xiv
Introduction
essentially given by a Dynkin diagram as in the case of Lie algebras; for example,
the group SL n (resp. SO 2n+1 Sp 2n , SO 2n ) is related to the Dynkin diagram A n−1
(resp. B n , C n , D n ). These are the classical groups; there are also exceptional groups
corresponding to the Dynkin diagrams E 6 , E 7 , E 8 , F 4 and G 2 .
The classification of real almost simple algebraic groups is due to Satake and
Tits; see §3.4 (and the appendix) of [168] for the complete list. The simplest case
is the classification of real algebraic groups G such that G × R C is isomorphic to
SL 2, R × R C. The list consists of SL 2,R and SL 1 (H) where H = R ⊕ Ri ⊕ Rj ⊕ Rij
is the Hamilton quaternion algebra and where SL 1 (H) is the special linear group of
H with respect to the quaternionic norm: for each R-algebra A, we have
SL 1 (H)(A) =
x + yi + j z + ik ∈ H ⊗ R A | x
2
+ y
2
+ z
2
+ t
2
= 1
.
Over our base field k, for each quaternion k-algebra Q 1 we have the k-group
SL 1 (Q) defined in a similar manner to SL 1 (H). We know that Q ∼ = Q if and only
if SL 1 (Q) ∼ = SL 1 (Q ). This provides an exhaustive list of isomorphism classes of
k-forms of SL 2 , that is of k-groups G such that G × k k ∼ = SL 2,k . In other words,
the classification of k-forms of SL 2 is the same as the classification of quaternion
k-algebras. For example, over R there are two isomorphism classes of quaternion
algebras, H and the matrix algebra M 2 (R), whose associated groups are the two
mentioned above. Over the field Q of rational numbers, there are infinitely many
isomorphism classes of forms of SL 2 .
Given two regular quadratic forms q, q of dimension 2n, it is known that the
algebraic groups SO(q) and SO(q ) are isomorphic if and only if q and q are
similar, that is, q = λq for λ ∈ k × [115, Th. 4.2 and 24.5]. It follows that
the classification of certain almost simple groups of type D n is equivalent to the
classification of quadratic forms (up to similarity). Over number fields, we have
a classification of quadratic forms (resp. of quadratic forms up to similarity) due
to Hasse (resp. Ono [141]; see also [54]). In small dimensions, we have a precise
classification of quadratic forms over a general field [106, §8], but this seems to be
out of reach in arbitrary dimensions. Thus, in general we are far from a precise list
of special orthogonal groups.
Let us focus on function fields of complex algebraic varieties. The first invariant
is their transcendence degree. If such a field F/C is of transcendence degree 1 (i.e.
the function field of an algebraic curve), Tsen’s theorem implies that all quaternion
F -algebras are isomorphic to M 2 (F ) [86, Th. 6.2.3, 6.2.8]. It follows that SL 2, F is
the only F -form of SL 2 . Also, F -quadratic forms are classified by their dimension
and their discriminant since all quaternionic norms are hyperbolic [155, Lemma
12.10]. In this case, forms of SO 2n are parameterized by quadratic (étale) extensions
of F . This is a general phenomenon for almost simple groups: Steinberg has shown
1 If char(k) = 2, the presentation of the k-algebra Q is Q = k ⊕ k i ⊕ k j ⊕ k ij with i 2 = a,
j 2 = b, ij + ji = 0 and a, b ∈ k × .
Introduction
essentially given by a Dynkin diagram as in the case of Lie algebras; for example,
the group SL n (resp. SO 2n+1 Sp 2n , SO 2n ) is related to the Dynkin diagram A n−1
(resp. B n , C n , D n ). These are the classical groups; there are also exceptional groups
corresponding to the Dynkin diagrams E 6 , E 7 , E 8 , F 4 and G 2 .
The classification of real almost simple algebraic groups is due to Satake and
Tits; see §3.4 (and the appendix) of [168] for the complete list. The simplest case
is the classification of real algebraic groups G such that G × R C is isomorphic to
SL 2, R × R C. The list consists of SL 2,R and SL 1 (H) where H = R ⊕ Ri ⊕ Rj ⊕ Rij
is the Hamilton quaternion algebra and where SL 1 (H) is the special linear group of
H with respect to the quaternionic norm: for each R-algebra A, we have
SL 1 (H)(A) =
x + yi + j z + ik ∈ H ⊗ R A | x
2
+ y
2
+ z
2
+ t
2
= 1
.
Over our base field k, for each quaternion k-algebra Q 1 we have the k-group
SL 1 (Q) defined in a similar manner to SL 1 (H). We know that Q ∼ = Q if and only
if SL 1 (Q) ∼ = SL 1 (Q ). This provides an exhaustive list of isomorphism classes of
k-forms of SL 2 , that is of k-groups G such that G × k k ∼ = SL 2,k . In other words,
the classification of k-forms of SL 2 is the same as the classification of quaternion
k-algebras. For example, over R there are two isomorphism classes of quaternion
algebras, H and the matrix algebra M 2 (R), whose associated groups are the two
mentioned above. Over the field Q of rational numbers, there are infinitely many
isomorphism classes of forms of SL 2 .
Given two regular quadratic forms q, q of dimension 2n, it is known that the
algebraic groups SO(q) and SO(q ) are isomorphic if and only if q and q are
similar, that is, q = λq for λ ∈ k × [115, Th. 4.2 and 24.5]. It follows that
the classification of certain almost simple groups of type D n is equivalent to the
classification of quadratic forms (up to similarity). Over number fields, we have
a classification of quadratic forms (resp. of quadratic forms up to similarity) due
to Hasse (resp. Ono [141]; see also [54]). In small dimensions, we have a precise
classification of quadratic forms over a general field [106, §8], but this seems to be
out of reach in arbitrary dimensions. Thus, in general we are far from a precise list
of special orthogonal groups.
Let us focus on function fields of complex algebraic varieties. The first invariant
is their transcendence degree. If such a field F/C is of transcendence degree 1 (i.e.
the function field of an algebraic curve), Tsen’s theorem implies that all quaternion
F -algebras are isomorphic to M 2 (F ) [86, Th. 6.2.3, 6.2.8]. It follows that SL 2, F is
the only F -form of SL 2 . Also, F -quadratic forms are classified by their dimension
and their discriminant since all quaternionic norms are hyperbolic [155, Lemma
12.10]. In this case, forms of SO 2n are parameterized by quadratic (étale) extensions
of F . This is a general phenomenon for almost simple groups: Steinberg has shown
1 If char(k) = 2, the presentation of the k-algebra Q is Q = k ⊕ k i ⊕ k j ⊕ k ij with i 2 = a,
j 2 = b, ij + ji = 0 and a, b ∈ k × .