(20.13) Examples. (1) GL1 (A) is connected.

(2) For a central simple algebra A, G = SL1 (A) is connected since F [G] is is

the quotient of a polynomial ring modulo the ideal generated by the irreducible

polynomial Nrd(X) ’ 1.

(3) µn is an example of a non-connected group scheme.

(20.14) Lemma (Homogeneity property of Hopf algebras). Let A be a Hopf al-

gebra which is ¬nitely generated over F = Falg . Then for any pair of maximal

ideals M , M ‚ A there exists an F -algebra automorphism ρ : A ’ A such that

ρ(M ) = M .

Proof : We may assume that M is the augmentation ideal in A. Let f be the

canonical projection A ’ A/M = F , and set ρ = (Id A — f ) —¦ c. One checks that the

map IdA — (f —¦ i) —¦ c is inverse to ρ, i.e., ρ ∈ AutF (A). Since (u — IdA ) —¦ c = IdA ,

it follows that u —¦ ρ = (u — f ) —¦ c = f —¦ (u — IdA ) —¦ c = f and ρ(M ) = ρ(ker f ) =

ker u = M .

Let nil(A) be the set of all nilpotent elements of A; it is an ideal of A, and equals

the intersection of all the prime ideals of A. The algebra A/ nil(A) is denoted by

Ared .

(20.15) Proposition. Let G be an algebraic group scheme over F and let A =

F [G]. Then the following conditions are equivalent:

(1) G is connected.

(2) A is connected.

(3) Ared is connected.

(4) Ared is a domain.

Proof : The implications (??) ” (??) ” (??) ⇐ (??) are easy.

For (??) ’ (??) we may assume that F = Falg . Since G is an algebraic group

the scheme A is ¬nitely generated. Hence the intersection of all maximal ideals

in A containing a given prime ideal P is P (Bourbaki [?, Ch.V, §3, no. 4, Cor. to

Prop. 8 (ii)]) and there is a maximal ideal containing exactly one minimal prime

ideal. By the Lemma above, each maximal ideal contains exactly one minimal

prime ideal. Hence any two di¬erent minimal prime ideals P and P are coprime:

332 VI. ALGEBRAIC GROUPS

P + P = A. Let P1 , P2 , . . . , Pn be all minimal prime ideals. Since Pi = nil(A),

we have Ared = A/ nil(A) A/Pi by the Chinese Remainder Theorem. By

assumption Ared is connected, hence n = 1 and Ared = A/P1 is a domain.

Constant and ´tale group schemes. Let H be a ¬nite (abstract) group.

e

Consider the algebra

A = Map(H, F )

of all functions H ’ F . For h ∈ H, let eh be the characteristic function of {h};

this map is an idempotent in A, and we have A = h∈H F · eh . A Hopf algebra

structure on A is given by

1 if h = 1,

c(eh ) = ex — e y , i(eh ) = eh’1 , u(eh ) =

0 if h = 1.

xy=h

The group scheme over F represented by A is denoted Hconst and called the constant

group scheme associated to H. For any connected F -algebra R ∈ Alg F , Hconst (R) =

H.

A group scheme G over F is said to be ´tale if F [G] is an ´tale F -algebra. For

e

e

example, constant group schemes are ´tale and, for any algebraic group scheme G,

e

the group scheme π0 (G) is ´tale. If G is ´tale, then G(Fsep ) is a ¬nite (discrete)

e e

group with a continuous action of “ = Gal(Fsep /F ). Conversely, given a ¬nite

group H with a continuous “-action by group automorphisms, we have a “-action on

the Fsep -algebra A = Map(H, Fsep ). Let Het be the ´tale group scheme represented

e

“

by the (´tale) Hopf algebra A . Subgroups of Het are ´tale and correspond to

e e

“-stable subgroups of H.

(20.16) Proposition. The two functors

´ Finite groups with

Etale group schemes

←’

continuous “-action

over F

G ’ G(Fsep )

Het ← H

are mutually inverse equivalences of categories. In this equivalence constant group

schemes correspond to ¬nite groups with trivial “-action.

Proof : This follows from Theorem (??).

Diagonalizable group schemes and group schemes of multiplicative

type. Let H be an (abstract) abelian group, written multiplicatively. We have a

structure of a Hopf algebra on the group algebra F H over F given by c(h) = h—h,

i(h) = h’1 and u(h) = 1. The group scheme represented by F H is said to be

diagonalizable and is denoted Hdiag . Clearly,

Hdiag (R) = Hom(H, R— ), R ∈ Alg F .

The group-like elements in F H are of the form h — h for h ∈ H. Hence the

character group (Hdiag )— is naturally isomorphic to H. For example, we have

Zdiag = Gm , (Z/nZ)diag = µn .

A group scheme G over F is said to be of multiplicative type if Gsep (= GFsep ) is

diagonalizable. In particular, diagonalizable group schemes are of multiplicative

§20. HOPF ALGEBRAS AND GROUP SCHEMES 333

type. Let G be of multiplicative type. The character group (Gsep )— has a natural

continuous action of “ = Gal(Fsep /F ). To describe this action we observe that the

group of characters (Gsep )— is isomorphic to the group of group-like elements in

Fsep [Gsep ]. The action is induced from the natural action on action on Fsep [Gsep ].

Conversely, given an abelian group H with a continuous “-action, the Hopf alge-

bra of “-stable elements in Fsep [Hdiag ] = Fsep H represents a group scheme of

multiplicative type which we denote Hmult . Clearly,

Hmult (R) = Hom“ H, (R —F Fsep )— .

(20.17) Proposition. The two contravariant functors

Group schemes of

Abelian groups with

multiplicative type ←’

continuous “-action

over F

(Gsep )—

G ’

Hmult ← H

de¬ne an equivalence of categories. Under this equivalence diagonalizable group

schemes correspond to abelian groups with trivial “-action.

An algebraic torus is a group scheme of multiplicative type Hmult where H is a

free abelian group of ¬nite rank. A torus T is said to be split if it is a diagonalizable

group scheme, i.e., T = Hdiag (Zn )diag = Gm — · · ·—Gm (n factors) is isomorphic

to the group scheme of diagonal matrices in GLn (F ). Any torus T is split over Fsep .

Cartier Duality. Let H be a ¬nite abelian (abstract) group with a continuous

“-action and let “ = Gal(Fsep /F ). One can associate two group schemes to H:

Het and Hmult . We discuss the relation between these group schemes. A group

scheme G over F is called ¬nite if dimF F [G] < ∞. The order of G is dimF F [G].

For example an ´tale group scheme G is ¬nite. Its order is the order of G(Fsep ).

e

Let G be a ¬nite commutative group scheme over F ; then A = F [G] is of ¬nite

dimension. Consider the dual F -vector space A— = HomF (A, F ). The duals of

the ¬ve structure maps on A, namely the unit map e : F ’ A, the multiplication

m : A —F A ’ A and the maps c, i, u de¬ning the Hopf algebra structure on A,

yield ¬ve maps which de¬ne a Hopf algebra structure on A— . The associated group

scheme is denoted GD and is called Cartier dual of G. Thus, F [GD ] = F [G]— and

GDD = G.

Elements of the group (GD )(F ) are represented by F -algebra homomorphisms

F [G]— ’ F which, as is easily seen, are given by group-like elements of F [G].

Hence, GD (F ) G— , the character group of G.

Cartier duality is an involutory contravariant functor D on the category of

¬nite commutative group schemes over F .

The restriction of D gives an equivalence of categories

´ Finite group schemes of

Etale commutative

←’

multiplicative type over F

group schemes over F

More precisely, if H is a ¬nite abelian (abstract) group with a continuous “-action,

then

(Het )D = Hmult , (Hmult )D = Het .

334 VI. ALGEBRAIC GROUPS

(20.18) Example.

(Z/nZ)D = µn , µD = Z/nZ.

n

(We write Z/nZ for (Z/nZ)const .)

§21. The Lie Algebra and Smoothness

Let M be an A-module. A derivation D of A into M is an F -linear map

D : A ’ M such that

D(ab) = a · D(b) + b · D(a).

We set Der(A, M ) for the A-module of all derivations of A into M .

21.A. The Lie algebra of a group scheme. Let G be an algebraic group

scheme over F and let A = F [G]. A derivation D ∈ Der(A, A) is said to be left-

invariant if c —¦ D = (id — D) —¦ c. The F -vector space of left-invariant derivations

is denoted Lie(G) and is called the Lie algebra of G. The Lie algebra structure

on Lie(G) is given by [D1 , D2 ] = D1 —¦ D2 ’ D2 —¦ D1 .

Denote by F [µ] the F -algebra of dual numbers, i.e., F [µ] = F ·1•F ·µ with multi-

plication given by µ2 = 0. There is a unique F -algebra homomorphism κ : F [µ] ’ F

G(κ)

with κ(µ) = 0. The kernel of G(F [µ]) ’ ’ G(F ) carries a natural F -vector space

’’

structure: addition is the multiplication in G(F [µ]) and scalar multiplication is de-

¬ned by the formula a · g = G( a )(g) for g ∈ G(F [µ]), a ∈ F , where a : F [µ] ’ F [µ]

is the F -algebra homomorphism de¬ned by a (µ) = aµ.

(21.1) Proposition. There exist natural isomorphisms between the following F -

vector spaces:

(1) Lie(G),

(2) Der(A, F ) where F is considered as an A-module via the co-unit map u : A ’ F ,

(3) (I/I 2 )— where I ‚ A is the augmentation ideal,

G(κ)

(4) ker G(F [µ]) ’ ’ G(F ) .

’’

Proof : (??) ” (??) If D ∈ Lie(G), then u —¦ D ∈ Der(A, F ). Conversely, for

d ∈ Der(A, F ) one has D = (Id — d) —¦ c ∈ Lie(G).

(??) ” (??) Any derivation d : A ’ F satis¬es d(I 2 ) = 0, hence the restriction

d|I induces an F -linear form on I/I 2 . Conversely, if f : I ’ F is an F -linear map

such that f (I 2 ) = 0, then d : A = F · 1 • I ’ F given by d(± + x) = f (x) is a

derivation.

(??) ” (??) An element f of ker G(κ) is an F -algebra homomorphism f : A ’

F [µ] of the form f (a) = u(a) + d(a) · µ where d ∈ Der(A, F ).

(21.2) Corollary. If G is an algebraic group scheme, then dimF Lie(G) < ∞.

Proof : Since A is noetherian, I is a ¬nitely generated ideal, hence I/I 2 is ¬nitely

generated over A/I = F .

(21.3) Proposition. Let G be an algebraic group scheme over F and let A = F [G].

Then Der(A, A) is a ¬nitely generated free A-module and

rankA Der(A, A) = dim Lie(G).

§21. THE LIE ALGEBRA AND SMOOTHNESS 335

Proof : Let G be an algebraic group scheme over F . The map

(I/I 2 )— ’ Der(A, A),

Der(A, F ) d ’ (id — d) —¦ c

used in the proof of Proposition (??) extends to an isomorphism of A-modules

(Waterhouse [?, 11.3.])

∼

A —F (I/I 2 )— ’ Der(A, A).

’

The Lie algebra structure on Lie(G) can be recovered as follows (see Waterhouse

2

[?, p. 94]). Consider the commutative F -algebra R = F [µ, µ ] with µ2 = 0 = µ .

From d, d ∈ Der(A, F ) we build two elements g = u+d·µ and g = u+d ·µ in G(R).

’1

A computation of the commutator of g and g in G(R) yields gg g ’1 g = u+d ·µµ

where d = [d, d ] in Lie(G).

Any homomorphism of group schemes f : G ’ H induces a commutative dia-

gram

fF [µ]

G(F [µ]) ’ ’ ’ H(F [µ])

’’

¦ ¦

¦ ¦H(κ)

G(κ)

fF

G(F ) ’’’

’’ H(F )

and hence de¬nes an F -linear map df : Lie(G) ’ Lie(H), which is a Lie algebra

homomorphism, called the di¬erential of f . If f is a closed embedding (i.e., G is

a subgroup of H) then df is injective and identi¬es Lie(G) with a Lie subalgebra

of Lie(H).

In the next proposition we collect some properties of the Lie algebra; we assume

that all group schemes appearing here are algebraic.

(21.4) Proposition. (1) For any ¬eld extension L/F , Lie(GL ) Lie(G) —F L.

(2) Let fi : Gi ’ H be group scheme homomorphisms, i = 1, 2. Then

Lie(G1 —H G2 ) = Lie(G1 ) —Lie(H) Lie(G2 ).

In particular :

(a) For a homomorphism f : G ’ H and a subgroup H ‚ H,

Lie f ’1 (H ) (df )’1 Lie(H ) .

(b) Lie(ker f ) = ker(df ).

(c) Lie(G1 — G2 ) = Lie(G1 ) — Lie(G2 ).

(3) For any ¬nite separable ¬eld extension L/F and any algebraic group scheme G

over L, Lie RL/F (G) = Lie(G) as F -algebras.

(4) Lie(G) = Lie(G0 ).

Reference: See Waterhouse [?, Chap. 12].

(21.5) Examples. (1) Let G = V, V a vector space over F . The elements of

ker G(κ) have the form v · µ with v ∈ V arbitrary. Hence Lie(G) = V with the

trivial Lie product. In particular, Lie(Ga ) = F .

(2) Let G = GL1 (A) where A is a ¬nite dimensional associative F -algebra. El-

ements of ker G(κ) are of the form 1 + a · µ, a ∈ A. Hence Lie GL1 (A) = A.

One can compute the Lie algebra structure using R = F [µ, µ ]: the commutator of

336 VI. ALGEBRAIC GROUPS

1 + a · µ and 1 + a · µ in G(R) is 1 + (aa ’ a a) · µµ , hence the Lie algebra structure

on A is given by [a, a ] = aa ’ a a. In particular,

Lie GL(V ) = End(V ), Lie(Gm ) = F.

(3) For a central simple algebra A over F , the di¬erential of the reduced norm

homomorphism Nrd : GL1 (A) ’ Gm is the reduced trace Trd : A ’ F since

Nrd(1 + a · µ) = 1 + Trd(a) · µ. Hence,

Lie SL1 (A) = { a ∈ A | Trd(a) = 0 } ‚ A.

(4) The di¬erential of the nth power homomorphism Gm ’ Gm is multiplication

by n : F ’ F since (1 + a · µ)n = 1 + na · µ. Hence

0 if char F does not divide n,

Lie(µn ) =

F otherwise.

(5) If G is an ´tale group scheme, then Lie(G) = 0 since Der(A, A) = 0 for any

e

´tale F -algebra A.

e

(6) Let H be an (abstract) abelian group with a continuous “-action and let G =

Hmult . An element in ker G(κ) has the form 1 + f · µ where f ∈ Hom“ (H, Fsep ).

Hence,

Lie(G) = Hom“ (H, Fsep ) = Hom“ (G— , Fsep ).

sep

(7) Let Sv ‚ GL(V ) be the stabilizer of 0 = v ∈ V . An element of ker Sv (κ) has

the form 1 + ± · µ where ± ∈ End(V ) and (1 + ± · µ)(v) = v, i.e., ±(v) = 0. Thus,

Lie(Sv ) = { ± ∈ End(V ) | ±(v) = 0 }.

(8) Let ρ : G ’ GL(V ) be a homomorphism, 0 = v ∈ V . Then

Lie AutG (v) = { x ∈ Lie(G) | (df )(x)(v) = 0 }.

(9) Let G = Autalg (A) where A is a ¬nite dimensional F -algebra and let

ρ : GL(A) ’ GL Hom(A —F A, A)

be as in Example (??), (??). By computing over F [µ], one ¬nds that the di¬erential

dρ : End(A) ’ End Hom(A —F A, A)

is given by the formula

(dρ)(±)(φ)(a — a ) = ± φ(a — a ) ’ φ ±(a) — a ’ φ a — ±(a ) ,

hence the condition (dρ)(±)(v) = 0, where v is the multiplication map, is equivalent

to ± ∈ Der(A, A), i.e.,

Lie Autalg (A) = Der(A, A).

(10) Let NU be the normalizer of a subspace U ‚ V (see Example (??.??)). Since

the condition (1 + ± · µ)(u + u · µ) ∈ U + U · µ, for ± ∈ End(V ), u, u ∈ U , is

equivalent to ±(u) ∈ U , we have

Lie(NU ) = { ± ∈ End(V ) | ±(U ) ‚ U }.

§21. THE LIE ALGEBRA AND SMOOTHNESS 337

The dimension. Let G be an algebraic group scheme over F . If G is con-

nected, then F [G]red is a domain (Proposition (??)). The dimension dim G of G

is the transcendence degree over F of the ¬eld of fractions of F [G]red . If G is not

connected, we de¬ne dim G = dim G0 .

(21.6) Examples. (1) dim V = dimF V .

(2) dim GL1 (A) = dimF A.

(3) dim SL1 (A) = dimF A ’ 1.

(4) dim Gm = dim Ga = 1.

(5) dim µn = 0.

The main properties of the dimension are collected in the following

(21.7) Proposition. (1) dim G = dim F [G] (Krull dimension).

(2) G is ¬nite if and only if dim G = 0.

(3) For any ¬eld extension L/F , dim(GL ) = dim G.

(4) dim(G1 — G2 ) = dim G1 + dim G2 .

(5) For any separable ¬eld extension L/F and any algebraic group scheme G over L,

dim RL/F (G) = [L : F ] · dim G.

(6) Let G be a connected algebraic group scheme with F [G] reduced (i.e., F [G] has

no nilpotent elements) and let H be a proper subgroup of G. Then dim H < dim G.

Proof : (??) follows from Matsumura [?, Th. 5.6]; (??) and (??) are immediate

consequences of the de¬nition.

(??) Set A = F [G]. Since A is noetherian and dim A = 0, A is artinian. But

A is also ¬nitely generated, hence dimF A < ∞.

(??) We may assume that the Gi are connected and F = Falg . Let Li be the

¬eld of fractions of F [Gi ]red . Since F = Falg , the ring L1 —F L2 is an integral domain

and the ¬eld of fractions of F [G1 — G2 ]red is the ¬eld of fractions E of L1 —F L2 .

Thus, dim(G1 —G2 ) = tr.degF (E) = tr.degF (L1 )+tr.degF (L2 ) = dim G1 +dim G2 .

(??) With the notation of (??) we have

dim RL/F (G) = dim RL/F (G)sep

dim G„ = [L : F ] · dim G.

= dim( G„ ) =

„ ∈X „ ∈X

Smoothness. Let S be a commutative noetherian local ring with maximal

ideal M and residue ¬eld K = S/M . It is known (see Matsumura [?, p. 78]) that

dimK (M/M 2 ) ≥ dim S.

The ring S is said to be regular if equality holds. Recall that regular local rings are

integral domains (Matsumura [?, Th. 19.4]).

(21.8) Lemma. For any algebraic group scheme G over F we have dim F Lie(G) ≥

dim G. Equality holds if and only if the local ring F [G]I is regular where I is the

augmentation ideal of F [G].

Proof : Let A = F [G]. The augmentation ideal I ‚ A is maximal with A/I = F .

Hence, for the localization S = AI with respect to the maximal ideal M = IS we

have S/M = F and

dimF Lie(G) = dimF (I/I 2 ) = dimF (M/M 2 ) ≥ dim S = dim A = dim G,

proving the lemma.

338 VI. ALGEBRAIC GROUPS

(21.9) Proposition. Let G be an algebraic group scheme over F and let A = F [G].

Then the following conditions are equivalent:

(1) AL is reduced for any ¬eld extension L/F .

(2) AFalg is reduced.

(3) dimF Lie(G) = dim G.

If F is perfect, these conditions are also equivalent to

(4) A is reduced.

Proof : (??) ’ (??) is trivial.

(??) ’ (??) We may assume that F = Falg and that G is connected (since

F [G0 ] is a direct factor of A and hence is reduced). By (??) A is an integral domain.

Let K be its ¬eld of fractions. The K-space of derivations Der(K, K) is isomorphic

to Der(A, A) —A K, hence by (??)

dimF Lie(G) = rankA Der(A, A) = dimK Der(K, K).

But the latter is known to equal tr.degF (K) = dim G.

(??) ’ (??) We may assume that L = F = Falg . By (??) the ring AI is

regular hence is an integral domain and is therefore reduced. By the homogeneity

property (see (??)) AM is reduced for every maximal ideal M ‚ A. Hence, A is

reduced.

Finally, assume F is perfect. Since the tensor product of reduced algebras over

a perfect ¬eld is reduced (see Bourbaki [?, Ch.V, §15, no. 5, Th´or`me 3]), it follows

ee

that AFalg is reduced if A is reduced. The converse is clear.

An algebraic group scheme G is said to be smooth if G satis¬es the equivalent

conditions of Proposition (??). Smooth algebraic group schemes are also called

algebraic groups.

(21.10) Proposition. (1) Let G be an algebraic group scheme over F and let L/F

be a ¬eld extension. Then GL is smooth if and only if G is smooth.

(2) If G1 , G2 are smooth then G1 — G2 is smooth.

(3) If char F = 0, all algebraic group schemes are smooth.

(4) An algebraic group scheme is smooth if and only if its connected component G 0

is smooth.

Proof : (??) and (??) follow from the de¬nition of smoothness and (??) (which is a

result due to Cartier) is given in Waterhouse [?, §11.4]. (??) follows from the proof

of (??).

(21.11) Examples. (1) GL1 (A), SL1 (A) are smooth for any central simple F -

algebra A.

´

(2) Etale group schemes are smooth.

(3) Hmult is smooth if and only if H has no p-torsion where p = char F .

Let F be a perfect ¬eld (for example F = Falg ), let G be an algebraic group

scheme over F and let A = F [G]. Since the ring Ared —F Ared is reduced, the

comultiplication c factors through

cred : Ared ’ Ared —F Ared ,

making Ared a Hopf algebra. The corresponding smooth algebraic group scheme

Gred is called the smooth algebraic group associated to G. Clearly Gred is a subgroup

of G and Gred (R) = G(R) for any reduced algebra R ∈ Alg F .

§22. FACTOR GROUPS 339

(21.12) Remark. The classical notion of an (a¬ne) algebraic group over an al-

gebraically closed ¬eld, as an a¬ne variety Spec A endowed with a group structure

corresponds to reduced ¬nitely generated Hopf algebras A, i.e., coincides with the

notion of a smooth algebraic group scheme. This is why we call such group schemes

algebraic groups. Therefore, for any algebraic group scheme G, one associates a

(classical) algebraic group (Galg )red over Falg . The notions of dimension, connected-

ness, Lie algebra, . . . given here then coincide with the classical ones (see Borel [?],

Humphreys [?]).

§22. Factor Groups

22.A. Group scheme homomorphisms.

The injectivity criterion. We will use the following

(22.1) Proposition. Let A ‚ B be Hopf algebras. Then B is faithfully ¬‚at over A.

Reference: See Waterhouse [?, §14.1].

A group scheme homomorphism f : G ’ H is said to be injective if ker f = 1,

or equivalently, if fR : G(R) ’ H(R) is injective for all R ∈ Alg F .

(22.2) Proposition. Let f : G ’ H be a homomorphism of algebraic group sche-

mes. The following conditions are equivalent:

(1) f is injective.

(2) f is a closed embedding (i.e., f — is surjective).

(3) falg : G(Falg ) ’ H(Falg ) is injective and df is injective.

Proof : (??) ’ (??) By replacing H by the image of f we may assume that

f — : A = F [H] ’ F [G] = B

is injective. The elements in G(B—A B) given by the two natural maps B B—A B

have the same image in H(B —A B), hence they are equal. But B is faithfully ¬‚at

over A, hence the equalizer of B B —A B is A. Thus, A = B.

The implication (??) ’ (??) is clear.

(??) ’ (??) Let N = ker f . We have Lie(N ) = ker(df ) = 0, hence by Lemma

(??) dim N ¤ dim Lie(N ) = 0 and N is ¬nite (Proposition (??)). Then it follows

from Proposition (??) that N is smooth and hence ´tale, N = Het where H =

e

N (Fsep ) (see ??). But N (Fsep ) ‚ N (Falg ) = ker(falg ) = 1, hence N = 1 and f is

injective.

The surjectivity criterion.

(22.3) Proposition. Let f : G ’ H be a homomorphism of algebraic group sche-

mes. If H is smooth, the following conditions are equivalent:

(1) f is surjective (i.e., f — is injective).

(2) falg : G(Falg ) ’ H(Falg ) is surjective.

Proof : (??) ’ (??) We may assume that F = Falg . Since B = F [G] is faithfully

¬‚at over A = F [H], any maximal ideal of A is the intersection with A of a maximal

ideal of B (Bourbaki [?, Ch.1, §3, no. 5, Prop. 8 (iv)]), or equivalently, any F -

algebra homomorphism A ’ F can be extended to B. (Note that we are not using

the smoothness assumption here.)

(??) ’ (??) Assume F = Falg . Any F -algebra homomorphism A ’ F factors

through f — , hence all maximal ideals in A contain ker f . But the intersection of

340 VI. ALGEBRAIC GROUPS

all maximal ideals in A is zero since A is reduced, therefore f — is injective and f is

surjective.

(22.4) Proposition. Let f : G ’ H be a surjective homomorphism of algebraic

group schemes.

(1) If G is connected (resp. smooth), then H is connected (resp. smooth).

(2) Let H be a subgroup of H. Then the restriction of f to f ’1 (H ) is a surjective

homomorphism f ’1 (H ) ’ H .

Proof : (??) is clear. For (??), let J ‚ A = F [H] be the Hopf ideal corresponding

to H . Hence the ideal J = f — (J) · B ‚ B = F [G] corresponds to f ’1 (H ),

and the homomorphism F [H ] = A/J ’ B/J = F [f ’1 (H )] is injective since

(f — )’1 (J ) = J (see Bourbaki [?, Ch.I, §3, no. 5, Prop. 8 (ii)]).

The isomorphism criterion. Propositions (??) and (??) imply that

(22.5) Proposition. Let f : G ’ H be a homomorphism of algebraic group sche-

mes with H smooth. Then the following conditions are equivalent:

(1) f is an isomorphism.

(2) f is injective and surjective.

(3) falg : G(Falg ) ’ H(Falg ) is an isomorphism and df is injective.

(22.6) Example. Let f : Gm ’ Gm be the pth -power homomorphism where p =

char F . Clearly, falg is an isomorphism, but f is not since df = 0.

Factor group schemes.

(22.7) Proposition. Let f : G ’ H be a surjective homomorphism of group sche-

mes with kernel N . Then any group scheme homomorphism f : G ’ H vanishing

on N factors uniquely through f .

Proof : Let A = F [H] and B = F [G]. The two natural homomorphisms B

B —A B, being elements in G(B —A B), have the same image in H(B —A B) and

hence they are congruent modulo N (B —A B). Hence the two composite maps

—

f

F [H ] ’ ’ B

’ B —A B

coincide. By the faithful ¬‚atness of B over A the equalizer of B B —A B is A,

—

thus the image of f lies in A.

The proposition shows that a surjective homomorphism f : G ’ H is uniquely

determined (up to isomorphism) by G and the normal subgroup N . We write

H = G/N and call H the factor group scheme G modulo N .

(22.8) Proposition. Let G be a group scheme and let N be a normal subgroup

in G. Then there is a surjective homomorphism G ’ H with the kernel N , i.e.,

the factor group scheme G/N exists.

Reference: See Waterhouse [?, §16.3].

§22. FACTOR GROUPS 341

Exact sequences. A sequence of homomorphisms of group schemes

f g

(22.9) 1’N ’ G’ H ’1

’’

is called exact if f induces an isomorphism of N with ker(g) and g is surjective or,

equivalently, f is injective and H G/ im(f ). For any group scheme homomor-

phism g : G ’ H we have an exact sequence 1 ’ ker(g) ’ G ’ im(g) ’ 1, i.e.,

im(g) G/ ker(g).

(22.10) Proposition. A sequence as in (??) with H smooth is exact if and only

if

fR gR

(1) 1 ’ N (R) ’’ G(R) ’’ H(R) is exact for every R ∈ Alg F and

(2) galg : G(Falg ) ’ H(Falg ) is surjective.

Proof : It follows from (??) that N = ker(g) and from Proposition (??) that g is

surjective.

(22.11) Proposition. Suppose that (??) is exact. Then

dim G = dim N + dim H.

Proof : We may assume that F = Falg and that G (hence also H) is connected.

Put A = F [H], B = F [G], C = F [N ]. We have a bijection of represented functors

∼

G — N ’ G —H G,

’ (g, n) ’ (g, gn).

By Yoneda™s lemma there is an F -algebra isomorphism B —A B B —F C. We

compute the Krull dimension of both sides. Denote by Quot(S) the ¬eld of fractions

of a domain S; let K = Quot(Ared ) and L = Quot(Bred ); then

dim(B —A B) = dim(Bred —Ared Bred ) = tr.degF Quot(L —K L)red

= 2 · tr.degK (L) + tr.degF (K) = 2 · tr.degF (L) ’ tr.degF (K)

= 2 · dim G ’ dim H.

On the other hand

dim(B —F C) = dim(G — N ) = dim G + dim N

by Proposition (??).

(22.12) Corollary. Suppose that in (??) N and H are smooth. Then G is also

smooth.

Proof : By Proposition (??)(b), ker(dg) = Lie(N ). Hence

dim Lie(G) = dim ker(dg) + dim im(dg) ¤ dim Lie(N ) + dim Lie(H),

= dim N + dim H = dim G,

and therefore, G is smooth.

A surjective homomorphism f : G ’ H is said to be separable if the di¬erential

df : Lie(G) ’ Lie(H) is surjective.

(22.13) Proposition. A surjective homomorphism f : G ’ H of algebraic groups

is separable if and only if ker(f ) is smooth.

342 VI. ALGEBRAIC GROUPS

Proof : Let N = ker(f ). By Propositions (??) and (??),

dim Lie(N ) = dim ker(df ) = dim Lie(G) ’ dim im(df ),

= dim G ’ dim im(df ) = dim N + dim H ’ dim im(df ),

= dim N + dim Lie(H) ’ dim im(df ).

Hence, N is smooth if and only if dim N = dim Lie(N ) if and only if dim Lie(H) ’

dim im(df ) = 0 if and only if df is surjective.

(22.14) Example. The natural surjection GL1 (A) ’ GL1 (A)/ Gm is separable.

f

(22.15) Proposition. Let 1 ’ N ’ G ’ H ’ 1 be an exact sequence of alge-

’

braic group schemes with N smooth. Then the sequence of groups

1 ’ N (Fsep ) ’ G(Fsep ) ’ H(Fsep ) ’ 1

is exact.

Proof : Since N = ker(f ), it su¬ces to prove only exactness on the right. We may

assume that F = Fsep . Let A = F [H], B = F [G], (so A ‚ B) and C = F [N ]. Take

any h ∈ H(F ) and consider the F -algebra D = B —A F where F is made into an

A-algebra via h. For any R ∈ Alg F with structure homomorphism ν : F ’ R, we

have

’1

HomAlg F (D, R) = { g ∈ HomAlg F (B, R) | g|A = ν —¦ h } = fR (ν —¦ h),

i.e., the F -algebra D represents the ¬ber functor

’1

R ’ P (R) := fR (ν —¦ h) ‚ G(R).

If there exists g ∈ G(F ) such that fF (g) = h, i.e. g ∈ P (F ), then there is a bijection

of functors „ : N ’ P , given by „ (R)(n) = n · (ν —¦ g) ∈ P (R). By Yoneda™s lemma

the F -algebras C and D representing the functors N and P are then isomorphic.

We do not know yet if such an element g ∈ P (F ) exists, but it certainly exists

over E = Falg since HomAlg E (DE , E) = … (a form of Hilbert Nullstellensatz).

Hence the E-algebras CE and DE are isomorphic. In particular, DE is reduced.

Then HomAlg F (D, F ) = … (see Borel [?, AG 13.3]), i.e., P (F ) = …, so h belongs to

the image of fF , and the described g exists.

Isogenies. A surjective homomorphism f : G ’ H of group schemes is called

an isogeny if N = ker(f ) is ¬nite, and is called a central isogeny if N (R) is central

in G(R) for every R ∈ Alg F .

(22.16) Example. The nth -power homomorphism Gm ’ Gm is a central isogeny.

Representations. Let G be a group scheme over F , with A = F [G]. A

representation of G is a group scheme homomorphism ρ : G ’ GL(V ) where V is

a ¬nite dimensional vector space over F . For any R ∈ Alg F the group G(R) then

acts on VR = V —F R by R-linear automorphisms; we write

g · v=ρR (g)(v), g ∈ G(R), v ∈ VR .

By taking R = A, we obtain an F -linear map

ρ : V ’ V —F A, ρ(v) = IdA · v

§22. FACTOR GROUPS 343

(where IdA ∈ G(A) is the “generic” element), such that the following diagrams

commute (see Waterhouse [?, §3.2])

ρ

V ’’’

’’ V —F A

¦ ¦

¦ ¦

(22.17) ρ Id—c

ρ—Id

V —F A ’ ’ ’ V —F A —F A,

’’

ρ

V ’ ’ ’ V —F A

’’

¦

¦

(22.18) Id—u

∼

V ’ ’ ’ V —F F.

’’

Conversely, a map ρ for some F -vector space V , such that the diagrams (??)

and (??) commute, yields a representation ρ : G ’ GL(V ) as follows: given g ∈

G(R), ρ(g) is the R-linear extension of the composite map

ρ Id—g

V ’ V —F A ’ ’ V —F R.

’ ’’

A ¬nite dimensional F -vector space V together with a map ρ as above is called an

A-comodule. There is an obvious notion of subcomodules.

A vector v ∈ V is said to be G-invariant if ρ(v) = v — 1. Denote by V G the

F -subspace of all G-invariant elements. Clearly, G(R) acts trivially on (V G ) —F R

for any R ∈ Alg F . For a ¬eld extension L/F one has (VL )GL (V G )L .

A representation ρ : G ’ GL(V ) is called irreducible if the A-comodule V has

no nontrivial subcomodules.

(22.19) Examples. (1) If dim V = 1, then GL(V ) = Gm . Hence a 1-dimensional

representation is simply a character.

(2) Let G be an algebraic group scheme over F . For any R ∈ Alg F the group G(R)

acts by conjugation on

ker G(R[µ]) ’ G(R) = Lie(G) —F R.

Hence we get a representation

Ad = AdG : G ’ GL Lie(G)

called the adjoint representation. When G = GL(V ) the adjoint representation

Ad : GL(V ) ’ GL End(V )

is given by conjugation: Ad(±)(β) = ±β±’1 .

Representations of diagonalizable groups. Let G = Hdiag be a diagonal-

izable group scheme, A = F [G] = F H . Let ρ : V ’ V —F A be the A-comodule

structure on a ¬nite dimensional vector space V corresponding to some represen-

tation ρ : G ’ GL(V ).

Write ρ(v) = fh (v) — h for uniquely determined F -linear maps fh : V ’ V .

h∈H

The commutativity of diagram (??) is equivalent to the conditions

fh if h = h ,

fh —¦ f h =

0 if h = h ,

344 VI. ALGEBRAIC GROUPS

and the commutativity of (??) gives fh (v) = v for all v ∈ V . Hence the maps fh

h∈H

induce a decomposition

(22.20) V= Vh , where Vh = im(fh ).

h∈H

A character h ∈ H = G— is called a weight of ρ if Vh = 0. A representation ρ of

a diagonalizable group is uniquely determined (up to isomorphism) by its weights

and their multiplicities mh = dim Vh .

§23. Automorphism Groups of Algebras

In this section we consider various algebraic group schemes related to algebras

and algebras with involution.

Let A be a separable associative unital F -algebra (i.e., A is a ¬nite product

of algebras which are central simple over ¬nite separable ¬eld extensions of F , or

equivalently, AF = A — F is semisimple for every ¬eld extension F of F ). Let L

be the center of A (which is an ´tale F -algebra). The kernel of the restriction ho-

e

momorphism Autalg (A) ’ Autalg (L) is denoted AutL (A). Since all L-derivations

of A are inner (see for example Knus-Ojanguren [?, p. 73-74]), it follows from

Example (??.??) that

Lie AutL (A) = DerL (A, A) = A/L.

We use the notation ad(a)(x) = [a, x] = ax ’ xa for the inner derivation ad(a)

associated to a ∈ A. Consider the group scheme homomorphism

Int : GL1 (A) ’ AutL (A), a ’ Int(a)

with kernel GL1 (L) = RL/F (Gm,L ). By Proposition (??) we have:

dim AutL (A) ≥ dim im(Int) = dim GL1 (A) ’ dim GL1 (L)

= dimF A ’ dimF L = dimF Lie AutL (A) .

The group scheme AutL (A) is smooth. This follows from Lemma (??) and

Proposition (??). By the Skolem-Noether theorem the homomorphism IntE is

surjective for any ¬eld extension E/F , hence Int is surjective by Proposition (??),

and AutL (A) is connected by Proposition (??). Thus we have an exact sequence

of connected algebraic groups

(23.1) 1 ’ GL1 (L) ’ GL1 (A) ’ AutL (A) ’ 1.

Assume now that A is a central simple algebra over F , i.e., L = F . We write

PGL1 (A) for the group Autalg (A), so that

PGL1 (A) GL1 (A)/ Gm , Lie PGL1 (A) = A/F,

and

PGL1 (A)(R) = AutR (AR ), R ∈ Alg F .

We say that an F -algebra R satis¬es the SN -condition if for any central simple

algebra A over F all R-algebra automorphisms of AR are inner. Fields and local

rings satisfy the SN -condition (see for example Knus-Ojanguren [?, p. 107]).

If R satis¬es the SN -condition then

PGL1 (A)(R) = (AR )— /R— .

(23.2)

§23. AUTOMORPHISM GROUPS OF ALGEBRAS 345

We set PGL(V ) = PGL1 End(V ) = GL(V )/ Gm and call PGL(V ) the projective

general linear group; we write PGL(V ) = PGL n if V = F n .

23.A. Involutions. In this part we rediscuss most of the groups introduced

in Chapter ?? from the point of view of group schemes. Let A be a separable F -

algebra with center K and F -involution σ. We de¬ne various group schemes over

F related to A. Consider the representation

ρ : GL1 (A) ’ GL(A), a ’ x ’ a · x · σ(a) .

The subgroup AutGL1 (A) (1) in GL1 (A) is denoted Iso(A, σ) and is called the group

scheme of isometries of (A, σ):

Iso(A, σ)(R) = { a ∈ A— | a · σR (a) = 1 }.

R

An element 1 + a · µ, a ∈ A lies in ker Iso(A, σ)(κ) if and only if

(1 + a · µ) 1 + σ(a) · µ = 1,

or equivalently, a + σ(a) = 0. Hence,

Lie Iso(A, σ) = Skew(A, σ) ‚ A.

Consider the adjoint representation

± ’ (β ’ ±β±’1 )

ρ : GL(A) ’ GL EndF (A) ,

and denote the intersection of the subgroups Autalg (A) and AutGL(A) (σ) of GL(A)

by Aut(A, σ):

Aut(A, σ)(R) = { ± ∈ AutR (AR ) | ± —¦ σR = σR —¦ ± }.

A derivation x = ad(a) ∈ Der(A, A) = Lie Autalg (A) lies in Lie Aut(A, σ) if

and only if (1 + x · µ) —¦ σ = σ —¦ (1 + x · µ) if and only if x —¦ σ = σ —¦ x if and only if

a + σ(a) ∈ K. Hence

Lie Aut(A, σ) = { a ∈ A | a + σ(a) ∈ K }/K.

Denote the intersection of Aut(A, σ) and AutK (A) by AutK (A, σ). If an F -

algebra R satis¬es the SN -condition, then

AutK (A, σ)(R) = { a ∈ A— | a · σR (a) ∈ KR }/KR .

— —

R

The inverse image of AutK (A, σ) with respect to the surjection

Int : GL1 (A) ’ AutK (A)

(see ??) is denoted Sim(A, σ) and called the group scheme of similitudes of (A, σ).

Clearly,

Sim(A, σ)(R) = { a ∈ A— | a · σ(a) ∈ KR }

—

R

Lie Sim(A, σ) = { a ∈ A | a + σ(a) ∈ K }.

By Proposition (??) we have an exact sequence of group schemes

(23.3) 1 ’ GL1 (K) ’ Sim(A, σ) ’ AutK (A, σ) ’ 1.

Let E be the F -subalgebra of K consisting of all σ-invariant elements. We have a

group scheme homomorphism

µ : Sim(A, σ) ’ GL1 (E), a ’ a · σ(a).

346 VI. ALGEBRAIC GROUPS

The map µalg is clearly surjective. Hence, by Proposition (??), we have an exact

sequence

µ

(23.4) 1 ’ Iso(A, σ) ’ Sim(A, σ) ’ GL1 (E) ’ 1.

’

Unitary involutions. Let K/F be an ´tale quadratic extension, B be a cen-

e

tral simple algebra over K of degree n with a unitary F -involution „ . We use the

following notation (and de¬nitions) for group schemes over F :

U(B, „ ) = Iso(B, „ ) Unitary group

GU(B, „ ) = Sim(B, „ ) Group of unitary similitudes

PGU(B, „ ) = AutK (B, „ ) Projective unitary group

Assume ¬rst that K is split, K F — F . Then B A — Aop and „ is the exchange

involution. Let b = (a1 , aop ) ∈ B. The condition b·„ b = 1 is equivalent to a1 a2 = 1.

2

Hence we have an isomorphism

∼

a ’ a, (a’1 )op .

GL1 (A) ’ U(B, „ ),

’

The homomorphism

φ ’ (φ, φop )

Autalg (A) ’ PGU(B, „ ),

is clearly an isomorphism. Hence,

PGU(B, „ ) PGL1 (A).

Thus the group schemes U(B, „ ) and PGU(B, „ ) are smooth and connected. This

also holds when K is not split, as one sees by scalar extension. Furthermore the sur-

jection Aut(B, „ ) ’ Autalg (K) Z/2Z induces an isomorphism π0 Aut(B, „ )

Z/2Z. Hence PGU(B, „ ) is the connected component of Aut(B, „ ) and is as a sub-

group of index 2.

The kernel of the reduced norm homomorphism Nrd : U(B, „ ) ’ GL 1 (K) is

denoted SU(B, „ ) and called the special unitary group. Clearly,

SU(B, „ )(R) = { b ∈ (B —F R)— | b · „R (b) = 1, NrdR (b) = 1 },

Lie SU(B, „ ) = { x ∈ Skew(B, „ ) | Trd(x) = 0 }.

The group scheme SU(B, „ ) is smooth and connected since, when K is split,

SU(B, „ ) = SL1 (A) (as the description given above shows). The kernel N of

the composition

f : SU(B, „ ) ’ U(B, „ ) ’ PGU(B, „ )

satis¬es

N (R) = { B ∈ (K —F R)— | b · „R (b) = 1, bn = 1 }.

In other words,

NK/F

N = ker RK/F (µn,K ) ’ ’ ’ µn,F ,

’’

hence N is a ¬nite group scheme of multiplicative type and is Cartier dual to Z/nZ

where the Galois group “ acts through Gal(K/F ) as x ’ ’x. Subgroups of N

correspond to (cyclic) subgroups of Z/nZ, which are automatically “-invariant.

Since falg is surjective, f is surjective by Proposition (??). Clearly, f is a

central isogeny and

PGU(B, „ ) SU(B, „ )/N.

§23. AUTOMORPHISM GROUPS OF ALGEBRAS 347

Symplectic involutions. Let A be a central simple algebra of degree n =

2m over F with a symplectic involution σ. We use the following notation (and

de¬nitions):

Sp(A, σ) = Iso(A, σ) Symplectic group

GSp(A, σ) = Sim(A, σ) Group of symplectic similitudes

PGSp(A, σ) = Aut(A, σ) Projective symplectic group

Assume ¬rst that A is split, A = EndF (V ), hence σ = σh where h is a nonsingular

alternating bilinear form on V . Then Sp(A, σ) = Sp(V, h), the symplectic group

of (V, h),

Sp(V, h)(R) = { ± ∈ GL(VR ) | hR ±(v), ±(v ) = hR (v, v ) for v, v ∈ VR }.

The associated classical algebraic group is connected of dimension m(2m + 1)

(Borel [?, 23.3]).

Coming back to the general case, we have

dim Lie Sp(A, σ) = dim Skew(A, σ) = m(2m + 1) = dim Sp(A, σ),

hence Sp(A, σ) is a smooth and connected group. It follows from the exactness of

µ

1 ’ Sp(A, σ) ’ GSp(A, σ) ’ Gm ’ 1

’

(see ??) and Corollary (??) that GSp(A, σ) is smooth.

The exactness of

1 ’ Gm ’ GSp(A, σ) ’ PGSp(A, σ) ’ 1

(see ??) implies that PGSp(A, σ) is smooth. Consider the composition

f : Sp(A, σ) ’ GSp(A, σ) ’ PGSp(A, σ)

whose kernel is µ2 . Clearly, falg is surjective, hence f is surjective and PGSp(A, σ)

is connected. Therefore, f is a central isogeny and PGSp(A, σ) Sp(A, σ)/µ2 .

In the split case the group PGSp(V, h) = PGSp(A, σ) is called the projective

symplectic group of (V, h).

Orthogonal involutions. Let A be a central simple algebra of degree n over F

with an orthogonal involution σ. We use the following notation

O(A, σ) = Iso(A, σ)

GO(A, σ) = Sim(A, σ)

PGO(A, σ) = Aut(A, σ)

Consider the split case A = EndF (V ), σ = σb , where b is a nonsingular symmetric

(non-alternating, if char F = 2) bilinear form. Then

O(A, σ)(R) = { ± ∈ GL(VR ) | b(±v, ±v ) = b(v, v ) for v, v ∈ VR }.

n(n’1)

The associated classical algebraic group has dimension (Borel [?]). On the

2

other hand,

n(n’1)

if char F = 2,

2

dim Lie O(A, σ) = dim Skew(A, σ) = n(n+1)

if char F = 2.

2

Hence O(A, σ) (and the other groups) are not smooth if char F = 2. To get smooth

groups also in characteristic 2 we use a di¬erent context, described in the next two

subsections.

348 VI. ALGEBRAIC GROUPS

Orthogonal groups. Let (V, q) be a quadratic form of dimension n over F

and let bq be the polar bilinear form of q on V . We recall that the form q is regular

if bq is a nonsingular bilinear form except for the case n is odd and char F = 2.

In this case bq is symplectic and is degenerate. The radical of q is the space V ⊥

and (in case charF = 2 and dimF V is odd) q is regular if dim rad(bq ) = 1, say

rad(q) = F · v, q(v) = 0.

We view q as an element of S 2 (V — ), the space of degree 2 elements in the

symmetric algebra S 2 (V — ). There is a natural representation

ρ : GL(V ) ’ GL S 2 (V — ) .

We set O(V, q) for the group AutGL(V ) (q) and call it the orthogonal group of (V, q):

O(V, q)(R) = { ± ∈ GL(VR ) | qR (±v) = qR (v) for v ∈ VR }.

The associated classical algebraic group has dimension n(n’1) (Borel [?, 23.6]). For

2

± ∈ End(V ), we have 1 + ± · µ ∈ O(V, q) if and only if bq (v, ±v) = 0 for all v ∈ V .

Hence

Lie O(V, q) = { ± ∈ End(V ) | bq (v, ±v) = 0 for v ∈ V } = o(V, q).

The dimensions are:

n(n’1)

if n is even or char F = 2,

2

dim Lie O(V, q) = n(n’1)

+ 1 if n is odd and char F = 2.

2

Hence, in the ¬rst case O(V, q) is a smooth group scheme. We consider now the

following cases:

(a) char F = 2 and n is even: we de¬ne

∆

O+ (V, q) = ker O(V, q) ’ Z/2Z

’

where ∆ is the Dickson invariant, i.e., ∆(±) = 0 for ± ∈ O(V, q)(R) if

± induces the identity automorphism of the center of the Cli¬ord algebra

and ∆(±) = 1 if not (see (??)). The associated classical algebraic group

is known to be connected (Borel [?, 23.6]). Hence, O+ (V, q) is a smooth

connected group scheme.

(b) char F = 2 or n is odd: we set

det

O+ (V, q) = ker O(V, q) ’’ Gm

’

where det is the determinant map. Here also the associated classical alge-

braic group is known to be connected (Borel [?]).

We get in each case

Lie O+ (V, q) = { ± ∈ End(V ) | tr(±) = 0, bq (v, ±v) = 0 for v ∈ V }.

If char F = 2 this Lie algebra coincides with Lie O(V, q) and O+ (V, q) is the

connected component of O(V, q). If char F = 2, then Lie O+ (V, q) Lie O(V, q)

hence

n(n ’ 1)

dim Lie O+ (V, q) =

2

and O (V, q) is a smooth connected group scheme. Thus in every case O+ (V, q) is

+

a connected algebraic group. Consider the conjugation homomorphism

GL1 C0 (V, q) ’ GL C(V, q) , x ’ Int(x)

§23. AUTOMORPHISM GROUPS OF ALGEBRAS 349

where C(V, q) = C0 (V, q) • C1 (V, q) is the Cli¬ord algebra. The inverse image of

the normalizer NV of the subspace V ‚ C(V, q) is “+ (V, q), the even Cli¬ord group

of (V, q),

“+ (V, q)(R) = { g ∈ C0 (V, q)— | q · VR · g ’1 = VR }.

R

It follows from Example (??.??) that

Lie “+ (V, q) = { x ∈ C0 (V, q) | [x, V ] ‚ V } = V · V ‚ C0 (V, q).

Let

χ : “+ (V, q) ’ O+ (V, q), x ’ Int(x)|V .

Clearly, ker χ = Gm ‚ “+ (V, q). Since χalg is surjective, we have by Proposi-

tion (??) an exact sequence

χ

1 ’ Gm ’ “+ (V, q) ’ O+ (V, q) ’ 1.

’

Hence by Corollary (??) “+ (V, q) is smooth.

The kernel of the spinor norm homomorphism

Sn : “+ (V, q) ’ Gm , x ’ x · σ(x)

is the spinor group of (V, q) and is denoted Spin(V, q). Thus,

Spin(V, q)(R) = { g ∈ C0 (V, q)— | g · VR · g ’1 = VR , g · σ(g) = 1 }

R

The di¬erential d(Sn) is given by

d(Sn)(uv) = uv + σ(uv) = uv + vu = bq (u, v).

In particular, Sn is separable and

Lie Spin(V, q) = { x ∈ V · V ‚ C0 (V, q) | x + σ(x) = 0 }.

Since Snalg is surjective, we have by Proposition (??) an exact sequence

Sn

1 ’ Spin(V, q) ’ “+ (V, q) ’ Gm ’ 1.

’

Hence by Proposition (??) Spin(V, q) is smooth. The classical algebraic group asso-

ciated to Spin(V, q) is known to be connected (Borel [?, 23.3]), therefore Spin(V, q)

is connected.

The kernel of the composition

χ

f : Spin(V, q) ’ “+ (V, q) ’ O+ (V, q)

’

is µ2 . Since falg is surjective, it follows by Proposition (??) that f is surjective.

Hence, f is a central isogeny and

O+ (V, q) Spin(V, q)/µ2 .

(23.5) Remark. The preceding discussion focuses on orthogonal groups of quad-

ratic spaces. Orthogonal groups of symmetric bilinear spaces may be de¬ned in a

similar fashion: every nonsingular symmetric nonalternating bilinear form b on a

vector space V may be viewed as an element of S 2 (V )— , and letting GL(V ) act on

S 2 (V )— we may set O(V, b) = AutGL(V ) (b).

If char F = 2 we may identify S 2 (V )— to S 2 (V — ) by mapping every symmetric

bilinear form b to its associated quadratic form qb de¬ned by qb (x) = b(x, x), hence

O(V, b) = O(V, qb ). If char F = 2 the group O(V, b) is not smooth, and if F is not

perfect there may be no associated smooth algebraic group, see Exercise ??.

350 VI. ALGEBRAIC GROUPS

Suppose F is perfect of characteristic 2. In that case, there is an associated

smooth algebraic group O(V, b)red . If dim V is odd, O(V, b)red turns out to be

isomorphic to the symplectic group of an alternating space of dimension dim V ’ 1,

see Exercise ??. If dim V is even, O(V, b)red contains a nontrivial solvable connected

normal subgroup, see Exercise ??; it is therefore not semisimple (see §?? for the

de¬nition of semisimple group).

23.B. Quadratic pairs. Let A be a central simple algebra of degree n = 2m

over F , and let (σ, f ) be a quadratic pair on A. Consider the homomorphism

Aut(A, σ) ’ GL Sym(A, σ)— , ± ’ (g ’ g —¦ ±).

The inverse image of the stabilizer Sf of f is denoted PGO(A, σ, f ) and is called

the projective orthogonal group:

PGO(A, σ, f )(R) = { ± ∈ Aut(A, σ) | fR —¦ ± = fR }.

If R satis¬es the SN -condition, then, setting (A, σ)+ = Sym(A, σ),

PGO(A, σ, f )(R) =

{ a ∈ A— | a · σR (a) ∈ R— , f (axa’1 ) = f (x) for x ∈ (AR , σR )+ }/R— .

R

In the split case A = End(V ), with q a quadratic form corresponding to the

quadratic pair (σ, f ), we write PGO(V, q) for this group. The inverse image of

PGO(A, σ, f ) under

Int : GL1 (A) ’ Autalg (A)

is the group of orthogonal similitudes and is denoted GO(A, σ, f ):

GO(A, σ, f )(R) =

{ a ∈ A— | a · σR (a) ∈ R— , f (axa’1 ) = f (x) for x ∈ (AR , σR )+ }.

R

One sees that 1 + a · µ ∈ GO(A, σ, f )(F [µ]) if and only if a + σ(a) ∈ F and

f (ax ’ xa) = 0 for all symmetric x. Thus

Lie GO(A, σ, f ) = { a ∈ A | a + σ(a) ∈ F , f (ax ’ xa) = 0 for x ∈ Sym(A, σ) }.

An analogous computation shows that

Lie PGO(A, σ, f ) = Lie GO(A, σ, f ) /F.

The kernel of the homomorphism

µ : GO(A, σ, f ) ’ Gm , a ’ a · σ(a)

is denoted O(A, σ, f ) and is called the orthogonal group,

O(A, σ, f )(R) = { a ∈ A— | a · σ(a) = 1, f (axa’1 ) = f (x) for x ∈ Sym(A, σ)R }.

R

Since for a ∈ A with a + σ(a) = 0 one has f (ax ’ xa) = f ax + σ(ax) = Trd(ax),

it follows that the condition f (ax ’ xa) = 0 for all x ∈ Sym(A, σ) is equivalent

to a ∈ Alt(A, σ). Thus

Lie O(A, σ, f ) = Alt(A, σ)

(and does not depend on f !).

In the split case we have O(A, σ, f ) = O(V, q), hence by ??, O(A, σ, f ) is

smooth.

The sequence

µ

1 ’ O(A, σ, f ) ’ GO(A, σ, f ) ’ Gm ’ 1

’

§23. AUTOMORPHISM GROUPS OF ALGEBRAS 351

is exact by Proposition (??), since µalg is surjective. It follows from Corollary (??)

that GO(A, σ, f ) is smooth. By Proposition (??), the natural homomorphism

GO(A, σ, f ) ’ PGO(A, σ, f ) is surjective, hence PGO(A, σ, f ) is smooth. There

is an exact sequence

1 ’ Gm ’ GO(A, σ, f ) ’ PGO(A, σ, f ) ’ 1.

The kernel of the composition

g : O(A, σ, f ) ’ GO(A, σ, f ) ’ PGO(A, σ, f )

is µ2 . Clearly, galg is surjective, hence g is surjective. Therefore, g is a central

isogeny and

PGO(A, σ, f ) O(A, σ, f )/µ2 .

Now comes into play the Cli¬ord algebra C(A, σ, f ). By composing the natural

homomorphism

PGO(A, σ, f ) ’ Autalg C(A, σ, f )

with the restriction map

Autalg C(A, σ, f ) ’ Autalg (Z) = Z/2Z

where Z is the center of C(A, σ, f ), we obtain a homomorphism PGO(A, σ, f ) ’

Z/2Z, the kernel of which we denote PGO+ (A, σ, f ). The inverse image of this

group in GO(A, σ, f ) is denoted GO+ (A, σ, f ) and the intersection of GO+ (A, σ, f )

with O(A, σ, f ) by O+ (A, σ, f ). In the split case O+ (A, σ, f ) = O+ (V, q), hence

O+ (A, σ, f ) is smooth and connected. In particular it is the connected component

of O(A, σ, f ). It follows from the exactness of

1 ’ µ2 ’ O+ (A, σ, f ) ’ PGO+ (A, σ, f ) ’ 1

that PGO+ (A, σ, f ) is also a connected algebraic group, namely the connected

component of PGO(A, σ, f ).

Let B(A, σ, f ) be the Cli¬ord bimodule. Consider the representation

c ’ x ’ (c — x · c’1 ) .

GL1 C(A, σ, f ) ’ GL B(A, σ, f ) ,

Let b : A ’ B(A, σ, f ) be the canonical map. Let “(A, σ, f ) be the inverse image

of the normalizer Nb(A) of the subspace b(A) ‚ B(A, σ, f ) and call it the Cli¬ord

group of (A, σ, f ),

“(A, σ, f )(R) = { c ∈ C(A, σ, f )— | c — b(A)R · c’1 = b(A)R }.

R

In the split case “(A, σ, f ) = “+ (V, q) is a smooth group and

Lie “(A, σ, f ) = V · V = c(A) ‚ C(A, σ, f ).

Hence “(A, σ, f ) is a smooth algebraic group and

Lie “(A, σ, f ) = c(A).

For any g ∈ “(A, σ, f )(R) one has g · σ(g) ∈ R— , hence there is a spinor norm

homomorphism

Sn : “(A, σ, f ) ’ Gm , g ’ g · σ(g).

We denote the kernel of Sn by Spin(A, σ, f ) and call it the spinor group of (A, σ, f ).

It follows from the split case (where Spin(A, σ, f ) = Spin(V, q)) that Spin(A, σ, f )

is a connected algebraic group.

352 VI. ALGEBRAIC GROUPS

Let χ : “(A, σ, f ) ’ O+ (A, σ, f ) be the homomorphism de¬ned by the formula

c’1 — (1)b · c = χ(c) · b, and let g be the composition

χ

Spin(A, σ, f ) ’ “(A, σ, f ) ’ O+ (A, σ, f ).

’

Clearly, ker g = µ2 and, since galg is surjective, g is surjective, hence g is a central

isogeny and

O+ (A, σ, f ) Spin(A, σ, f )/µ2 .

Consider the natural homomorphism

C : PGO+ (A, σ, f ) ’ AutZ C(A, σ, f ), σ .

If n = deg A with n > 2, then c(A)R generates the R-algebra C(A, σ, f )R for any

R ∈ Alg F . Hence CR is injective and C is a closed embedding by Proposition (??).

By (??), there is an exact sequence

Int

1 ’ GL1 (Z) ’ Sim C(A, σ, f ), σ ’’ AutZ C(A, σ, f ), σ ’ 1.

’

Let „¦(A, σ, f ) be the group Int’1 (im C), which we call the extended Cli¬ord group.

Note that “(A, σ, f ) ‚ „¦(A, σ, f ) ‚ Sim C(A, σ, f ), σ . By Proposition (??) we

have a commutative diagram with exact rows:

O+ (A, σ, f )

1 ’’’

’’ Gm ’ ’ ’ “(A, σ, f ) ’ ’ ’

’’ ’’ ’’’ 1

’’

¦ ¦ ¦

¦ ¦ ¦

1 ’ ’ ’ GL1 (Z) ’ ’ ’ „¦(A, σ, f ) ’ ’ ’ PGO+ (A, σ, f ) ’ ’ ’ 1.

’’ ’’ ’’ ’’

The ¬rst two vertical maps are injective. By Corollary (??), the group „¦(A, σ, f )

is smooth.

(23.6) Remark. If char F = 2, the involution σ is orthogonal and f is prescribed.

We then have,

O(A, σ, f ) = O(A, σ)

GO(A, σ, f ) = GO(A, σ)

PGO(A, σ, f ) = PGO(A, σ).

§24. Root Systems

In this section we recall basic results from the theory of root systems and refer

to Bourbaki [?] for details. Let V be an R-vector space of positive ¬nite dimension.

An endomorphism s ∈ End(V ) is called a re¬‚ection with respect to ± ∈ V , ± = 0 if

(a) s(±) = ’±,

(b) there is a hyperplane W ‚ V such that s|W = Id.

We will use the natural pairing

V — — V ’ R, χ — v ’ χ, v = χ(v).

A re¬‚ection s with respect to ± is given by the formula s(v) = v ’ χ, v ± for a

uniquely determined linear form χ ∈ V — with χ|W = 0 and χ, ± = 2. Note that

a ¬nite set of vectors which spans V is preserved as a set by at most one re¬‚ection

with respect to any given ± (see Bourbaki [?, Chapter VI, § 1, Lemme 1]).

A ¬nite subset ¦ ‚ V = 0 is called a (reduced ) root system if

(a) 0 ∈ ¦ and ¦ spans V .

§24. ROOT SYSTEMS 353

(b) If ± ∈ ¦ and x± ∈ ¦ for x ∈ R, then x = ±1.

(c) For each ± ∈ ¦ there is a re¬‚ection s± with respect to ± such that s± (¦) =

¦.

(d) For each ±, β ∈ ¦, s± (β) ’ β is an integral multiple of ±.

The elements of ¦ are called roots. The re¬‚ection s± in (??) is uniquely determined

by ±. For ± ∈ ¦, we de¬ne ±— ∈ V — by

s± (v) = v ’ ±— , v · ±.

Such ±— are called coroots. The set ¦— = {±— ∈ V } forms the dual root system

in V — . Clearly, ±— , β ∈ Z for any ±, β ∈ ¦ and ±— , ± = 2.

An isomorphism of root systems (V, ¦) and (V , ¦ ) is an isomorphism of vector

spaces f : V ’ V such that f (¦) = ¦ . The automorphism group Aut(V, ¦) is a

¬nite group. The subgroup W (¦) of Aut(V, ¦) generated by all the re¬‚ections s± ,

± ∈ ¦, is called the Weyl group of ¦.

Let ¦i be a root system in Vi , i = 1, 2, . . . , n, and V = V1 • V2 • · · · • Vn ,

¦ = ¦1 ∪ ¦2 ∪ · · · ∪ ¦n . Then ¦ is a root system in V , called the sum of the ¦i .

We write ¦ = ¦1 + ¦2 + · · · + ¦n . A root system ¦ is called irreducible if ¦ is not

isomorphic to the sum ¦1 + ¦2 of some root systems. Any root system decomposes

uniquely into a sum of irreducible root systems.

Let ¦ be a root system in V . Denote by Λr the (additive) subgroup of V

generated by all roots ± ∈ ¦; Λr is a lattice in V , called the root lattice. The lattice

Λ = { v ∈ V | ±— , v ∈ Z for ± ∈ ¦ }

in V , dual to the root lattice generated by ¦— ‚ V — , is called the weight lattice.

Clearly, Λr ‚ Λ and Λ/Λr is a ¬nite group. The group Aut(V, ¦) acts naturally on

Λ, Λr , and Λ/Λr , and W (¦) acts trivially on Λ/Λr .

A subset Π ‚ ¦ of the root system ¦ is a system of simple roots or a base of a

root system if for any ± ∈ ¦,

±= nβ · β

β∈Π

for some uniquely determined nβ ∈ Z and either nβ ≥ 0 for all β ∈ Π or nβ ¤ 0 for

all β ∈ Π. In particular, Π is a basis of V . For a system of simple roots Π ‚ ¦ and

w ∈ W (¦) the subset w(Π) is also a system of simple roots. Every root system has

a base and the Weyl group W (¦) acts simply transitively on the set of bases of ¦.

Let ¦ be a root system in V and Π ‚ ¦ be a base. We de¬ne a graph, called

the Dynkin diagram of ¦, which has Π as its set of vertices. The vertices ± and

β are connected by ±— , β · β — , ± edges. If ±— , β > β — , ± , then all the edges

between ± and β are directed, with ± the origin and β the target. This graph does

not depend (up to isomorphism) on the choice of a base Π ‚ ¦, and is denoted

Dyn(¦). The group of automorphisms of Dyn(¦) embeds into Aut(V, ¦), and

Aut(V, ¦) is a semidirect product of W (¦) (a normal subgroup) and Aut Dyn(¦) .

In particular, Aut Dyn(¦) acts naturally on Λ/Λr .

Two root systems are isomorphic if and only if their Dynkin diagrams are iso-

morphic. A root system is irreducible if and only if its Dynkin diagram is connected.

The Dynkin diagram of a sum ¦1 + · · · + ¦n is the disjoint union of the Dyn(¦i ).

Let Π ‚ ¦ be a system of simple roots. The set

Λ+ = { χ ∈ Λ | ±— , χ ≥ 0 for ± ∈ Π }

354 VI. ALGEBRAIC GROUPS

is the cone of dominant weights in Λ (relative to Π). We introduce a partial ordering

on Λ: χ > χ if χ ’ χ is sum of simple roots. For any » ∈ Λ/Λr there exists a

unique minimal dominant weight χ(») ∈ Λ+ in the coset ». Clearly, χ(0) = 0.

24.A. Classi¬cation of irreducible root systems. There are four in¬nite

families An , Bn , Cn , Dn and ¬ve exceptional irreducible root systems E6 , E7 , E8 ,

F4 , G2 . We refer to Bourbaki [?] for the following datas about root systems.

Type An , n ≥ 1. Let V = Rn+1 /(e1 + e2 + · · · + en+1 )R where {e1 , . . . , en+1 }

is the canonical basis of Rn+1 . We denote by ei the class of ei in V .

Root system : ¦ = { ei ’ ej | i = j }, n(n + 1) roots.

Root lattice : Λr = { ai ei | ai = 0 }.

ei Z, Z/(n + 1)Z.

Weight lattice : Λ= Λ/Λr

Simple roots : Π = {e1 ’ e2 , e2 ’ e3 , . . . , en ’ en+1 }.

Dynkin diagram : c c ppp c

1 2 n

Aut Dyn(¦) : {1} if n = 1, {1, „ } if n ≥ 2.

Dominant weights : Λ+ = { ai · ei ∈ Λ | a1 ≥ a2 ≥ · · · ≥ an+1 }.

Minimal weights : e1 + e2 + · · · + ei , i = 1, 2, . . . , n + 1.

Type Bn , n ≥ 1. Let V = Rn with canonical basis {ei }.

: ¦ = { ±ei , ±ei ± ej | i > j }, 2n2 roots.

Root system

: Λ r = Zn .

Root lattice

: Λ = Λr + 1 (e1 + e2 + · · · + en )Z, Z/2Z.

Weight lattice Λ/Λr

2

Simple roots : Π = {e1 ’ e2 , e2 ’ e3 , . . . , en’1 ’ en , en }.

Dynkin diagram : c> c

c c ppp

1 2 n’1 n

Aut Dyn(¦) : {1}.

Dominant weights : Λ+ = { ai ei ∈ Λ | a1 ≥ a2 ≥ · · · ≥ an ≥ 0 }.

1

Minimal weights : 0, 2 (e1 + e2 + · · · + en ).

Type Cn , n ≥ 1. Let V = Rn with canonical basis {ei }.

: ¦ = { ±2ei , ±ei ± ej | i > j }, 2n2 roots.

Root system

ai ei | ai ∈ Z,

Root lattice : Λr = { ai ∈ 2Z }.

: Λ = Zn , Z/2Z.

Weight lattice Λ/Λr

Simple roots : Π = {e1 ’ e2 , e2 ’ e3 , . . . , en’1 ’ en , 2en }.

Dynkin diagram : c< c

c c ppp

1 2 n’1 n

Aut Dyn(¦) : {1}.

Dominant weights : Λ+ = { ai ei ∈ Λ | a1 ≥ a2 ≥ · · · ≥ an ≥ 0 }.

Minimal weights : 0, e1 .

§25. SPLIT SEMISIMPLE GROUPS 355

Type Dn , n ≥ 3. (For n = 2 the de¬nition works but yields A1 + A1 .) Let

V = Rn with canonical basis {ei }.

Root system : ¦ = { ±ei ± ej | i > j }, 2n(n ’ 1) roots.

ai ei | ai ∈ Z,

Root lattice : Λr = { ai ∈ 2Z }.

: Λ = Zn + 1 (e1 + e2 + · · · + en )Z,

Weight lattice 2

Z/2Z • Z/2Z if n is even,

Λ/Λr

Z/4Z if n is odd.

Simple roots : Π = {e1 ’ e2 , . . . , en’1 ’ en , en’1 + en }.

c n’1

c

Dynkin diagram : c c ppp

n’2 d

d cn

1 2

Aut Dyn(¦) : S3 if n = 4, {1, „ } if n = 3 or n > 4.

Dominant weights : Λ+ = { ai ei ∈ Λ | a1 ≥ a2 ≥ · · · ≥ an , an’1 + an ≥ 0 }.

1

Minimal weights : 0, e1 , 2 (e1 + e2 + · · · + en’1 ± en ).

Exceptional types.

Z/3Z.

E6 : Aut Dyn(¦) = {1, „ }, Λ/Λr

c c c c c

c

Z/2Z.

E7 : Aut Dyn(¦) = {1}, Λ/Λr

c c c c c c

c

E8 : Aut Dyn(¦) = {1}, Λ/Λr = 0.

c c c c c c c

c

F4 : Aut Dyn(¦) = {1}, Λ/Λr = 0. c> c

c c

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: