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Spin(A, σ, f ) ’ GL1 C(A, σ, f ) ’ GL1 C ± (A, σ, f )
where C(A, σ, f ) = C + (A, σ, f ) — C ’ (A, σ, f ).
Classi¬cation of simple groups of type F4 . Consider the groupoid F4 =
F4 (F ) of exceptional Jordan algebras of dimension 27 over F (see §?? below if
char F = 2 and §?? if char F = 2), where morphisms are F -algebra isomorphisms.
Denote by F 4 = F 4 (F ) the groupoid of simple groups of type F4 over F , where
morphisms are group isomorphisms. By Theorem (??) we have a functor
S : F4 (F ) ’ F 4 (F ), S(J) = Autalg (J).
(26.18) Theorem. The functor S : F4 (F ) ’ F 4 (F ) is an equivalence of cate-
gories.
Proof : Let “ = Gal(Fsep /F ). The ¬eld extension functor j : F4 (F ) ’ F4 (Fsep )
is clearly a “-embedding. We ¬rst show that the functor j satis¬es the descent
condition. Let J be some object in F4 (F ) (a split one, for example). If char F = 2,
consider the F -vector space
W = HomF (J —F J, J),
the multiplication element w ∈ W and the natural representation
ρ : GL(J) ’ GL(W ).
By Proposition (??) the “-embedding
i : A(ρsep , w) ’ A(ρsep , w)
satis¬es the descent condition. We have the functor
T = T(F ) : A(ρsep , w) ’ F4 (F )
taking w ∈ A(ρsep , w) to the F -vector space J with the Jordan algebra structure
de¬ned by w . A morphism from w to w is an element of GL(J) and it de¬nes an
isomorphism of the corresponding Jordan algebra structures on J. The functor T
has an evident “-extension
T = T(Fsep ) : A(ρsep , w) ’ F4 (Fsep ),
which is clearly an equivalence of groupoids. Since the functor i satis¬es the descent
condition, so does the functor j.
For the proof of the theorem it su¬ces by Proposition (??) to show that for any
J ∈ F4 (F ) the functor S for a separably closed ¬eld F induces a group isomorphism
Autalg (J) ’ Aut Autalg (J) .
This follows from the fact that automorphisms of simple groups of type F4 are inner
(Theorem (??)).
If char F = 2, an exceptional Jordan algebra of dimension 27 is (see §?? below)
a datum (J, N, #, T, 1) consisting of a space J of dimension 27, a cubic form N : J ’
F , the adjoint # : J ’ J of N , which is a quadratic map, a bilinear trace form T ,
and a distinguished element 1, satisfying certain properties (given in §??). In this
case we consider the F -vector space
W = S 3 (J — ) • S 2 (J — ) — J • S 2 (J — ) • F
and complete the argument as in the preceding cases.
§26. SEMISIMPLE GROUPS OVER AN ARBITRARY FIELD 373


Classi¬cation of simple groups of type G2 . Consider the groupoid G2 =
G2 (F ) of Cayley algebras over F , where morphisms are F -algebra isomorphisms.
Denote by G 2 = G 2 (F ) the groupoid of simple groups of type G2 over F , where
morphisms are group isomorphisms. By Theorem (??) there is a functor
S : G2 (F ) ’ G 2 (F ), S(C) = Autalg (C).
(26.19) Theorem. The functor S : G2 (F ) ’ G 2 (F ) is an equivalence of cate-
gories.
Proof : Let “ = Gal(Fsep /F ). The ¬eld extension functor j : G2 (F ) ’ G2 (Fsep )
is clearly a “-embedding. We ¬rst show that the functor j satis¬es the descent
condition. Let C be some object in G2 (F ) (a split one, for example). Consider
the F -vector space W = HomF (C —F C, C), the multiplication element w ∈ W ,
and the natural representation ρ : GL(C) ’ GL(W ). By Proposition (??) the
“-embedding i : A(ρsep , w) ’ A(ρsep , w) satis¬es the descent condition. We have a
functor
T = T(F ) : A(ρsep , w) ’ G2 (F )
which takes w ∈ A(ρsep , w) to the F -vector space C with the Cayley algebra
structure de¬ned by w . A morphism from w to w is an element of GL(C) and it
de¬nes an isomorphism between the corresponding Cayley algebra structures on C.
The functor T has an evident “-extension
T = T(Fsep ) : A(ρsep , w) ’ G2 (Fsep ),
which is clearly an equivalence of groupoids. Since the functor i satis¬es the descent
condition, so does the functor j.
For the proof of the theorem it su¬ces by Proposition (??) to show that for any
J ∈ G2 (F ) the functor S for a separably closed ¬eld F induces a group isomorphism
Autalg (C) ’ Aut Autalg (C) .
This follows from the fact that automorphisms of simple groups of type G2 are
inner (Theorem (??)).
26.B. Algebraic groups of small dimension. Some Dynkin diagrams of
small ranks coincide:
(26.20) A1 = B 1 = C 1
(26.21) D2 = A 1 + A 1
(26.22) B2 = C 2
(26.23) A3 = D 3
We describe explicitly the corresponding isomorphisms for adjoint groups (ana-
logues for algebras are in §??):
A1 = B1 = C1 . Let (V, q) be a regular quadratic form of dimension 3 over a
¬eld F . Then C0 (V, q) is a quaternion algebra over F . The canonical homomor-
phism
O+ (V, q) ’ PGL1 C0 (V, q) = PGSp C0 (V, q), σ q
is injective (see §??) and hence is an isomorphism of adjoint simple groups of
types B1 , A1 and C1 since by dimension count its image has the same dimension
374 VI. ALGEBRAIC GROUPS


as the target group, and since these groups are connected they must coincide, by
Propositions (??) and (??). (We will use this argument several times below.)
Let Q be a quaternion algebra over F and let Q0 = { x ∈ Q | TrdQ (x) = 0 }.
For x ∈ Q0 , we have x2 ∈ F , and the squaring map s : Q0 ’ F is a quadratic form
of discriminant 1 on Q0 (see §??). Consider the conjugation homomorphism
f : GL1 (Q) ’ O+ (Q0 , s).
Since Q0 generates Q, ker(f ) = Gm and the injection
PGSp(Q, σ) = PGL1 (Q) ’ O+ (Q0 , s)
is an isomorphism of adjoint simple groups of types C1 , A1 , and B1 .
D2 = A1 + A1 . Let A be a central simple algebra over F of degree 4 with a
quadratic pair (σ, f ). Then C(A, σ, f ) is a quaternion algebra over a quadratic ´tale
e
extension Z of F . We have the canonical injection
PGO+ (A, σ, f ) ’ AutZ C(A, σ, f ) = RZ/F PGL1 C(A, σ, f )
which is an isomorphism between adjoint groups of type D2 and those of type
A1 + A 1 .
Conversely, let Q be a quaternion algebra over an ´tale quadratic extension
e
Z/F . The norm A = NZ/F (Q) is a central simple algebra of degree 4 over F with
a canonical quadratic pair (σ, f ) (see §??). We have the natural homomorphism
g : RZ/F GL1 (Q) ’ GO+ (A, σ, f ), x ∈ Q — ’ x — x ∈ A— .
R R

One checks that x — x ∈ R— if and only if x ∈ ZR , hence g ’1 (Gm ) = RZ/F (Gm,Z ).
By factoring out these subgroups we obtain an injective homomorphism
RZ/F PGL1 (Q) ’ PGO+ (A, σ, f )
which is actually an isomorphism from an adjoint group of type A1 + A1 to one of
type D2 .
B2 = C2 . Let (V, q) be a regular quadratic form of dimension 5. Then C0 (V, q)
is a central simple algebra of degree 4 with (canonical) symplectic involution „ .
There is a canonical injective homomorphism (see §??)
O+ (V, q) ’ PGSp C0 (V, q), „
which is in fact an isomorphism from an adjoint simple groups of type B2 to one
of type C2 .
Conversely, for a central simple algebra A of degree 4 over F with a symplectic
involution σ, the F -vector space
Symd(A, σ)0 = { x ∈ Symd(A, σ) | TrpA (x) = 0 }
admits the quadratic form sσ (x) = x2 ∈ F (see §??). Consider the conjugation
homomorphism
f : GSp(A, σ) ’ O+ Symd(A, σ)0 , sσ , a ’ Int(a).
Since Symd(A, σ) generates A, one has ker(f ) = Gm . Hence, the injection
PGSp(A, σ) ’ O+ V, q
is an isomorphism from an adjoint simple group of type C2 to one of type B2 .
§27. TITS ALGEBRAS OF SEMISIMPLE GROUPS 375


A3 = D3 . Let A be a central simple algebra of degree 6 over F with an orthog-
onal pair (σ, f ). Then C(A, σ, f ) is a central simple algebra of degree 4 over an ´tale
e
quadratic extension Z/F with a unitary involution σ. The natural homomorphism
PGO+ (A, σ, f ) ’ PGU C(A, σ, f ), σ
is injective (see §??) and hence is an isomorphism from an adjoint simple group of
type D3 to one of type A3 .
Conversely, let B be a central simple algebra of degree 4 over an ´tale quadratic
e
extension Z/F with a unitary involution „ . Then the discriminant algebra D(B, „ )
is a central simple algebra of degree 6 over F with canonical quadratic pair („ , f ).
Consider the natural homomorphism
GU(B, „ ) ’ GO+ D(B, „ ), „ , f .
One checks (in the split case) that g ’1 (Gm ) = GL1 (Z). By factoring out these
subgroups we obtain an injection
PGU(B, „ ) ’ PGO+ D(B, „ ), „ , f
which is an isomorphism from an adjoint simple group of type A3 to one of type D3 .

§27. Tits Algebras of Semisimple Groups
The Cli¬ord algebra, the discriminant algebra, the »-powers of a central simple
algebra all arise as to be so-called Tits algebras of the appropriate semisimple
groups. In this section we de¬ne Tits algebras and classify them for simple groups
of the classical series.
For this we need some results on the classi¬cation of representations of split
semisimple groups. Let G be a split semisimple group over F . Choose a split
maximal torus T ‚ G. Fix a system of simple roots in ¦(G), so we have the
corresponding cone Λ+ ‚ Λ of dominant weights.
Let ρ : G ’ GL(V ) be a representation. By the representation theory of diag-
onalizable groups (??) one can associate to the representation ρ|T a ¬nite number
of weights, elements of T — . If ρ is irreducible, among the weights there is a largest
(with respect to the ordering on Λ). It lies in Λ+ and is called the highest weight
of ρ (Humphreys [?]).
(27.1) Theorem. The map

Isomorphism classes of
T — © Λ+
←’
irreducible representations of G

taking the class of a representation ρ to its highest weight, is a bijection.
Reference: Tits [?, Th.2.5]

(27.2) Remark. If G is a simply connected group (i.e., T — = Λ), then T — © Λ+ =
Λ+ .
(27.3) Remark. The classi¬cation of irreducible representations of a split semi-
simple groups does not depend on the base ¬eld in the sense that an irreducible
representation remains irreducible over an arbitrary ¬eld extension and any irre-
ducible representation over an extension comes from the base ¬eld.
376 VI. ALGEBRAIC GROUPS


27.A. De¬nition of the Tits algebras. Let G be a semisimple (not neces-
sarily split) group over F and let T ‚ G be a maximal torus. Choose a system of

simple roots Π ‚ ¦ = ¦(G). The group “ acts on Tsep and is the identity on ¦, Λ,
Λr (but not Π).

There is another action of “ on Tsep , called the —-action, which is de¬ned as
follows. Take any γ ∈ “. Since the Weyl group W acts simply transitively on the
set of systems of simple roots and γΠ is clearly a system of simple roots, there is a
unique w ∈ W such that w(γΠ) = Π. We set γ — ± = w(γ±) ∈ Π for any ± ∈ Π.
This action, de¬ned on Π, extends to an action on Λ which is the identity on Π, ¦,
Λr , Λ+ . Note that since W acts trivially on Λ/Λr , the —-action on Λ/Λr coincides
with the usual one.
Choose a ¬nite Galois extension F ‚ L ‚ Fsep splitting T and hence G. The —-
action of “ then factors through Gal(L/F ). Let ρ : GL ’ GL(V ) be an irreducible
representation over L (so V is an L-vector space) with highest weight » ∈ Λ+ © T —
(see Theorem (??)). For any γ ∈ “ we can de¬ne the L-space γ V as V as an abelian
group and with the L-action x —¦ v = γ ’1 (x) · v, for all x ∈ L, v ∈ V . Then v ’ v
viewed as a map V ’ γ V is γ-semilinear. Denote it iγ .
Let A = F [G] and let ρ : V ’ V —L AL be the comodule structure for ρ (see
p. ??). The composite
i’1 iγ —(γ—Id)
ρ
γ
γV ’ ’ V ’ V —L (L —F A) ’ ’ ’ ’ γ V —L (L —F A)
’ ’ ’ ’ ’’
gives the comodule structure for some irreducible representation

γρ: GL ’ GL(γ V ).
(Observe that the third map is well-de¬ned because both iγ and γ — Id are γ-
semilinear.) Clearly, the weights of γ ρ are obtained from the weights of ρ by ap-
plying γ. Hence, the highest weight of γ ρ is γ — ».
Assume now that » ∈ Λ+ © T — is invariant under the —-action. Consider the
conjugation representation
g ’ ± ’ ρ(g) —¦ ± —¦ ρ(g)’1 .
Int(ρ) : G ’ GL EndF (V ) ,
Let EndG (V ) be the subalgebra of G-invariant elements in EndF (V ) under Int(ρ).
Then
(27.4) EndG (V ) —F L EndGL (V —F L) EndGL γ∈Gal(L/F ) γ V

since V —F L is L-isomorphic to γ V via v — x ’ (γ ’1 x · v)γ . Since the represen-
tation γ ρ is of highest weight γ — » = », it follows from Theorem (??) that γ ρ ρ,
i.e., all the G-modules γ V are isomorphic to V . Hence, the algebras in (??) are
isomorphic to
EndGL (V n ) = Mn EndGL (V )
where n = [L : F ].
(27.5) Lemma. EndGL (V ) L.
Proof : Since ρ is irreducible, EndGL (V ) is a division algebra over L by Schur™s
lemma. But ρalg remains irreducible by Remark (??), hence EndGL (V ) —L Falg is
also a division algebra and therefore EndGL (V ) = L.
§27. TITS ALGEBRAS OF SEMISIMPLE GROUPS 377


It follows from the lemma that EndG (V ) —F L Mn (L), hence EndG (V ) is a
central simple algebra over F of degree n. Denote its centralizer in EndF (V ) by
A» . This is a central simple algebra over F of degree dimL V . It is clear that A»
is independent of the choice of L. The algebra A» is called the Tits algebra of the
group G corresponding to the dominant weight ».
Since the image of ρ commutes with EndG (V ), it actually lies in A» . Thus we
obtain a representation
ρ : G ’ GL1 (A» ).
By the double centralizer theorem (see (??)), the centralizer of EndG (V ) —F L
in EndF (V ) —F L is A» —F L. On the other hand it contains EndL (V ) (where the
image of ρ lies). By dimension count we have
A» —F L = EndL (V )
and hence the representation (ρ )L is isomorphic to ρ. Thus, ρ can be considered
as a descent of ρ from L to F . The restriction of ρ to the center C = C(G) ‚ G
is given by the restriction of » on C, i.e., is the character of the center C given by
the class of » in C — = T — /Λr ‚ Λ/Λr .
The following lemma shows the uniqueness of the descent ρ .
(27.6) Lemma. Let µi : G ’ GL1 (Ai ), i = 1, 2 be two homomorphisms where
the Ai are central simple algebras over F . Assume that the representations (µ i )sep
are isomorphic and irreducible. Then there is an F -algebra isomorphism ± : A 1 ’
A2 such that GL1 (±) —¦ µ1 = µ2 .
Proof : Choose a ¬nite Galois ¬eld extension L/F splitting G and the Ai , Ai —F L

EndL (Vi ). An L-isomorphism V1 ’ V2 of GL -representations gives rise to an

algebra isomorphism

± : EndF (V1 ) ’ EndF (V2 )

taking EndG (V1 ) to EndG (V2 ). Clearly, Ai lies in the centralizer of EndG (Vi ) in
EndF (Vi ). By dimension count Ai coincides with the centralizer, hence ±(A1 ) =
A2 .

Let π : G ’ G be a central isogeny with G simply connected. Then the Tits
algebra built out of a representation ρ of GL is the Tits algebra of the group GL
corresponding to the representation ρ —¦ πL . Hence, in order to classify Tits algebras
one can restrict to simply connected groups.
Assume that G is a simply connected semisimple group. For any » ∈ Λ/Λr
consider the corresponding (unique) minimal weight χ(») ∈ Λ+ . The uniqueness
shows that χ(γ») = γ — χ(») for any γ ∈ “. Hence, if » ∈ (Λ/Λr )“ , then clearly
χ(») ∈ Λ“ (with respect to the —-action); the Tits algebra Aχ(») is called a minimal
+
Tits algebra and is denoted simply by A» . For example, if » = 0, then A» = F .
(27.7) Theorem. The map
β : (Λ/Λr )“ ’ Br F, » ’ [A» ]
is a homomorphism.
Reference: Tits [?, Cor. 3.5].
378 VI. ALGEBRAIC GROUPS


If » ∈ Λ/Λr is not necessarily “-invariant, let

“0 = { γ ∈ “ | γ(») = » } ‚ “

and F» = (Fsep )“0 . Then » ∈ (Λ/Λr )“0 and one gets a Tits algebra A» , which is a
central simple algebra over F» , for the group GF» . The ¬eld F» is called the ¬eld
of de¬nition of ».

27.B. Simply connected classical groups. We give here the classi¬cation
of the minimal Tits algebras of the absolutely simple simply connected groups of
classical type.
Type An , n ≥ 1. Let ¬rst G = SL1 (A) where A is a central simple algebra of
degree n + 1 over F . Then C = µn+1 , C — = Z/(n + 1)Z with the trivial “-action.
For any i = 0, 1, . . . , n, consider the natural representation

ρi : G ’ GL1 (»i A).

In the split case ρi is the i-th exterior power representation with the highest weight
e1 + e2 + · · · + ei in the notation of §??, which is a minimal weight. Hence, the
»-powers »i A, for i = 0, 1, . . . , n, (see §??) are the minimal Tits algebras of G.
Now let G = SU(B, „ ) where B is a central simple algebra of degree n + 1 with
a unitary involution over a quadratic separable ¬eld extension K/F . The group
“ acts on C — = Z/(n + 1)Z by x ’ ’x through Gal(K/F ). The only nontrivial
element in (C — )“ is » = n+1 + (n + 1)Z (n should be odd). There is a natural
2
homomorphism

ρ : G ’ GL1 D(B, „ )

which in the split case is the external n+1 -power. Hence, the discriminant algebra
2
(see §??) D(B, „ ) is the minimal Tits algebra corresponding to » for the group G.
The ¬elds of de¬nition Fµ of the other nontrivial characters µ = i + (n + 1)Z ∈
C — , (i = (n+1) ), coincide with K. Hence, by extending the base ¬eld to K one sees
2
that Aµ »i B.
Type Bn , n ≥ 1. Let G = Spin(V, q), here (V, q) is a regular quadratic form
of dimension 2n + 1. Then C = µ2 , C — = Z/2Z = {0, »}. The embedding

G ’ GL1 C0 (V, q)

in the split case is the spinor representation with highest weight 1 (e1 +e2 +· · ·+en )
2
in the notation of §??, which is a minimal weight. Hence, the even Cli¬ord algebra
C0 (V, q) is the minimal Tits algebra A» .
Type Cn , n ≥ 1. Let G = Sp(A, σ) where A is a central simple algebra of
degree 2n with a symplectic involution σ. Then C = µ2 , C — = Z/2Z = {0, »}. The
embedding

G ’ GL1 (A)

in the split case is the representation with highest weight e1 in the notation of §??,
which is a minimal weight. Hence, A is the minimal Tits algebra A» .
§27. TITS ALGEBRAS OF SEMISIMPLE GROUPS 379


Type Dn , n ≥ 2, n = 4. Let G = Spin(A, σ, f ) where A is a central simple
algebra of degree 2n with a quadratic pair (σ, f ), C — = {0, », »+ , »’ } where »
factors through O+ (A, σ, f ). The composition
Spin(A, σ, f ) ’ GO+ (A, σ, f ) ’ GL1 (A)
in the split case is the representation with highest weight e1 in the notation of §??,
which is a minimal weight. Hence, A is the minimal Tits algebra A» .
Assume further that the discriminant of σ is trivial (i.e., the center Z of the
Cli¬ord algebra is split). The group “ then acts trivially on C — . The natural
compositions
Spin(A, σ, f ) ’ GL1 C(A, σ, f ) ’ GL1 C ± (A, σ, f )
in the split case are the representations with highest weights 1 (e1 + · · · + en’1 ± en )
2
which are minimal weights. Hence, C ± (A, σ, f ) are the minimal Tits algebras A»± .
If disc(σ) is not trivial then “ interchanges »+ and »’ , hence the ¬eld of
de¬nition of »± is Z. By extending the base ¬eld to Z one sees that A»± =
C(A, σ, f ). Again, the case of D4 is exceptional, because of triality, and we give on
p. ?? a description of the minimal Tits algebra in this case.
27.C. Quasisplit groups. A semisimple group G is called quasisplit if there
is a maximal torus T ‚ G and a system Π of simple roots in the root system ¦
of G with respect to T which is “-invariant with respect to the natural action, or

equivalently, if the —-action on Tsep coincides with the natural one. For example,
split groups are quasisplit.
Let G be a quasisplit semisimple group. The natural action of “ on the
system Π of simple roots, which is invariant under “, de¬nes an action of “ on
Dyn(G) = Dyn(¦) by automorphisms of the Dynkin diagram. Simply connected
and adjoint split groups are classi¬ed by their Dynkin diagrams. The following
statement generalizes this result for quasisplit groups.
(27.8) Proposition. Two quasisplit simply connected (resp. adjoint) semisimple
groups G and G are isomorphic if and only if there is a “-bijection between Dyn(G)
and Dyn(G ). For any Dynkin diagram D and any (continuous) “-action on D
there is a quasisplit simply connected (resp. adjoint) semisimple group G and a
“-bijection between Dyn(G) and D.

The “-action on Dyn(G) is trivial if and only if “ acts trivially on Tsep , hence T
and G are split. Therefore, if Aut(Dyn(G)) = 1 (i.e., Dyn(G) has only irreducible
components Bn , Cn , E7 , E8 , F4 , G2 ) and G is quasisplit, then G is actually split.
(27.9) Example. The case An , n > 1. A non-trivial action of the Galois group
“ on the cyclic group Aut(An ) of order two factors through the Galois group of
a quadratic ¬eld extension L/F . The corresponding quasisplit simply connected
simple group of type An is isomorphic to SU(V, h), where (V, h) is a non-degenerate
hermitian form over L/F of dimension n + 1 and maximal Witt index.
(27.10) Example. The case Dn , n > 1, n = 4. As in the previous example, to give
a nontrivial “-action on Dn is to give a quadratic Galois ¬eld extension L/F . The
corresponding quasisplit simply connected simple group of type Dn is isomorphic
to Spin(V, q), where (V, q) is a non-degenerate quadratic form of dimension 2n and
Witt index n ’ 1 with the discriminant quadratic extension L/F .
380 VI. ALGEBRAIC GROUPS


Exercises
1. If L is an ´tale F -algebra, then Autalg (L) is an ´tale group scheme correspond-
e e
ing to the ¬nite group AutFsep (Lsep ) with the natural Gal(Fsep /F )-action.
2. Let G be an algebraic group scheme. Prove that the following statements are
equivalent:
(a) G is ´tale,
e
0
(b) G = 1,
(c) G is smooth and ¬nite,
(d) Lie(G) = 0.
3. Prove that Hdiag is algebraic if and only if H is a ¬nitely generated abelian
group.
4. Let H be a ¬nitely generated abelian group, and let H ‚ H be the subgroup
of elements of order prime to char F . Prove that (Hdiag )0 (H/H )diag and
π0 (Hdiag ) Hdiag .

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