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B —F L = B —K K SB(B) —K SB(ι B) ,
hence BL is split. Therefore, the split case considered above shows that the F -
algebra
op
A = D(B, „ ) —F K, NrdB (u) —F D B, Int(u) —¦ „
F
is split by L. However, the kernel of the scalar extension map Br(F ) ’ Br(L) is
the image under the norm map of the kernel of the scalar extension map Br(K) ’
Br K SB(B) (see Merkurjev-Tignol [?, Corollary 2.12]). The latter is known
to be generated by the Brauer class of B (see for instance Merkurjev-Tignol [?,
Corollary 2.7]), and NK/F (B) splits since B has an involution of the second kind
(see (??)). Therefore, the map Br(F ) ’ Br(L) is injective, hence A is split.

§11. Trace Form Invariants
The invariants of involutions de¬ned in this section are symmetric bilinear forms
derived from the reduced trace. Let A be a central simple algebra over an arbitrary
¬eld F and let σ be an involution of any kind on A. Our basic object of study is
the form
T(A,σ) : A — A ’ F
de¬ned by
T(A,σ) (x, y) = TrdA σ(x)y for x, y ∈ A.
Since σ TrdA σ(y)x = TrdA σ(x)y , by (??) and (??), the form T(A,σ) is sym-
metric bilinear if σ is of the ¬rst kind and hermitian with respect to σ|F if σ is
of the second kind. It is nonsingular in each case, since the bilinear trace form
TA (x, y) = TrdA (xy) is nonsingular, as is easily seen after scalar extension to a
splitting ¬eld of A.
More generally, for any u ∈ Sym(A, σ) we set
T(A,σ,u) (x, y) = TrdA σ(x)uy for x, y ∈ A.
The form T(A,σ,u) also is symmetric bilinear if σ is of the ¬rst kind and hermitian
if σ is of the second kind, and it is nonsingular if and only if u is invertible.
How much information on σ can be derived from the form T(A,σ) is suggested
by the following proposition, which shows that T(A,σ) determines σ — σ if σ is of
the ¬rst kind. To formulate a more general statement, we denote by ι = σ|F the
restriction of σ to F , by ιA = { ιa | a ∈ A } the conjugate algebra of A (see §??)
and by ισ the involution on ιA de¬ned by
ι
σ(ιa) = ι σ(a) for a ∈ A.

(11.1) Proposition. Under the isomorphism σ— : A — ι A ’ EndF (A) such that

σ— (a — ι b)(x) = axσ(b),
the involution σ — ισ corresponds to the adjoint involution with respect to the form
T(A,σ) . More generally, for all u ∈ Sym(A, σ)©A— , the involution Int(u’1 )—¦σ — ισ
§11. TRACE FORM INVARIANTS 133


corresponds to the adjoint involution with respect to the form T(A,σ,u) under the
isomorphism σ— .
Proof : The proposition follows by a straightforward computation: for a, b, x, y ∈ A,
T(A,σ,u) σ— (a — b)(x), y = TrdA bσ(x)σ(a)uy
and
T(A,σ,u) x, σ— Int(u’1 ) —¦ σ(a) — σ(b) (y) = TrdA σ(x)u u’1 σ(a)u yb .
The equality of these expressions proves the proposition. (Note that the ¬rst part
(i.e., the case where u = 1) was already shown in (??)).
On the basis of this proposition, we de¬ne below the signature of an involution
σ as the square root of the signature of T(A,σ) . We also show how the form T(A,σ)
can be used to determine the discriminant of σ (if σ is of orthogonal type and
char F = 2) or the Brauer class of the discriminant algebra of (A, σ) (if σ is of the
second kind).

11.A. Involutions of the ¬rst kind. In this section, σ denotes an involution
+ ’
of the ¬rst kind on a central simple algebra A over a ¬eld F . We set Tσ and Tσ
for the restrictions of the bilinear trace form T(A,σ) to Sym(A, σ) and Skew(A, σ)
respectively; thus
+
Tσ (x, y) = TrdA σ(x)y = TrdA (xy) for x, y ∈ Sym(A, σ)

Tσ (x, y) = TrdA σ(x)y = ’ TrdA (xy) for x, y ∈ Skew(A, σ).
Also let TA denote the symmetric bilinear trace form on A:
TA (x, y) = TrdA (xy) for x, y ∈ A,
+ ’
so that Tσ (x, y) = TA (x, y) for x, y ∈ Sym(A, σ) and Tσ (x, y) = ’TA (x, y) for x,
y ∈ Skew(A, σ).
(11.2) Lemma. Alt(A, σ) is the orthogonal space of Sym(A, σ) in A for each of
the bilinear forms T(A,σ) and TA . Consequently,
+ ’
(1) if char F = 2, the form Tσ = Tσ is singular;
+ ’
(2) if char F = 2, the forms Tσ and Tσ are nonsingular and there are orthogonal
sum decompositions

+ ’
A, T(A,σ) = Sym(A, σ), Tσ • Skew(A, σ), Tσ ,

+ ’
A, TA = Sym(A, σ), Tσ • Skew(A, σ), ’Tσ .

Proof : For x ∈ A and y ∈ Sym(A, σ), we have TrdA σ(x)y = TrdA σ(yx) =
TrdA (xy), hence
TA x ’ σ(x), y = TrdA (xy) ’ TrdA σ(x)y = 0.
This shows Alt(A, σ) ‚ Sym(A, σ)⊥ (the orthogonal space for the form TA ); the
equality Alt(A, σ) = Sym(A, σ)⊥ follows by dimension count. Since
T(A,σ) x ’ σ(x), y = ’TA x ’ σ(x), y for x ∈ A, y ∈ Sym(A, σ),
the same arguments show that Alt(A, σ) is the orthogonal space of Sym(A, σ) for
the form T(A,σ) .
134 II. INVARIANTS OF INVOLUTIONS


(11.3) Examples. (1) Quaternion algebras. Let Q = (a, b)F be a quaternion al-
gebra with canonical involution γ over a ¬eld F of characteristic di¬erent from 2.
Let Q0 denote the vector space of pure quaternions, so that Q0 = Skew(Q, γ). A
direct computation shows that the elements i, j, k of the usual quaternion basis
+ ’
are orthogonal for T(Q,γ) , hence Tγ and Tγ have the following diagonalizations:
+ ’
Tγ = 2 and Tγ = 2 · ’a, ’b, ab .
Now, let σ = Int(i) —¦ γ. Then Skew(Q, σ) = i · F , and Sym(Q, σ) has 1, j, k as
orthogonal basis. Therefore,
+ ’
Tσ = 2 · 1, b, ’ab and Tσ = ’2a .
(2) Biquaternion algebras. Let A = (a1 , b1 )F —F (a2 , b2 )F be a tensor product of
two quaternion F -algebras and σ = γ1 — γ2 , the tensor product of the canonical
involutions. Let (1, i1 , j1 , k1 ) and (1, i2 , j2 , k2 ) denote the usual quaternion bases of
(a1 , b1 )F and (a2 , b2 )F respectively. The element 1 and the products ξ — · where
ξ and · independently range over i1 , j1 , k1 , and i2 , j2 , k2 , respectively, form an
+
orthogonal basis of Sym(A, σ) for the form Tσ . Similarly, i1 — 1, j1 — 1, k1 — 1,
1 — i2 , 1 — j2 , 1 — k2 form an orthogonal basis of Skew(A, σ). Therefore,
+
Tσ = 1 ⊥ a1 , b1 , ’a1 b1 · a2 , b2 , ’a2 b2
and

Tσ = ’a1 , ’b1 , a1 b1 , ’a2 , ’b2 , a2 b2 .
(Note Tσ is not an Albert form of A as discussed in §??, unless ’1 ∈ F —2 ).



As a further example, consider the split orthogonal case in characteristic dif-
ferent from 2. If b is a nonsingular symmetric bilinear form on a vector space V ,
we consider the forms bS2 and b§2 de¬ned on the symmetric square S 2 V and the
2
exterior square V respectively by
bS2 (x1 · x2 , y1 · y2 ) = b(x1 , y1 )b(x2 , y2 ) + b(x1 , y2 )b(x2 , y1 ),
b§2 (x1 § x2 , y1 § y2 ) = b(x1 , y1 )b(x2 , y2 ) ’ b(x1 , y2 )b(x2 , y1 )
for x1 , x2 , y1 , y2 ∈ V . (The form b§2 has already been considered in §??). Assuming
2
V in V — V by mapping x1 · x2 to 1 (x1 — x2 +
char F = 2, we embed S 2 V and 2
1
x2 — x1 ) and x1 § x2 to 2 (x1 — x2 ’ x2 — x1 ) for x1 , x2 ∈ V .
(11.4) Proposition. Suppose char F = 2 and let (A, σ) = EndF (V ), σb . The

standard identi¬cation •b : V — V ’ A of (??) induces isometries of bilinear

spaces

V — V, b — b ’ A, T(A,σ) ,


S 2 V, 1 bS2 ’ Sym(A, σ), Tσ ,
+

2
2 ∼ ’
1
V, 2 b§2 ’ Skew(A, σ), Tσ .

Proof : As observed in (??), we have σ •b (x1 — x2 ) = •b (x2 — x1 ) and
TrdA •b (x1 — x2 ) = b(x2 , x1 ) = b(x1 , x2 ) for x1 , x2 ∈ V .
Therefore,
T(A,σ) •b (x1 — x2 ), •b (y1 — y2 ) = b(x1 , y1 )b(x2 , y2 ) for x1 , x2 , y1 , y2 ∈ V ,
proving the ¬rst isometry. The other isometries follow by similar computations.
§11. TRACE FORM INVARIANTS 135


Diagonalizations of bS2 and b§2 are easily obtained from a diagonalization of b:
if b = ±1 , . . . , ±n , then

bS2 = n 2 ⊥ (⊥1¤i<j¤n ±i ±j ) and b§2 = ⊥1¤i<j¤n ±i ±j .

Therefore, det bS2 = 2n (det b)n’1 and det b§2 = (det b)n’1 .

(11.5) Proposition. Let (A, σ) be a central simple algebra with involution of or-
thogonal type over a ¬eld F of characteristic di¬erent from 2. If deg A is even,
then

det Tσ = det Tσ = 2deg A/2 det σ.
+ ’


Proof : By extending scalars to a splitting ¬eld L in which F is algebraically closed
(so that the induced map F — /F —2 ’ L— /L—2 is injective), we reduce to considering
the case where A is split. If (A, σ) = EndF (V ), σb , then det σ = det b by (??),
and the computations above, together with (??), show that

det Tσ = det Tσ = 2’n(n’1)/2 (det b)n’1 = 2n/2 det b in F — /F —2 ,
+ ’


where n = deg A.

As a ¬nal example, we compute the form T(A,σ,u) for a quaternion algebra with
orthogonal involution. This example is used in §?? (see (??)). In the following
statement, we denote by W F the Witt ring of nonsingular bilinear forms over F .

(11.6) Proposition. Let Q be a quaternion algebra over a ¬eld F of arbitrary
characteristic, let σ be an orthogonal involution on Q and v ∈ Sym(Q, σ) © Q — .
For all s ∈ Q— such that σ(s) = s = ’γ(s),

T(Q,σ,v) TrdQ (v) · NrdQ (vs), disc σ if TrdQ (v) = 0;
T(Q,σ,v) = NrdQ (vs), disc σ = 0 in W F if TrdQ (v) = 0.

Proof : Let γ be the canonical (symplectic) involution on Q and let u ∈ Skew(Q, γ)
F be such that σ = Int(u) —¦ γ. The discriminant disc σ is therefore represented in
F — /F —2 by ’ NrdQ (u) = u2 . Since σ(v) = v, we have v = uγ(v)u’1 , hence
TrdQ (vu) = 0. A computation shows that 1, u are orthogonal for the form T(Q,σ,v) .
Since further T(Q,σ,v) (1, 1) = TrdQ (v) and T(Q,σ,v) (u, u) = TrdQ (v) NrdQ (u), the
subspace spanned by 1, u is totally isotropic if TrdQ (v) = 0, hence T(Q,σ,v) is
metabolic in this case. If TrdQ (v) = 0, a direct calculation shows that for all
s ∈ Q— as above, 1, u, γ(v)s, γ(v)su is an orthogonal basis of Q which yields the
diagonalization

T(Q,σ,v) TrdQ (v) · 1, NrdQ (u), ’ NrdQ (vs), ’ NrdQ (vsu) .

To complete the proof, we observe that if TrdQ (v) = 0, then v and s both anti-
commute with u, hence vs ∈ F [u] and therefore NrdQ (vs) is a norm from F [u]; it
follows that

NrdQ (vs), u2 = 0 in W F.
136 II. INVARIANTS OF INVOLUTIONS


The signature of involutions of the ¬rst kind. Assume now that the base
¬eld F has an ordering P , so char(F ) = 0. (See Scharlau [?, §2.7] for background
information on ordered ¬elds.) To every nonsingular symmetric bilinear form b
there is classically associated an integer sgnP b called the signature of b at P (or
with respect to P ): it is the di¬erence m+ ’m’ where m+ (resp. m’ ) is the number
of positive (resp. negative) entries in any diagonalization of b.
Our goal is to de¬ne the signature of an involution in such a way that in the
split case A = EndF (V ), the signature of the adjoint involution with respect to a
symmetric bilinear form b is the absolute value of the signature of b:
sgnP σb = |sgnP b| .
(Note that σb = σ’b and sgnP (’b) = ’ sgnP b, so sgnP b is not an invariant of σb ).
(11.7) Proposition. For any involution σ of the ¬rst kind on A, the signature of
the bilinear form T(A,σ) at P is a square in Z. If A is split: A = EndF (V ) and
σ = σb is the adjoint involution with respect to some nonsingular bilinear form b
on V , then
(sgnP b)2 if σ is orthogonal,
sgnP T(A,σ) =
0 if σ is symplectic.

Proof : When A is split and σ is orthogonal, (??) yields an isometry T(A,σ) b — b
from which the formula for sgnP T(A,σ) follows. When A is split and σ is symplectic,
we may ¬nd an isomorphism (A, σ) EndF (V ), σb for some vector space V and
some nonsingular skew-symmetric form b on V . The same argument as in (??)
yields an isometry (A, T(A,σ) ) (V — V, b — b). In this case, b — b is hyperbolic.
Indeed, if U ‚ V is a maximal isotropic subspace for b, then dim U = 1 dim V and
2
1
U — V is an isotropic subspace of V — V of dimension 2 dim(V — V ). Therefore,
sgnP T(A,σ) = 0.
In the general case, consider a real closure FP of F for the ordering P . Since
the signature at P of a symmetric bilinear form over F does not change under scalar
extension to FP , we may assume F = FP . The Brauer group of F then has order 2,
the nontrivial element being represented by the quaternion algebra Q = (’1, ’1) F .
Since the case where A is split has already been considered, we may assume for the
rest of the proof that A is Brauer-equivalent to Q. According to (??), we then have
(A, σ) EndQ (V ), σh
for some (right) vector space V over Q, and some nonsingular form h on V , which
is hermitian with respect to the canonical involution γ on Q if σ is symplectic, and
skew-hermitian with respect to γ if σ is orthogonal.
Let (ei )1¤i¤n be an orthogonal basis of V with respect to h, and let
h(ei , ei ) = qi ∈ Q— for i = 1, . . . , n.
Thus qi ∈ F if σ is symplectic and qi is a pure quaternion if σ is orthogonal. For
i, j = 1, . . . , n, write Eij ∈ EndQ (V ) for the endomorphism which maps ej to ei
and maps ek to 0 if k = j. Thus Eij corresponds to the matrix unit eij under the
isomorphism EndQ (V ) Mn (Q) induced by the choice of the basis (ei )1¤i¤n .
A direct veri¬cation shows that for i, j = 1, . . . , n and q ∈ Q,
’1
σ(Eij q) = Eji qj γ(q)qi .
§11. TRACE FORM INVARIANTS 137


Therefore, for i, j, k, = 1, . . . , n and q, q ∈ Q,
0 if i = k or j = ,
T(A,σ) (Eij q, Ek q ) = ’1
TrdQ qj γ(q)qi q if i = k and j = .
We thus have an orthogonal decomposition of EndQ (V ) with respect to the form
T(A,σ) :
(11.8) EndQ (V ) = ⊥1¤i,j¤n Eij · Q.
Suppose ¬rst that σ is orthogonal, so h is skew-hermitian and qi is a pure quaternion
for i = 1, . . . , n. Fix a pair of indices (i, j). If qi qj is a pure quaternion, then
Eij and Eij qi span an isotropic subspace of Eij · Q, so Eij · Q is hyperbolic. If
qi qj is not pure, pick a nonzero pure quaternion h ∈ Q which anticommutes with
’1
qj qi qj . Since Q = (’1, ’1)F and F is real-closed, the square of every nonzero
pure quaternion lies in ’F —2 . For i = 1, . . . , n, let qi = ’±2 for some ±i ∈ F — ;
2
i
let also h2 = ’β 2 with β ∈ F . Then Eij (±j β + qj h) and Eij qi (±j β + qj h) span a
2-dimensional isotropic subspace of Eij · Q, so again Eij · Q is hyperbolic. We have
thus shown that the form T(A,σ) is hyperbolic on EndQ (V ) when σ is orthogonal,
hence sgnP T(A,σ) = 0 in this case.
If σ is symplectic, then qi ∈ F — for all i = 1, . . . , n, hence
’1 ’1
T(A,σ) (Eij q, Eij q) = TrdQ γ(q)q qj qi = 2 NrdQ (q)qj qi
for all i, j = 1, . . . , n. From (??) it follows that
T(A,σ) 2 · N Q · q 1 , . . . , qn · q 1 , . . . , qn
where NQ is the reduced norm form of Q. Since Q = (’1, ’1)F , we have NQ 41,
hence the preceding relation yields
2
sgnP T(A,σ) = 4 sgnP q1 , . . . , qn .


(11.9) Remark. In the last case, the signature of the F -quadratic form q1 , . . . , qn
is an invariant of the hermitian form h on V : indeed, the form h induces a quadratic
form hF on V , regarded as an F -vector space, by
hF (x) = h(x, x) ∈ F,
since h is hermitian. Then
hF 4 q 1 , . . . , qn ,
so sgnP hF = 4 sgnP q1 , . . . , qn . Let
sgnP h = sgnP q1 , . . . , qn .
The last step in the proof of (??) thus shows that if A = EndQ (V ) and σ = σh for
some hermitian form h on V (with respect to the canonical involution on Q), then
sgnP T(A,σ) = 4(sgnP h)2 .
(11.10) De¬nition. The signature at P of an involution σ of the ¬rst kind on A
is de¬ned by

sgnP σ = sgnP T(A,σ) .
138 II. INVARIANTS OF INVOLUTIONS


By (??), sgnP σ is an integer. Since sgnP T(A,σ) ¤ dim A and sgnP T(A,σ) ≡
dim T(A,σ) mod 2, we have
0 ¤ sgnP σ ¤ deg A and sgnP σ ≡ deg A mod 2.
From (??), we further derive:
(11.11) Corollary. Let FP be a real closure of F for the ordering P .
(1) Suppose A is not split by FP ;
(a) if σ is orthogonal, then sgnP σ = 0;
(b) if σ is symplectic and σ — IdFP = σh for some hermitian form h over the
quaternion division algebra over FP , then sgnP σ = 2 |sgnP h|.
(2) Suppose A is split by FP ;
(a) if σ is orthogonal and σ — IdFP = σb for some symmetric bilinear form b
over FP , then sgnP σ = |sgnP b|;
(b) if σ is symplectic, then sgnP σ = 0.
11.B. Involutions of the second kind. In this section we consider the case
of central simple algebras with involution of the second kind (B, „ ) over an arbitrary
¬eld F . Let K be the center of B and ι the nontrivial automorphism of K over F .
The form T(B,„ ) is hermitian with respect to ι. Let T„ be its restriction to the space
of symmetric elements. Thus,
T„ : Sym(B, „ ) — Sym(B, „ ) ’ F
is a symmetric bilinear form de¬ned by
T„ (x, y) = TrdB „ (x)y = TrdB (xy) for x, y ∈ Sym(B, „ ).
Since multiplication in B yields a canonical isomorphism of K-vector spaces
B = Sym(B, „ ) —F K,
the hermitian form T(B,„ ) can be recaptured from the bilinear form T„ :
T(B,„ ) xi ± i , y j βj = ι(±i )T„ (xi , yj )βj
i j i,j
for xi , yj ∈ Sym(B, „ ) and ±i , βj ∈ K. Therefore, the form T„ is nonsingular.
Moreover, there is no loss of information if we focus on the bilinear form T„ instead
of the hermitian form T(B,„ ).
(11.12) Examples. (1) Quaternion algebras. Suppose char F = 2 and let Q0 =
(a, b)F be a quaternion algebra over F , with canonical involution γ0 . De¬ne an
involution „ of the second kind on Q = Q0 —F K by „ = γ0 — ι. (According to (??),
every involution of the second kind on a quaternion K-algebra is of this type for a
suitable quaternion F -subalgebra). Let K F [X]/(X 2 ’ ±) and let z ∈ K satisfy
z 2 = ± (and ι(z) = ’z). If (1, i, j, k) is the usual quaternion basis of Q0 , the
elements 1, iz, jz, kz form an orthogonal basis of Sym(Q, „ ) with respect to T„ ,
hence
T„ = 2 · 1, a±, b±, ’ab± .
If char F = 2, Q0 = [a, b)F and K = F [X]/(X 2 + X + ±), let (1, i, j, k) be the
usual quaternion basis of Q0 and let z ∈ K be an element such that z 2 + z = ±
and ι(z) = z + 1. A computation shows that the elements z + i, 1 + z + i + j,
1 + z + i + kb’1 and 1 + z + i + j + kb’1 form an orthogonal basis of Sym(B, „ ) for
the form T„ , with respect to which T„ has the diagonalization
T„ = 1, 1, 1, 1 .
§11. TRACE FORM INVARIANTS 139


(2) Exchange involution. Suppose (B, „ ) = (E — E op , µ) where µ is the exchange
involution:
µ(x, y op ) = (y, xop ) for x, y ∈ E.
The space of symmetric elements is canonically isomorphic to E:
Sym(B, „ ) = { (x, xop ) | x ∈ E } = E
and since TrdB (x, y op ) = TrdE (x), TrdE (y) , the form T„ is canonically isometric
to the reduced trace bilinear form on E:
T„ (x, xop ), (y, y op ) = TrdE (xy) = TE (x, y) for x, y ∈ E.
As a further example, we consider the case of split algebras. Let V be a (¬nite
dimensional) K-vector space with a nonsingular hermitian form h. De¬ne a K-
vector space ι V by
ι
V = { ιv | v ∈ V }
with the operations
ι
v + ι w = ι (v + w) (ι v)± = ι vι(±) for v, w ∈ V , ± ∈ K.
(Compare with §?? and §??). The hermitian form h induces on the vector space
V —K ι V a nonsingular hermitian form h — ι h de¬ned by
(h — ι h)(v1 — ι v2 , w1 — ι w2 ) = h(v1 , w1 )ι h(v2 , w2 ) for v1 , v2 , w1 , w2 ∈ V .
Let s : V —K ι V ’ V —K ι V be the switch map
s(v1 — ι v2 ) = v2 — ι v1 for v1 , v2 ∈ V .
The norm of V is then de¬ned as the F -vector space of s-invariant elements
(see (??)):
NK/F (V ) = { x ∈ V —K ι V | s(x) = x }.
Since (h — ι h)(v2 — ι v1 , w2 — ι w1 ) = ι (h — ι h)(v1 — ι v2 , w1 — ι w2 ) , it follows that
(h — ι h) s(x), s(y) = ι (h — ι h)(x, y) for x, y ∈ V —K ι V .
Therefore, the restriction of the form h — ι h to the F -vector space NK/F (V ) is a
symmetric bilinear form
NK/F (h) : NK/F (V ) — NK/F (V ) ’ F.
The following proposition follows by straightforward computation:
(11.13) Proposition. Let z ∈ K F and let (ei )1¤i¤n be an orthogonal K-basis
of V . For i, j = 1, . . . , n, let Vi = (ei — ei ) · F ‚ V —K ι V and let
Vij = (ei — ιej + ej — ιei ) · F • ei z — ιej + ej ι(z) — ιei · F ‚ V —K ι V.
There is an orthogonal decomposition of NK/F (V ) for the bilinear form NK/F (h):
⊥ ⊥

NK/F (V ) = Vi • Vij .
1¤i¤n 1¤i<j¤n

Moreover, Vi 1 for all i. If char F = 2, then Vij is hyperbolic; if char F = 2,
then if K F [X]/(X 2 ’ ±) we have
Vij 2h(ei , ei )h(ej , ej ) · 1, ’± .
140 II. INVARIANTS OF INVOLUTIONS


Therefore, letting δi = h(ei , ei ) for i = 1, . . . , n,

n2 1 if char F = 2,
NK/F (h)
n 1 ⊥ 2 · 1, ’± · ⊥1¤i<j¤n δi δj if char F = 2.

Consider now the algebra B = EndK (V ) with the adjoint involution „ = σh
with respect to h.

(11.14) Proposition. The standard identi¬cation •h : V —K ι V ’ B of (??) is

an isometry of hermitian spaces

(V —F ι V, h — ι h) ’ (B, T(B,„ ) )

and induces an isometry of bilinear spaces

NK/F (V ), NK/F (h) ’ Sym(B, „ ), T„ .

Proof : For x = •h (v1 — ι v2 ) and y = •h (w1 — ι w2 ) ∈ B,
T(B,„ ) (x, y) = TrdB •h (v2 — ι v1 ) —¦ •h (w1 — ι w2 ) = h(v1 , w1 )ι h(v2 , w2 ) ,
hence
T(B,„ ) •h (ξ), •h (·) = (h — ι h)(ξ, ·) for ξ, · ∈ V — ι V .
Therefore, the standard identi¬cation is an isometry

•h : (V — ι V, h — ι h) ’ (B, T(B,„ ) ).

Since the involution „ corresponds to the switch map s, this isometry restricts to an
isometry between the F -subspaces of invariant elements under s on the one hand
and under „ on the other.

(11.15) Remark. For u ∈ Sym(B, „ )©B — , the form hu (x, y) = h u(x), y on V is
hermitian with respect to the involution „u = Int(u’1 ) —¦ „ . The same computation
as above shows that •h is an isometry of hermitian spaces

(V — ι V, hu — ι h) ’ (B, T(B,„,u) )

where (hu — ι h)(v1 — ι v2 , w1 — ι w2 ) = hu (v1 , w1 )ι h(v2 , w2 ) for v1 , v2 , w1 , w2 ∈ V .
In particular, since the Gram matrix of hu with respect to any basis of V is the
product of the Gram matrix of h by the matrix of u, it follows that det T(B,„,u) =
(det u)dim V det(h — ι h), hence
det T(B,„,u) = (det u)dim V · N (K/F ) ∈ F — /N (K/F ).
(11.16) Corollary. Let (B, „ ) be a central simple algebra of degree n with involu-
tion of the second kind over F . Let K be the center of B.
(1) The determinant of the bilinear form T„ is given by

1 · F —2 if char F = 2,
det T„ =
(’±)n(n’1)/2 · F —2 F [X]/(X 2 ’ ±).
if char F = 2 and K

(2) For u ∈ Sym(B, „ ) © B — , the determinant of the hermitian form T(B,„,u) is

det T(B,„,u) = NrdB (u)deg B · N (K/F ) ∈ F — /N (K/F ).
§11. TRACE FORM INVARIANTS 141


Proof : (??) As in (??), the idea is to extend scalars to a splitting ¬eld L of B in
which F is algebraically closed, and to conclude by (??). The existence of such a
splitting ¬eld has already been observed in (??): we may take for L the function
¬eld of the (Weil) transfer of the Severi-Brauer variety of B if K is a ¬eld, or the
function ¬eld of the Severi-Brauer variety of E if B E — E op .
(??) For the same splitting ¬eld L as above, the extension of scalars map
Br(F ) ’ Br(L) is injective, by Merkurjev-Tignol [?, Corollary 2.12] (see the proof
of (??)). Therefore, the quaternion algebra
K, det T(B,„,u) NrdB (u)deg B F
splits, since Remark (??) shows that it splits over L.
The same reduction to the split case may be used to relate the form T„ to the
discriminant algebra D(B, „ ), which is de¬ned when the degree of B is even. In
the next proposition, we assume char F = 2, so that the bilinear form T„ de¬nes a
nonsingular quadratic form
Q„ : Sym(B, „ ) ’ F
by
Q„ (x) = T„ (x, x) for x ∈ Sym(B, „ ).
(11.17) Proposition. Let (B, „ ) be a central simple algebra with involution of
the second kind over a ¬eld F of characteristic di¬erent from 2, and let K be
F [X]/(X 2 ’ ±). Assume that the degree of (B, „ ) is
the center of B, say K
even: deg(B, „ ) = n = 2m. Then the (full ) Cli¬ord algebra of the quadratic space
Sym(B, „ ), Q„ and the discriminant algebra D(B, „ ) are related as follows:
C Sym(B, „ ), Q„ ∼ D(B, „ ) —F ’±, 2m (’1)m(m’1)/2 ,
F
where ∼ is Brauer-equivalence.
Proof : Suppose ¬rst that K is a ¬eld. By extending scalars to the function ¬eld L
of the transfer of the Severi-Brauer variety of B, we reduce to considering the split
case. For, L splits B and the scalar extension map Br(F ) ’ Br(L) is injective, as
observed in (??).
We may thus assume that B is split: let B = EndK (V ) and „ = σh for some
nonsingular hermitian form h on V . If (ei )1¤i¤n is an orthogonal basis of V and
h(ei , ei ) = δi for i = 1, . . . , n, then (??) yields
D(B, „ ) ∼ ±, (’1)n(n’1)/2 d = ±, (’1)m d
(11.18) F F
where we have set d = δ1 . . . δn . On the other hand, (??) and (??) yield
Q„ n 1 ⊥ 2 · 1, ’± · q,
where q = ⊥1¤i<j¤n δi δj . From known formulas for the Cli¬ord algebra of a direct
sum (see for instance Lam [?, Chapter 5, §2]), it follows that
2(’1)m · 1, ’± · q .
(11.19) C(Q„ ) C n1 —F C
Let IF be the fundamental ideal of even-dimensional forms in the Witt ring W F
and let I n F = (IF )n . Let d ∈ F — be a representative of disc(q). Since n = 2m,
we have
d ≡ (’1)m(m’1)/2 d mod F —2 .
142 II. INVARIANTS OF INVOLUTIONS


hence
mod I 2 F
1, ’d if m is even,
q≡
mod I 2 F
d if m is odd.
Therefore, the form 2(’1)m · 1, ’± · q is congruent modulo I 3 F to
1, ’± · 1, ’d if m is even,
1, ’± · ’2d if m is odd.
Since quadratic forms which are congruent modulo I 3 F have Brauer-equivalent
Cli¬ord algebras (see Lam [?, Chapter 5, Cor. 3.4]) it follows that
(±, d )F if m is even,
2(’1)m · 1, ’± · q ∼
C
(±, ’2d )F if m is odd.
On the other hand,
—m(m’1)/2
C n1 ∼ (’1, ’1)F ,
hence the required equivalence follows from (??) and (??).
To complete the proof, consider the case where K F — F . Then, there is a
central simple F -algebra E of degree n = 2m such that (B, „ ) (E—E op , µ) where µ
is the exchange involution. As we observed in (??), we then have Sym(B, „ ), Q„
(E, QE ) where QE (x) = TrdE (x2 ) for x ∈ E. Moreover, D(B, „ ) »m E ∼ E —m .
Since ± ∈ F —2 and (’1, 2)F is split, the proposition reduces to
—m(m’1)/2
C(E, QE ) ∼ E —m —F (’1, ’1)F .
This formula has been proved by Saltman (unpublished), Serre [?, Annexe, p. 167],
Lewis-Morales [?] and Tignol [?].
Algebras of odd degree. When the degree of B is odd, no discriminant of
(B, „ ) is de¬ned. However, we may use the fact that B is split by a scalar extension
of odd degree, together with Springer™s theorem on the behavior of quadratic forms
under odd-degree extensions, to get some information on the form T„ . Since the
arguments rely on Springer™s theorem, we need to assume char F = 2 in this section.
We may therefore argue in terms of quadratic forms instead of symmetric bilinear
forms, associating to the bilinear form T„ the quadratic form Q„ (x) = T„ (x, x).
(11.20) Lemma. Suppose char F = 2. Let L/F be a ¬eld extension of odd degree
and let q be a quadratic form over F . Let qL be the quadratic form over L derived
from q by extending scalars to L, and let ± ∈ F — F —2 . If qL 1, ’± · h for
some quadratic form h over L, of determinant 1, then there is a quadratic form t
of determinant 1 over F such that
q 1, ’± · t.
√ √
Proof : Let K = F ( ±) and M = L · K = L( ±). Let qan be an anisotropic
form over F which is Witt-equivalent to q. The form (qan )M is Witt-equivalent
to the form 1, ’± · h M , hence it is hyperbolic. Since the ¬eld extension M/K
has odd degree, Springer™s theorem on the behavior of quadratic forms under ¬eld
extensions of odd degree (see Scharlau [?, Theorem 2.5.3]) shows that (qan )K is
hyperbolic, hence, by Scharlau [?, Remark 2.5.11],
qan = 1, ’± · t0
§11. TRACE FORM INVARIANTS 143


for some quadratic form t0 over F . Let dim q = 2d, so that dim h = d, and let w
be the Witt index of q, so that
(11.21) q wH ⊥ 1, ’± · t0 ,
where H is the hyperbolic plane. We then have dim t0 = d ’ w, hence
det q = (’1)w (’±)d’w · F —2 ∈ F — /F —2 .
On the other hand, the relation qL 1, ’± · h yields
det qL = (’±)d · L—2 ∈ L— /L—2 .
Therefore, ±w ∈ F — becomes a square in L; since the degree of L/F is odd, this
implies that ±w ∈ F —2 , hence w is even. Letting t1 = w H ⊥ t0 , we then derive
2
from (??):
q 1, ’± · t1 .
It remains to prove that we may modify t1 so as to satisfy the determinant condition.
Since dim t1 = d, we have
1, ’(’1)d(d’1)/2 det t1 mod I 2 F if d is even,
t1 ≡
(’1)d(d’1)/2 det t1 mod I 2 F if d is odd.
We may use these relations to compute the Cli¬ord algebra of q 1, ’± · t1 (up
to Brauer-equivalence): in each case we get the same quaternion algebra:
C(q) ∼ ±, (’1)d(d’1)/2 det t1 .
F
On the other hand, since det h = 1 we derive from qL 1, ’± · h:
C(qL ) ∼ ±, (’1)d(d’1)/2 .
L
It follows that the quaternion algebra (±, det t1 )F is split, since it splits over the
L/F of odd degree. Therefore, if δ ∈ F — is a representative of det t1 ∈
extension
F — /F —2 , we have δ ∈ N (K/F ). Let β ∈ F — be a represented value of t1 , so that
t1 t 2 ⊥ β for some quadratic form t2 over F , and let t = t2 ⊥ δβ . Then
det t = δ · det t1 = 1.
On the other hand, since δ is a norm from the extension K/F we have 1, ’± · δβ
1, ’± · β , hence
1, ’± · t 1, ’± · t1 q.


(11.22) Proposition. Let B be a central simple K-algebra of odd degree n =
2m ’ 1 with an involution „ of the second kind. Then, there is a quadratic form q „
of dimension n(n ’ 1)/2 and determinant 1 over F such that
Q„ n 1 ⊥ 2 · 1, ’± · q„ .
Proof : Suppose ¬rst K = F — F . We may then assume (B, „ ) = (E — E op , µ) where
QE where QE (x) = TrdE (x2 ), as
µ is the exchange involution. In that case Q„
observed in (??). Since ± ∈ F —2 , we have to show that this quadratic form is Witt-
equivalent to n 1 . By Springer™s theorem, it su¬ces to prove this relation over an
odd-degree ¬eld extension. Since the degree of E is odd, we may therefore assume
E is split: E = Mn (F ). In that case, the relation is easy to check. (Observe that
the upper-triangular matrices with zero diagonal form a totally isotropic subspace).
144 II. INVARIANTS OF INVOLUTIONS


For the rest of the proof, we may thus assume K is a ¬eld. Let D be a division
K-algebra Brauer-equivalent to B and let θ be an involution of the second kind
on D. Let L be a ¬eld contained in Sym(D, θ) and maximal for this property. The
¬eld M = L·K is then a maximal sub¬eld of D, since otherwise the centralizer CD M
contains a symmetric element outside M , contradicting the maximality of L. We
have [L : F ] = [M : K] = deg D, hence the degree of L/F is odd, since D is Brauer-
equivalent to the algebra B of odd degree. Moreover, the algebra B — F L = B —K M
splits, since M is a maximal sub¬eld of D. By (??) and (??) the quadratic form
(Q„ )L obtained from Q„ by scalar extension to L has the form
(11.23) (Q„ )L n 1 ⊥ 2 · 1, ’± · h
where h = ⊥1¤i<j¤n ai aj for some a1 , . . . , an ∈ L— . Therefore, the Witt index
of the form (Q„ )L ⊥ n ’1 is at least n:
w (Q„ )L ⊥ n ’1 ≥ n.
By Springer™s theorem the Witt index of a form does not change under an odd-
degree scalar extension. Therefore,
w Q„ ⊥ n ’1 ≥ n,
and it follows that Q„ contains a subform isometric to n 1 . Let
Q„ n 1 ⊥q
for some quadratic form q over F . Relation (??) shows that
(q)L 2 · 1, ’± · h.
Since det h = 1, we may apply (??) to the quadratic form 2 · q, obtaining a
quadratic form q„ over F , of determinant 1, such that 2 · q 1, ’± · q„ ; hence
Q„ n 1 ⊥ 2 · 1, ’± · q„ .


In the case where n = 3, we show in Chapter ?? that the form q„ classi¬es the
involutions „ on a given central simple algebra B.
The signature of involutions of the second kind. Suppose√ that P is an
ordering of F which does not extend to K; this means that K = F ( ±) for some
± < 0. If (V, h) is a hermitian space over K (with respect to ι), the signature
sgnP h may be de¬ned just as in the case of quadratic spaces (see Scharlau [?,
Examples 10.1.6]). Indeed, we may view V as an F -vector space and de¬ne a
quadratic form h0 : V ’ F by
h0 (x) = h(x, x) for x ∈ V ,
since h is hermitian. If (ei )1¤i¤n is an orthogonal K-basis of V for h and z ∈ F
is such that z 2 = ±, then (ei , ei z)1¤i¤n is an orthogonal F -basis of V for h0 .
Therefore, if h(ei , ei ) = δi , then
h0 = 1, ’± · δ1 , . . . , δn ,
hence the signature of the F -quadratic form δ1 , . . . , δn is an invariant for h, equal
to 1 sgnP h0 . We let
2
1
sgnP h = sgnP h0 .
2
EXERCISES 145


Note that if ± > 0, then sgnP h0 = 0. This explains why the signature is meaningful
only when ± < 0.
(11.24) Proposition. Let (B, „ ) be a central simple algebra with involution of the
second kind over F , with center K. Then, the signature of the hermitian form
T(B,„ ) on B is the square of an integer. Moreover, if B is split: B = EndK (V ) and
„ = σh for some hermitian form h on V , then
sgnP T(B,„ ) = (sgnP h)2 .
Proof : If B is split, (??) yields an isometry
(V —K ι V, h — ι h)
(B, T(B,„ ) )
from which the equation sgnP T(B,„ ) = (sgnP h)2 follows.
In order to prove the ¬rst statement, we may extend scalars from F to a real
closure FP since signatures do not change under scalar extension to a real closure.
However, K — FP is algebraically closed since ± < 0, hence B is split over FP .
Therefore, the split case already considered shows that the signature of T(B,„ ) is a
square.
(11.25) De¬nition. For any involution „ of the second kind on B, we set

sgnP „ = sgnP T(B,„ ) .
The proposition above shows that if FP is a real closure of F for P and if „ —IdFP =
σh for some hermitian form h over FP , then
sgnP „ = |sgnP h| .



Exercises
1. Let (A, σ) be a central simple algebra with involution of any kind over a ¬eld F .
Show that for any right ideals I, J in A,
(I + J)⊥ = I ⊥ © J ⊥ and (I © J)⊥ = I ⊥ + J ⊥ .
Use this observation to prove that all the maximal isotropic right ideals in A
have the same reduced dimension.
Hint: If J is an isotropic ideal and I is an arbitrary right ideal, show
that rdim J ’ rdim(I ⊥ © J) ¤ rdim I ’ rdim(I © J). If I also is isotropic and
rdim I ¤ rdim J, use this relation to show I ⊥ © J ‚ I, and conclude that
I + (I ⊥ © J) is an isotropic ideal which strictly contains I.
2. (Bayer-Fluckiger-Shapiro-Tignol [?]) Let (A, σ) be a central simple algebra with
orthogonal involution over a ¬eld F of characteristic di¬erent from 2. Show that
(A, σ) is hyperbolic if and only if
(A, σ) M2 (F ) — A0 , σh — σ0
for some central simple algebra with orthogonal involution (A0 , σ0 ), where σh is
the adjoint involution with respect to some hyperbolic 2-dimensional symmetric
bilinear form. Use this result to give examples of central simple algebras with
involution (A, σ), (B, „ ), (C, ν) such that (A, σ) — (B, „ ) (A, σ) — (C, ν) and
(B, „ ) (C, ν).
146 II. INVARIANTS OF INVOLUTIONS


Let (σ, f ) be a quadratic pair on a central simple algebra A over a ¬eld F
of characteristic 2. Show that (A, σ, f ) is hyperbolic if and only if
(A, σ, f ) M2 (F ) — A0 , γh — σ0 , fh—
for some central simple algebra with involution of the ¬rst kind (A0 , σ0 ), where
(γh , fh— ) is the quadratic pair on M2 (F ) associated with a hyperbolic 2-dimen-
sional quadratic form.
Hint: If e ∈ A is an idempotent such that σ(e) = 1 ’ e, use (??) to ¬nd
a symmetric element t ∈ A— such that tσ(e)t’1 = e, and show that e, etσ(e),
σ(e)te and σ(e) span a subalgebra isomorphic to M2 (F ).
3. Let (A, σ) be a central simple F -algebra with involution of orthogonal type
and let K ‚ A be a sub¬eld containing F . Suppose K consists of symmetric
elements, so that the restriction σ = σ|CA K of σ to the centralizer of K in A
is an involution of orthogonal type. Prove that disc σ = NK/F (disc σ ).
4. Let Q = (a, b)F be a quaternion algebra over a ¬eld F of characteristic di¬erent
from 2. Show that the set of discriminants of orthogonal involutions on Q is
the set of represented values of the quadratic form a, b, ’ab .
5. (Tits [?]) Let (A, σ) be a central simple algebra of even degree with involution
of the ¬rst kind. Assume that σ is orthogonal if char F = 2, and that it is sym-
plectic if char F = 2. For any a ∈ Alt(A, σ) © A— whose reduced characteristic
polynomial is separable, let
H = { x ∈ Alt(A, σ) | xa = ax }.
Show that a’1 H is an ´tale subalgebra of A of dimension deg A/2. (The
e
space H is called a Cartan subspace in Tits [?].)
Hint: See Lemma (??).
6. Let (A, σ) be a central simple algebra with orthogonal involution over a ¬eld F
of characteristic di¬erent from 2. For brevity, write C for C(A, σ) its Cli¬ord
algebra, Z for Z(A, σ) the center of C and B for B(A, σ) the Cli¬ord bimodule.
Endow A—F C with the C-bimodule structure such that c1 ·(a—c)·c2 = a—c1 cc2
for a ∈ A and c, c1 , c2 ∈ C.
(a) Show that there is an isomorphism of C-bimodules ψ : B —C B ’ A —F C
which in the split case satis¬es
ψ (v1 — c1 ) — (v2 — c2 ) = (v1 — v2 ) — c1 c2
under the standard identi¬cations A = V — V , B = V — C1 (V, q) and
C = C0 (V, q).
(b) De¬ne a hermitian form H : B — B ’ A —F Z by
H(x, y) = IdA — (ι —¦ TrdC ) ψ(x — y ω ) for x, y ∈ B.
Show that the natural isomorphism ν of (??) is an isomorphism of algebras
with involution

ν : (ι C op , ι σ op ) —Z (C, σ) ’ EndA—Z (B), σH .

7. To each permutation π ∈ Sk , associate a permutation π — of {0, 1, . . . , k ’ 1} by
composing the following bijections:
π ’1
+1
{0, . . . , k ’ 1} ’’ {1, . . . , k} ’ ’ {1, . . . , k} ’’ {0, . . . , k ’ 1}
’ ’
EXERCISES 147


where the last map carries k to 0 and leaves every i between 1 and k ’ 1
invariant. Consider the decomposition of π — into disjoint cycles (including the
cycles of length 1):
π — = (0, ±1 , . . . , ±r )(β1 , . . . , βs ) · · · (γ1 , . . . , γt ).
Since the map Sandk : A—k ’ HomF (A—k’1 , A) is bijective (see (??)), there is
a unique element xπ ∈ A—k such that for b1 , . . . , bk’1 ∈ A:

Sandk (xπ )(b1 — · · · — bk’1 ) =
b±1 · · · b±r TrdA (bβ1 · · · bβs ) · · · TrdA (bγ1 · · · bγt ).
Show that xπ = gk (π).
Show that s2 = k! sk .
8. k
9. Show by a direct computation that if A is a quaternion algebra, the canonical
involution on A = »1 A is the quaternion conjugation.
10. Let (B, „ ) be a central simple F -algebra with unitary involution. Assume that
deg(B, „ ) is divisible by 4 and that char F = 2, so that the canonical involution
„ on D(B, „ ) has orthogonal type. Show that disc „ = 1 if deg(B, „ ) is not
a power of 2 and that disc „ = ± · F —2 if deg(B, „ ) is a power of 2 and K
F [X]/(X 2 ’ ±).
Hint: Reduce to the split case by scalar extension to some splitting ¬eld
of B in which F is algebraically closed (for instance the function ¬eld of the
Weil transfer of the Severi-Brauer variety of B). Let deg(B, „ ) = n = 2m.
Using the same notation as in (??), de¬ne a map v ∈ EndF ( m V0 ) as follows:
consider a partition of the subsets S ‚ {1, . . . , n} of cardinality m into two
classes C, C such that the complement of every S ∈ C lies in C and vice-
versa; then set v(eS ) = eS if S ∈ C, v(eS ) = ’eS if S ∈ C . Show that

v — ± ∈ Skew D(B, „ ), „ and use this element to compute disc „ .
11. Let K/F be a quadratic extension with non-trivial automorphism ι, and let
± ∈ F — , β ∈ K — . Assume F contains a primitive 2m-th root of unity ξ and
consider the algebra B of degree 2m over K generated by two elements i, j
subject to the following conditions:
i2m = ± j 2m = ι(β)/β ji = ξij.
(a) Show that there is a unitary involution „ on B such that „ (i) = i and
„ (j) = j ’1 .
(b) Show that D(B, „ ) ∼ ±, NK/F (β) F —F K, (’1)m ± F .
Hint: Let X = RK/F SB(B) be the transfer of the Severi-Brauer variety
of B. The algebra B splits over K —F F (X), but the scalar extension map
Br(F ) ’ Br F (X) is injective (see Merkurjev-Tignol [?]); so it su¬ces to
prove the claim when B is split.
12. Let V be a vector space of dimension n over a ¬eld F . Fix k with 1 ¤ k ¤ n’1,
k n
and let = n ’ k. The canonical pairing § : V— V’ V induces an
isomorphism
k
V )— ’
( V
which is uniquely determined up to a factor in F — , hence the pairing also
induces a canonical isomorphism
k
V )— ’ EndF (
ψk, : EndF ( V ).
148 II. INVARIANTS OF INVOLUTIONS


Our aim in this exercise is to de¬ne a corresponding isomorphism for non-split
algebras.
Let A be a central simple F -algebra of degree n. For 2 ¤ k ¤ n, set:
sgn(π)gk (π) ∈ A—k
sk =
π∈Sk

(as in §??), and extend this de¬nition by setting s1 = 1. Let = n ’ k, where
1 ¤ k ¤ n ’ 1.
(a) Generalize (??) by showing that sn ∈ A—n · (sk — s ).
We may thus consider the right ideal
f ∈ EndA—n A—n (sk — s ) A—n sf = {0} ‚ »k A — » A.
I= n

(b) Using exercise ?? of Chapter ??, show that this right ideal de¬nes a canon-
ical isomorphism

•k, : »k Aop ’ » A.

Show that if A = EndF V , then •k, = ψk, under the canonical identi¬-
k—
cations »k Aop = EndF ( V ) and » A = EndF ( V ).
13. (Wadsworth, unpublished) The aim of this exercise is to give examples of central
simple algebras with unitary involution whose discriminant algebra has index 4.
Let F0 be an arbitrary ¬eld of characteristic di¬erent from 2 and let K =
F0 (x, y, z) be the ¬eld of rational fractions in three independent indeterminates
over F0 . Denote by ι the automorphism of K which leaves F0 (x, y) elementwise
invariant and maps z to ’z, and let F = F0 (x, y, z 2 ) be the invariant sub¬eld.
Consider the quaternion algebras Q0 = (x, y)F and Q = Q0 —F K, and de¬ne an
involution θ on Q by θ = γ0 — ι where γ0 is the quaternion conjugation on Q0 .
Finally, let B = Mn (Q) for an arbitrary odd integer n > 1, and endow B with
the involution — de¬ned by
t
(aij )—
1¤i,j¤n = θ(aij ) .
1¤i,j¤n

(a) Show that D(B, — ) ∼ D(Q, θ)—n ∼ D(Q, θ) ∼ Q0 .
Let c1 , . . . , cn ∈ Sym(Q, θ) © Q— and d = diag(c1 , . . . , cn ) ∈ B. De¬ne another
involution of unitary type on B by „ = Int(d) —¦ — .
(a) Show that
D(B, „ ) ∼ D(B, — ) —F z 2 , NrdB (d) F
2
∼ (x, y)F — z , NrdQ (c1 ) · · · NrdQ (cn ) .
F
(b) Show that the algebra D(B, „ ) has index 4 if c1 = z 2 + zi, c2 = z 2 + zj
and c3 = · · · = cn = 1.
14. (Yanchevski˜ [?, Proposition 1.4]) Let σ, σ be involutions on a central simple
±
algebra A over a ¬eld F of characteristic di¬erent from 2. Show that if σ and
σ have the same restriction to the center of A and Sym(A, σ) = Sym(A, σ ),
then σ = σ .
Hint: If σ and σ are of the ¬rst kind, use (??).
15. Let (A, σ) be a central simple algebra with involution of the ¬rst kind over a
¬eld F of arbitrary characteristic. Show that a nonsingular symmetric bilinear
form on Symd(A, σ) may be de¬ned as follows: for x, y ∈ Symd(A, σ), pick
y ∈ A such that y = y + σ(y ), and let T (x, y) = TrdA (xy ). Mimic this
construction to de¬ne a nonsingular symmetric bilinear form on Alt(A, σ).
NOTES 149


Notes
§??. On the same model as Severi-Brauer varieties, varieties of isotropic ideals,
known as Borel varieties, or homogeneous varieties, or twisted ¬‚ag varieties, are
associated to an algebra with involution. These varieties can also be de¬ned as va-
rieties of parabolic subgroups of a certain type in the associated simply connected
group: see Borel-Tits [?]; their function ¬elds are the generic splitting ¬elds investi-
gated by Kersten and Rehmann [?]. In particular, the variety of isotropic ideals of
reduced dimension 1 in a central simple algebra with orthogonal involution (A, σ)
of characteristic di¬erent from 2 may be regarded as a twisted form of a quadric:
after scalar extension to a splitting ¬eld L of A, it yields the quadric q = 0 where q
is a quadratic form whose adjoint involution is σL . These twisted forms of quadrics
are termed involution varieties by Tao [?], who studied their K-groups to obtain
index reduction formulas for their function ¬elds. Tao™s results were generalized
to arbitrary Borel varieties by Merkurjev-Panin-Wadsworth [?], [?]. The Brauer
group of a Borel variety is determined in Merkurjev-Tignol [?].
The notion of index in (??) is inspired by Tits™ de¬nition of index for a semi-
simple linear algebraic group [?, (2.3)]. Hyperbolic involutions are de¬ned in Bayer-
Fluckiger-Shapiro-Tignol [?]. Example (??) is borrowed from Dejai¬e [?] where a
notion of orthogonal sum for algebras with involution is investigated.
§??. The discriminant of an orthogonal involution on a central simple alge-
bra of even degree over a ¬eld of characteristic di¬erent from 2 ¬rst appeared in
Jacobson [?] as the center of the (generalized, even) Cli¬ord algebra. The approach
in Tits [?] applies also in characteristic 2; it is based on generalized quadratic forms
instead of quadratic pairs. For involutions, the more direct de¬nition presented here
is due to Knus-Parimala-Sridharan [?]. Earlier work of Knus-Parimala-Sridharan [?]
used another de¬nition in terms of generalized pfa¬an maps.
A short, direct proof of (??) is given in Kersten [?, (3.1)]; the idea is to split
the algebra by a scalar extension in which the base ¬eld is algebraically closed.
The set of determinants of orthogonal involutions on a central simple algebra A
of characteristic di¬erent from 2 has been investigated by Parimala-Sridharan-
Suresh [?]. It turns out that, except in the case where A is a quaternion algebra
(where the set of determinants is easily determined, see Exercise ??), the set of
determinants is the group of reduced norms of A modulo squares:

det σ = Nrd(A— ) · F —2 .
σ

§??. The ¬rst de¬nition of Cli¬ord algebra for an algebra with orthogonal
involution of characteristic di¬erent from 2 is due to Jacobson [?]; it was obtained
by Galois descent. A variant of Jacobson™s construction was proposed by Seip-
Hornix [?] for the case of central simple algebras of Schur index 2. Her de¬nition
also covers the characteristic 2 case. Our treatment owes much to Tits [?]. In
particular, the description of the center of the Cli¬ord algebra in §?? and the proof
of (??) closely follow Tits™ paper. Other proofs of (??) were given by Allen [?,
Theorem 3] and Van Drooge (thesis, Utrecht, 1967).
If deg A is divisible by 8, the canonical involution σ on C(A, σ, f ) is part of a
canonical quadratic pair (σ, f ). If A is split and the quadratic pair (σ, f ) is hyper-
bolic, we may de¬ne this canonical pair as follows: representing A = EndF H(U )
150 II. INVARIANTS OF INVOLUTIONS


we have as in (??)
C(A, σ, f ) = C0 H(U ) End( U ) — End( U ) ‚ End( U ).
0 1
r
Let m = dim U . For ξ ∈ U , let ξ [r] be the component of ξ in U . Fix a nonzero
r
linear form s : U ’ F which vanishes on U for r < m and de¬ne a quadratic
form q§ : U ’ F by

§ ξ [m’r] + q(ξ [m/2] )
[r]
q§ (ξ) = s r<m/2 ξ

m/2 m
where q : U’ U is the canonical quadratic map of (??) and is the
involution on U which is the identity on U (see the proof of (??)). For i = 0,
1, let qi be the restriction of q§ to i U . The pair (q0 , q1 ) may be viewed as a
quadratic form
(q0 , q1 ) : U— U ’ F — F.
0 1

The canonical quadratic pair on End( 0 U )—End( 1 U ) is associated to this quad-
ratic form. In the general case, the canonical quadratic pair on C(A, σ, f ) can be
de¬ned by Galois descent. The canonical involution on the Cli¬ord algebra of a
central simple algebra with hyperbolic involution (of characteristic di¬erent from 2)
has been investigated by Garibaldi [?].
Cli¬ord algebras of tensor products of central simple algebras with involution
have been determined by Tao [?]. Let (A, σ) = (A1 , σ1 )—F (A2 , σ2 ) where A1 , A2 are
central simple algebras of even degree over a ¬eld F of characteristic di¬erent from 2,
and σ1 , σ2 are involutions which are either both orthogonal or both symplectic,
so that σ is an orthogonal involution of trivial discriminant, by (??). It follows
from (??) that the Cli¬ord algebra C(A, σ) decomposes into a direct product of
two central simple F -algebras: C(A, σ) = C + (A, σ) — C ’ (A, σ). Tao proves in [?,
Theorems 4.12, 4.14, 4.16]:
(a) Suppose σ1 , σ2 are orthogonal and denote by Q the quaternion algebra
Q = (disc σ1 , disc σ2 )F .
(i) If deg A1 or deg A2 is divisible by 4, then one of the algebras C± (A, σ)
is Brauer-equivalent to A —F Q and the other one to Q.
(ii) If deg A1 ≡ deg A2 ≡ 2 mod 4, then one of the algebras C± (A, σ) is
Brauer-equivalent to A1 —F Q and the other one to A2 —F Q.
(b) Suppose σ1 , σ2 are symplectic.
(i) If deg A1 or deg A2 is divisible by 4, then one of the algebras C± (A, σ)
is split and the other one is Brauer-equivalent to A.
(ii) If deg A1 ≡ deg A2 ≡ 2 mod 4, then one of the algebras C± (A, σ) is
Brauer-equivalent to A1 and the other one to A2 .
§??. In characteristic di¬erent from 2, the bimodule B(A, σ) is de¬ned by
Galois descent in Merkurjev-Tignol [?]. The fundamental relations in (??) between
a central simple algebra with orthogonal involution and its Cli¬ord algebra have
been observed by several authors: (??) was ¬rst proved by Jacobson [?, Theorem 4]
in the case where Z = F —F . In the same special case, proofs of (??) and (??) have
been given by Tits [?, Proposition 7], [?, 6.2]. In the general case, these relations
have been established by Tamagawa [?] and by Tao [?]. See (??) for a cohomological
proof of the fundamental relations in characteristic di¬erent from 2 and Exercise ??
of Chapter ?? for another cohomological proof valid in arbitrary characteristic.
Note that the bimodule B(A, σ) carries a canonical hermitian form which may
NOTES 151


be used to strengthen (??) into an isomorphism of algebras with involution: see
Exercise ??.
§??. The canonical representation of the symmetric group Sk in the group of
invertible elements of A—k was observed by Haile [?, Lemma 1.1] and Saltman [?].
Note that if k = ind A, (??) shows that A—k is split; therefore the exponent of A
divides its index. Indeed, the purpose of Saltman™s paper is to give an easy direct
proof (also valid for Azumaya algebras) of the fact that the Brauer group is torsion.
Another approach to the »-construction, using Severi-Brauer varieties, is due to
Suslin [?].
m
The canonical quadratic map on V , where V is a 2m-dimensional vector
space over a ¬eld of characteristic 2 (see (??)), is due to Papy [?]. It is part of a
general construction of reduced p-th powers in exterior algebras of vector spaces
over ¬elds of characteristic p.
The discriminant algebra D(B, „ ) also arises from representations of classical
algebraic groups of type 2An : see Tits [?]. If the characteristic does not divide
2 deg B, its Brauer class can be obtained by reduction modulo 2 of a cohomological
invariant t(B, „ ) called the Tits class, see (??). This invariant has been investigated
by Qu´guiner [?], [?]. In [?, Proposition 11], Qu´guiner shows that (??) can be
e e
derived from (??) if char F = 2; she also considers the analogue of (??) where
the involution „0 is symplectic instead of orthogonal, and proves that D(B, „ ) is
—m
Brauer-equivalent to B0 in this case. (Note that Qu´guiner™s “determinant class
e
modulo 2” di¬ers from the Brauer class of D(B, „ ) by the class of the quaternion
algebra (K, ’1)F if deg B ≡ 2 mod 4.)
§??. The idea to consider the form T(A,σ) as an invariant of the involution
σ dates back to Weil [?]. The relation between the determinant of an orthogonal
+
involution σ and the determinant of the bilinear form Tσ (in characteristic di¬erent
from 2) was observed by Lewis [?] and Qu´guiner [?], who also computed the Hasse
e
invariant s(Qσ ) of the quadratic form Qσ (x) = TrdA σ(x)x associated to T(A,σ) .
The result is the following: for an involution σ on a central simple algebra A of
degree n,
±
 n [A] + (’1, det σ)F if n is even and σ is orthogonal,
2
n n
s(Qσ ) = 2 [A] + 2 (’1, ’1)F if n is even and σ is symplectic,


0 if n is odd.
In Lewis™ paper [?], these relations are obtained by comparing the Hasse invariant
of Qσ and of QA (x) = TrdA (x2 ) through (??). Qu´guiner [?] also gives the com-
e
putation of the Hasse invariant of the quadratic forms Q+ and Q’ which are the
σ σ
restrictions of Qσ to Sym(A, σ) and Skew(A, σ) respectively. Just as for Qσ , the
result only depends on the parity of n and on the type and discriminant of σ.
The signature of an involution of the ¬rst kind was ¬rst de¬ned by Lewis-
Tignol [?]. The corresponding notion for involutions of the second kind is due to
Qu´guiner [?].
e
Besides the classical invariants considered in this chapter, there are also “higher
cohomological invariants” de¬ned by Rost (to appear) by means of simply connected
algebraic groups, with values in Galois cohomology groups of degree 3. See §?? for
a general discussion of cohomological invariants. Some special cases are considered
in the following chapters: see §?? for the case of symplectic involutions on central
simple algebras of degree 4 and §?? for the case of unitary involutions on central
152 II. INVARIANTS OF INVOLUTIONS


simple algebras of degree 3. (In the same spirit, see §?? for an H 3 -invariant of
Albert algebras.) Another particular instance dates back to Jacobson [?]: if A is a
central simple F -algebra of index 2 whose degree is divisible by 4, we may represent
A = EndQ (V ) for some vector space V of even dimension over a quaternion F -
algebra Q. According to (??), every symplectic involution σ on A is adjoint to
some hermitian form h on V with respect to the canonical involution of Q. Assume
char F = 2 and let h = ±1 , . . . , ±n be a diagonalization of h; then ±1 , . . . , ±n ∈ F —
and the element (’1)n/2 ±1 · · · ±n ·NrdQ (Q— ) ∈ F — / NrdQ (Q— ) is an invariant of σ.
There is an alternate description of this invariant, which emphasizes the relation
with Rost™s cohomological approach: we may associate to σ the quadratic form
qσ = 1, ’(’1)n/2 ±1 · · · ±n — nQ ∈ I 3 F where nQ is the reduced norm form of Q,
or the cup product (’1)n/2 ±1 · · · ±n ∪ [Q] ∈ H 3 (F, µ2 ), see (??).
CHAPTER III


Similitudes

In this chapter, we investigate the automorphism groups of central simple alge-
bras with involution. The inner automorphisms which preserve the involution are
induced by elements which we call similitudes, and the automorphism group of a
central simple algebra with involution is the quotient of the group of similitudes by
the multiplicative group of the center. The various groups thus de¬ned are natu-
rally endowed with a structure of linear algebraic group; they may then be seen as
twisted forms of orthogonal, symplectic or unitary groups, depending on the type of
the involution. This point of view will be developed in Chapter ??. Here, however,
we content ourselves with a more elementary viewpoint, considering the groups of
rational points of the corresponding algebraic groups.
After a ¬rst section which contains general de¬nitions and results valid for all
types, we then focus on quadratic pairs and unitary involutions, where additional
information can be derived from the algebra invariants de¬ned in Chapter ??. In
the orthogonal case, we also use the Cli¬ord algebra and the Cli¬ord bimodule to
de¬ne Cli¬ord groups and spin groups.

§12. General Properties
To motivate our de¬nition of similitude for an algebra with involution, we ¬rst
consider the split case, where the algebra consists of endomorphisms of bilinear or
hermitian spaces.

12.A. The split case. We treat separately the cases of bilinear, hermitian
and quadratic spaces, although the basic de¬nitions are the same, to emphasize the
special features of these various cases.
Bilinear spaces. Let (V, b) be a nonsingular symmetric or alternating bilinear
space over an arbitrary ¬eld F . A similitude of (V, b) is a linear map g : V ’ V for
which there exists a constant ± ∈ F — such that
(12.1) b g(v), g(w) = ±b(v, w) for v, w ∈ V .
The factor ± is called the multiplier of the similitude g. A similitude with mul-
tiplier 1 is called an isometry. The similitudes of the bilinear space (V, b) form a
group denoted Sim(V, b) , and the map
µ : Sim(V, b) ’ F —
which carries every similitude to its multiplier is a group homomorphism. By
de¬nition, the kernel of this map is the group of isometries of (V, b), which we write
Iso(V, b). We also de¬ne the group PSim(V, b) of projective similitudes by
PSim(V, b) = Sim(V, b)/F — .
153
154 III. SIMILITUDES


Speci¬c notations for the groups Sim(V, b), Iso(V, b) and PSim(V, b) are used ac-
cording to the type of b. If b is symmetric nonalternating, we set
O(V, b) = Iso(V, b), GO(V, b) = Sim(V, b) and PGO(V, b) = PSim(V, b);
if b is alternating, we let
Sp(V, b) = Iso(V, b), GSp(V, b) = Sim(V, b) and PGSp(V, b) = PSim(V, b).
Note that condition (??), de¬ning a similitude of (V, b) with multiplier ±, can be
rephrased as follows, using the adjoint involution σb :
(12.2) σb (g) —¦ g = ±IdV .
By taking the determinant of both sides, we obtain (det g)2 = ±n where n = dim V .
It follows that the determinant of an isometry is ±1 and that, if n is even,
det g = ±µ(g)n/2 for g ∈ Sim(V, b).
A ¬rst di¬erence between the orthogonal case and the symplectic case shows up in
the following result:
(12.3) Proposition. If b is a nonsingular alternating bilinear form on a vector
space V of dimension n (necessarily even), then
det g = µ(g)n/2 for g ∈ GSp(V, b).
Proof : Let g ∈ GSp(V, b) and let G, B denote the matrices of g and b respectively
with respect to some arbitrary basis of V . The matrix B is alternating and we have
Gt BG = µ(g)B.
By taking the pfa¬an of both sides, we obtain, by known formulas for pfa¬ans (see
Artin [?, Theorem 3.28]; compare with (??)):
det G pf B = µ(g)n/2 pf B,
hence det g = µ(g)n/2 .
By contrast, if b is symmetric and char F = 2, every hyperplane re¬‚ection is an
isometry with determinant ’1 (see (??)), hence it satis¬es det g = ’µ(g)n/2 .
We set
O+ (V, b) = { g ∈ O(V, b) | det g = 1 }.
Of course, O+ (V, b) = O(V, b) if char F = 2.
Similarly, if dim V = n is even, we set
GO+ (V, b) = { g ∈ GO(V, b) | det g = µ(g)n/2 },
and
PGO+ (V, b) = GO+ (V, b)/F — .
The elements in GO+ (V, b), O+ (V, b) are called proper similitudes and proper isome-
tries respectively.
If dim V is odd, there is a close relationship between similitudes and isometries,
as the next proposition shows:
(12.4) Proposition. Suppose that (V, b) is a nonsingular symmetric bilinear space
of odd dimension over an arbitrary ¬eld F ; then
GO(V, b) = O+ (V, b) · F — O+ (V, b) — F — O+ (V, b).
PGO(V, b)
and
§12. GENERAL PROPERTIES 155


Proof : If g is a similitude of (V, b) with multiplier ± ∈ F — , then by taking the
b we get ± ∈ F —2 . If ± = ±2 ,
determinant of both sides of the isometry ± · b 1
’1
then ±1 g is an isometry. Moreover, after changing the sign of ±1 if necessary, we
may assume that det(±’1 g) = 1. The factorization g = (±’1 g) · ±1 shows that
1 1
GO(V, b) = O+ (V, b) · F — , and the other isomorphisms are clear.
Hermitian spaces. Suppose (V, h) is a nonsingular hermitian space over a
quadratic separable ¬eld extension K of F (with respect to the nontrivial automor-
phism of K/F ). A similitude of (V, h) is an invertible linear map g : V ’ V for
which there exists a constant ± ∈ F — , called the multiplier of g, such that
(12.5) h g(v), g(w) = ±h(v, w) for v, w ∈ V .
As in the case of bilinear spaces, we write Sim(V, h) for the group of similitudes of
(V, h); let
µ : Sim(V, h) ’ F —
be the group homomorphism which carries every similitude to its multiplier; write
Iso(V, h) for the kernel of µ, whose elements are called isometries, and let
PSim(V, h) = Sim(V, h)/K — .
We also use the following more speci¬c notation:
U(V, h) = Iso(V, h), GU(V, h) = Sim(V, h), PGU(V, h) = PSim(V, h).
Condition (??) can be rephrased as
σh (g) —¦ g = ±IdV .
By taking the determinant of both sides, we obtain
NK/F (det g) = µ(g)n , where n = dim V .
This relation shows that the determinant of every isometry has norm 1. Set
SU(V, h) = { g ∈ U(V, h) | det g = 1 }.
Quadratic spaces. Let (V, q) be a nonsingular quadratic space over an arbi-
trary ¬eld F . A similitude of (V, q) is an invertible linear map g : V ’ V for which
there exists a constant ± ∈ F — , called the multiplier of g, such that
q g(v) = ±q(v) for v ∈ V .
The groups Sim(V, q), Iso(V, q), PSim(V, q) and the group homomorphism
µ : Sim(V, q) ’ F —
are de¬ned as for nonsingular bilinear forms. We also use the notation
O(V, q) = Iso(V, q), GO(V, q) = Sim(V, q), PGO(V, q) = PSim(V, q).
It is clear from the de¬nitions that every similitude of (V, q) is also a similitude of
its polar bilinear space (V, bq ), with the same multiplier, hence
GO(V, bq ) if char F = 2,
GO(V, q) ‚ Sim(V, b) =
GSp(V, bq ) if char F = 2,
and the reverse inclusion also holds if char F = 2.
For the rest of this section, we assume therefore char F = 2. If dim V is odd,
the same arguments as in (??) yield:
156 III. SIMILITUDES


(12.6) Proposition. Suppose (V, q) is a nonsingular symmetric quadratic space
of odd dimension over a ¬eld F of characteristic 2; then
GO(V, q) = O(V, q) · F — O(V, q) — F — PGO(V, q) O(V, q).
and
We omit the proof, since it is exactly the same as for (??), using the determinant
of q de¬ned in (??).
If dim V is even, we may again distinguish proper and improper similitudes, as
we now show.
By using a basis of V , we may represent the quadratic map q by a quadratic
form, which we denote again q. Let M be a matrix such that
q(X) = X t · M · X.
Since q is nonsingular, the matrix W = M + M t is invertible. Let g be a similitude
of V with multiplier ±, and let G be its matrix with respect to the chosen basis of V .
The equation q(G · X) = ±q(X) shows that the matrices Gt M G and ±M represent
the same quadratic form. Therefore, Gt M G ’ ±M is an alternating matrix. Let
R ∈ Mn (K) be such that
Gt M G ’ ±M = R ’ Rt .
(12.7) Proposition. The element tr(±’1 W ’1 R) ∈ K depends only on the simil-
itude g, and not on the choice of basis of V nor on the choices of matrices M
and R. It equals 0 or 1.
Proof : With a di¬erent choice of basis of V , the matrix G is replaced by G =
P ’1 GP for some invertible matrix P ∈ GLn (K), and the matrix M is replaced by
t
a matrix M = P t M P + U ’ U t for some matrix U . Then W = M + M is related
to W by W = P t W P . Suppose R, R are matrices such that
t t
Gt M G ’ ±M = R ’ Rt
(12.8) and G M G ’ ±M = R ’ R .
By adding each side to its transpose, we derive from these equations:
t
Gt W G = ±W
(12.9) and G W G = ±W .
In order to prove that tr(±’1 W ’1 R) depends only on the similitude g, we have
’1
to show tr(W ’1 R) = tr(W R ). By substituting for M its expression in terms
t t
of M , we derive from (??) that R ’ R = R ’ R , where
R = P t RP + P t Gt (P ’1 )t U P ’1 GP ’ ±U,
(12.10)
’1
hence R = R + S for some symmetric matrix S ∈ Mn (K). Since W =
’1 ’1 ’1 ’1 t ’1
W MW + (W M W ) , it follows that W is alternating. By (??),
alternating matrices are orthogonal to symmetric matrices for the trace bilinear
’1 ’1
form, hence tr(g R ) = tr(g R ). In view of (??) we have
’1
R ) = tr(P ’1 W ’1 RP ) + tr P ’1 W ’1 Gt (P ’1 )t U P ’1 GP
(12.11) tr(W
+ ± tr P ’1 W ’1 (P ’1 )t U .
By (??), W ’1 Gt = ±G’1 W ’1 , hence the second term on the right side of (??)
equals
± tr P ’1 G’1 W ’1 (P ’1 )t U P ’1 GP = ± tr W ’1 (P ’1 )t U P ’1 .
§12. GENERAL PROPERTIES 157


Therefore, the last two terms on the right side of (??) cancel, and we get
’1 ’1
R ) = tr(W ’1 R),
tr(W R ) = tr(W
proving that tr(±’1 W ’1 R) depends only on the similitude g.
In order to prove that this element is 0 or 1, we compute s2 (W ’1 M ), the coef-
¬cient of X n’2 in the characteristic polynomial of W ’1 M (see (??)). By (??), we
have G’1 W ’1 = ±’1 W ’1 Gt , hence G’1 W ’1 M G = ±’1 W ’1 Gt M G, and there-
fore
s2 (W ’1 M ) = s2 (±’1 W ’1 Gt M G).
On the other hand, (??) also yields Gt M G = ±M + R ’ Rt , hence by substituting
this in the right side of the preceding equation we get
s2 (W ’1 M ) = s2 (W ’1 M + ±’1 W ’1 R ’ ±’1 W ’1 Rt ).
By (??), we may expand the right side to get
s2 (W ’1 M ) = s2 (W ’1 M ) + tr(±’1 W ’1 R) + tr(±’1 W ’1 R)2 .
Therefore, tr(±’1 W ’1 R) + tr(±’1 W ’1 R)2 = 0, hence
tr(±’1 W ’1 R) = 0, 1.


(12.12) De¬nition. Let (V, q) be a nonsingular quadratic space of even dimension
over a ¬eld F of characteristic 2. Keep the same notation as above. In view of the
preceding proposition, we set
∆(g) = tr(±’1 W ’1 R) ∈ {0, 1} for g ∈ GO(V, q).
Straightforward veri¬cations show that ∆ is a group homomorphism
∆ : GO(V, q) ’ Z/2Z,
called the Dickson invariant. We write GO+ (V, q) for the kernel of this homomor-
phism. Its elements are called proper similitudes, and the similitudes which are
mapped to 1 under ∆ are called improper. We also let
O+ (V, q) = { g ∈ O(V, q) | ∆(g) = 0 } and PGO+ (V, q) = GO+ (V, q)/F — .
(12.13) Example. Let dim V = n = 2m. For any anisotropic vector v ∈ V , the
hyperplane re¬‚ection ρv : V ’ V is de¬ned in arbitrary characteristic by
ρv (x) = x ’ vq(v)’1 bq (v, x) for x ∈ V .
This map is an isometry of (V, q). We claim that it is improper.
This is clear if char F = 2, since the matrix of ρv with respect to an orthogonal
basis whose ¬rst vector is v is diagonal with diagonal entries (’1, 1, . . . , 1), hence
det ρv = ’1.
If char F = 2, we compute ∆(ρv ) by means of a symplectic basis (e1 , . . . , en )
of (V, bq ) such that e1 = v. With respect to that basis, the quadratic form q is
represented by the matrix
« 
M1 0
q(e2i’1 ) 0
¬ ·
..
M =  where Mi = ,
. 1 q(e2i )
0 Mm
158 III. SIMILITUDES


and the map ρv is represented by
« 
G1 0
1 q(e1 )’1
¬ ·
..
G=  where G1 = , Gi = I, i ≥ 2.
. 0 1
0 Gm
As a matrix R such that Gt M G + M = R + Rt we may take
« 
R1 0
01
¬ ·
..
R=  where R1 = , Ri = I, i ≥ 2.
. 00
0 Rm
It is readily veri¬ed that tr(W ’1 R) = 1, hence ∆(ρv ) = 1, proving the claim.
12.B. Similitudes of algebras with involution. In view of the charac-
terization of similitudes of bilinear or hermitian spaces by means of the adjoint
involution (see (??)), the following de¬nition is natural:
(12.14) De¬nition. Let (A, σ) be a central simple F -algebra with involution. A
similitude of (A, σ) is an element g ∈ A such that
σ(g)g ∈ F — .
The scalar σ(g)g is called the multiplier of g and is denoted µ(g). The set of all
similitudes of (A, σ) is a subgroup of A— which we call Sim(A, σ), and the map µ
is a group homomorphism
µ : Sim(A, σ) ’ F — .
It is then clear that similitudes of bilinear spaces are similitudes of their endo-
morphism algebras:
Sim EndF (V ), σb = Sim(V, b)
if (V, b) is a nonsingular symmetric or alternating bilinear space. There is a corre-
sponding result for hermitian spaces.
Similitudes can also be characterized in terms of automorphisms of the algebra
with involution. Recall that an automorphism of (A, σ) is an F -algebra automor-
phism which commutes with σ:
AutF (A, σ) = { θ ∈ AutF (A) | σ —¦ θ = θ —¦ σ }.
Let K be the center of A, so that K = F if σ is of the ¬rst kind and K is a quadratic
´tale F -algebra if σ is of the second kind. De¬ne Aut K (A, σ) = AutF (A, σ) ©
e
AutK (A).
(12.15) Theorem. With the notation above,
AutK (A, σ) = { Int(g) | g ∈ Sim(A, σ) }.
There is therefore an exact sequence:
Int
1 ’ K — ’ Sim(A, σ) ’’ AutK (A, σ) ’ 1.

Proof : By the Skolem-Noether theorem, every automorphism of A over K has the
form Int(g) for some g ∈ A— . Since
σ —¦ Int(g) = Int σ(g)’1 —¦ σ,
§12. GENERAL PROPERTIES 159


the automorphism Int(g) commutes with σ if and only if σ(g)’1 ≡ g mod K — , i.e.,
σ(g)g ∈ K — . Since σ(g)g is invariant under σ, the latter condition is also equivalent
to σ(g)g ∈ F — .

Let PSim(A, σ) be the group of projective similitudes, de¬ned as
PSim(A, σ) = Sim(A, σ)/K — .
In view of the preceding theorem, the map Int de¬nes a natural isomorphism

PSim(A, σ) ’ AutK (A, σ).

Speci¬c notations for the groups Sim(A, σ) and PSim(A, σ) are used according
to the type of σ, re¬‚ecting the notations for similitudes of bilinear or hermitian
spaces:
±
GO(A, σ) if σ is of orthogonal type,

Sim(A, σ) = GSp(A, σ) if σ is of symplectic type,


GU(A, σ) if σ is of unitary type,

and
±
PGO(A, σ) if σ is of orthogonal type,

PSim(A, σ) = PGSp(A, σ) if σ is of symplectic type,


PGU(A, σ) if σ is of unitary type.
Similitudes with multiplier 1 are isometries; they make up the group Iso(A, σ):
Iso(A, σ) = { g ∈ A— | σ(g) = g ’1 }.
We also use the following notation:
±
O(A, σ) if σ is of orthogonal type,

Iso(A, σ) = Sp(A, σ) if σ is of symplectic type,


U(A, σ) if σ is of unitary type.
For quadratic pairs, the corresponding notions are de¬ned as follows:
(12.16) De¬nition. Let (σ, f ) be a quadratic pair on a central simple F -algebra A.
An automorphism of (A, σ, f ) is an F -algebra automorphism θ of A such that
σ—¦θ =θ—¦σ and f —¦ θ = f.
A similitude of (A, σ, f ) is an element g ∈ A— such that σ(g)g ∈ F — and f (gsg ’1 ) =
f (s) for all s ∈ Sym(A, σ). Let GO(A, σ, f ) be the group of similitudes of (A, σ, f ),
let
PGO(A, σ, f ) = GO(A, σ, f )/F —
and write AutF (A, σ, f ) for the group of automorphisms of (A, σ, f ). The same
arguments as in (??) yield an exact sequence
Int
1 ’ F — ’ GO(A, σ, f ) ’’ AutF (A, σ, f ) ’ 1,

hence also an isomorphism

PGO(A, σ, f ) ’ AutF (A, σ, f ).

160 III. SIMILITUDES


For g ∈ GO(A, σ, f ) we set µ(g) = σ(g)g ∈ F — . The element µ(g) is called the
multiplier of g and the map
µ : GO(A, σ, f ) ’ F —
is a group homomorphism. Its kernel is denoted O(A, σ, f ).
It is clear from the de¬nition that GO(A, σ, f ) ‚ Sim(A, σ). If char F = 2, the
map f is the restriction of 1 TrdA to Sym(A, σ), hence the condition f (gsg ’1 ) =
2
f (s) for all s ∈ Sym(A, σ) holds for all g ∈ GO(A, σ). Therefore, we have in this
case
GO(A, σ, f ) = GO(A, σ), PGO(A, σ, f ) = PGO(A, σ) and O(A, σ, f ) = O(A, σ).
In particular, if (V, q) is a nonsingular quadratic space over F and (σq , fq ) is the
associated quadratic pair on EndF (V ) (see (??)),
GO EndF (V ), σq , fq = GO EndF (V ), σq = GO(V, q).
There is a corresponding result if char F = 2:
(12.17) Example. Let (V, q) be a nonsingular quadratic space of even dimension
over a ¬eld F of characteristic 2, and let (σq , fq ) be the associated quadratic pair
on EndF (V ). We claim that
GO EndF (V ), σq , fq = GO(V, q),
hence also PGO EndF (V ), σq , fq = PGO(V, q) and O EndF (V ), σq , fq = O(V, q).
In order to prove these equalities, observe ¬rst that the standard identi¬cation
•q of (??) associated with the polar of q satis¬es the following property: for all
g ∈ EndF (V ), and for all v, w ∈ V ,
g —¦ •q (v — w) —¦ σq (g) = •q g(v) — g(w) .
Therefore, if g ∈ GO EndF (V, σq , fq ) and ± = µ(g) ∈ F — , the condition
fq g —¦ •q (v — v) —¦ g ’1 = fq —¦ •q (v — v) for v ∈ V
amounts to
q g(v) = ±q(v) for v ∈ V ,
which means that g is a similitude of the quadratic space (V, q), with multiplier ±.
This shows GO EndF (V ), σq , fq ‚ GO(V, q).
For the reverse inclusion, observe that if g is a similitude of (V, q) with multiplier
±, then σq (g)g = ± since g also is a similitude of the associated bilinear space (V, bq ).
Moreover, the same calculation as above shows that
fq g —¦ •q (v — v) —¦ g ’1 = fq —¦ •q (v — v) for v ∈ V .
Since Sym EndF (V ), σq , fq is spanned by elements of the form •q (v —v), it follows
that fq (gsg ’1 ) = fq (s) for all s ∈ Sym(A, σ), hence g ∈ GO EndF (V ), σq , fq . This
proves the claim.
We next determine the groups of similitudes for quaternion algebras.
(12.18) Example. Let Q be a quaternion algebra with canonical (symplectic)
involution γ over an arbitrary ¬eld F . Since γ(q)q ∈ F for all q ∈ Q, we have
Sim(Q, γ) = GSp(Q, γ) = Q— .
Therefore, γ commutes with all the inner automorphisms of Q. (This observation
also follows from the fact that γ is the unique symplectic involution of Q: for
§12. GENERAL PROPERTIES 161


every automorphism θ, the composite θ —¦ γ —¦ θ ’1 is a symplectic involution, hence
θ —¦ γ —¦ θ’1 = γ).
Let σ be an orthogonal involution on Q; by (??) we have
σ = Int(u) —¦ γ
for some invertible quaternion u such that γ(u) = ’u and u ∈ F . Since γ commutes
with all automorphisms of Q, an inner automorphism Int(g) commutes with σ if
and only if it commutes with Int(u), i.e., gu ≡ ug mod F — . If » ∈ F — is such that
gu = »ug, then by taking the reduced norm of both sides of this equation we obtain
»2 = 1, hence gu = ±ug. The group of similitudes of (Q, σ) therefore consists of
the invertible elements which commute or anticommute with u. If char F = 2, we
thus obtain
GO(Q, σ) = F (u)— .
If char F = 2, let v be any invertible element which anticommutes with u; then
GO(Q, σ) = F (u)— ∪ F (u)— · v .
Finally, we consider the case of quadratic pairs on Q. We assume that char F =
2 since, if the characteristic is di¬erent from 2, the similitudes of a quadratic pair
(σ, f ) are exactly the similitudes of the orthogonal involution σ. Since char F =
2, every involution which is part of a quadratic pair is symplectic, hence every
quadratic pair on Q has the form (γ, f ) for some linear map f : Sym(Q, γ) ’ F .
Take any ∈ Q satisfying
f (s) = TrdQ ( s) for s ∈ Sym(Q, γ)
(see (??)). The element is uniquely determined by the quadratic pair (γ, f ) up to
the addition of an element in Alt(Q, γ) = F , and it satis¬es TrdQ ( ) = 1, by (??)
and (??). Therefore, there exists an element v ∈ Q— such that v ’1 v = + 1. We
claim that
GO(Q, γ, f ) = F ( )— ∪ F ( )— · v .
Since GSp(Q, γ) = Q— , an element g ∈ Q— is a similitude of (Q, γ, f ) if and only
if f (gsg ’1 ) = f (s) for all s ∈ Sym(Q, γ). By de¬nition of , this condition can be
rephrased as
TrdQ ( gsg ’1 ) = TrdQ ( s) for s ∈ Sym(Q, γ).
Since the left-hand expression equals TrdQ (g ’1 gs), this condition is also equivalent
to
TrdQ ( ’ g ’1 g)s = 0 for s ∈ Sym(Q, γ);

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