeed, for any γ ∈ “ one has

γ(w ) = ρ(γβ ’1 )(w ) = ρ(β ’1 —¦ fγ )(w ) = ρ(β ’1 )(w ) = w

since ρ(fγ )(w ) = w . Finally, the equation (??) shows that the functor i satis¬es

the descent condition.