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(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

 
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:

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