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