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);