„v,d (x) = x + v(d ’ 1)h(v, v)’1 h(v, x).

Prove: „v,d is an isometry of (V, h), NrdEnd(V ) („v,d ) = NrdD (d), and show that

the group of isometries of (V, h) is generated by elements of the form „v,d .

Hint: For the last part, see the proof of Witt™s theorem in Scharlau [?,

Theorem 7.9.5].

6. (Notation as in the preceding exercise.) Suppose is of the ¬rst kind. Show

that if d ∈ D— is such that dsd = s for some s ∈ D — such that s = ±s, then

NrdD (d) = 1, except if D is split and d = ’1.

1’d 1’e

Hint: If d = ’1, set e = 1+d . Show that s’1 es = ’e and d = 1+e .

7. Use the preceding two exercises to prove the following special case of (??) due

to Kneser: assuming char F = 2, if (A, σ) is a central simple F -algebra with

orthogonal involution which contains an improper isometry, then A is split.

8. (Dieudonn´) Let (V, q) be a 4-dimensional hyperbolic quadratic space over the

e

¬eld F with two elements, and let (e1 , . . . , e4 ) be a basis of V such that q(e1 x1 +

· · · + e4 x4 ) = x1 x2 + x3 x4 . Consider the map „ : V ’ V such that „ (e1 ) = e3 ,

„ (e2 ) = e4 , „ (e3 ) = e1 and „ (e4 ) = e2 . Show that „ is a proper isometry of

(V, q) which is not a product of hyperplane re¬‚ections. Consider the element

γ = e2 · (e1 + e3 ) + (e1 + e3 ) · e4 ∈ C0 (V, q).

Show that γ ∈ “+ (V, q), χ(γ) = „ , and that γ is not a product of vectors in V .

9. Let (A, σ) be a central simple algebra with involution (of any type) over a

¬eld F of characteristic di¬erent from 2. Let

U = { u ∈ A | σ(u)u = 1 }

denote the group of isometries of (A, σ) and let

U 0 = { u ∈ U | 1 + u ∈ A— }.

Let also

Skew(A, σ)0 = { a ∈ A | σ(a) = ’a and 1 + a ∈ A— }.

Show that U is generated (as a group) by the set U 0 . Show that U 0 (resp.

Skew(A, σ)0 ) is a Zariski-open subset of U (resp. Skew(A, σ)) and that the

1’a

map a ’ 1+a de¬nes a bijection from Skew(A, σ)0 onto U 0 . (This bijection is

known as the Cayley parametrization of U .)

202 III. SIMILITUDES

10. Let (A, σ, f ) be a central simple algebra of degree 2m with quadratic pair and

let g ∈ GO(A, σ, f ). Show that

± m is odd and g is improper,

1 if

m is even and g is proper,

Sn µ(g)’1 g 2 =

m is odd and g is proper,

µ(g) · F —2 if

m is even and g is improper.

and that

µ GO+ (A, σ, f ) · F —2 if m is odd,

+

Sn O (A, σ, f ) ⊃

µ GO’ (A, σ, f ) · F —2 if m is even,

where GO’ (A, σ, f ) is the coset of improper similitudes of (A, σ, f ).

Hint: Use the arguments of (??).

11. Let (A, σ, f ) be a central simple algebra of degree deg A ≡ 2 mod 4 with a

quadratic pair. Show that

{ c ∈ Sim C(A, σ, f ), σ | σ(c) — Ab · c = Ab } = “(A, σ, f ).

Hint: σ(c)c ∈ F — for all c ∈ Sim(A, σ, f ).

12. Let (B, „ ) be a central simple F -algebra with unitary involution. Let ι be the

nontrivial automorphism of the center K of B and assume that char F = 2.

(a) (Merkurjev [?, Proposition 6.1]) Show that

NrdB U(B, „ ) = { zι(z)’1 | z ∈ NrdB (B — ) }.

In particular, the subgroup NrdB U (B, „ ) ‚ K — depends only on the

Brauer class of B.

Hint: (Suresh [?, Theorem 5.1.3]) For u ∈ U(B, „ ), show that there ex-

ists x ∈ K such that v = x + uι(x) is invertible. Then u = v„ (v)’1

and Nrd(u) = zι(z)’1 with z = Nrd(v). To prove the reverse inclu-

sion, let (B, „ ) = EndD (V ), σh for some hermitian space (V, h) over a

division algebra D with unitary involution θ. By considering endomor-

phisms which have a diagonal matrix representation with respect to an

orthogonal basis of V , show that NrdD U(D, θ) ‚ NrdB U(B, „ ) . Fi-

nally, for d ∈ D— , show by dimension count that d · Sym(D, θ) © F +

Skew(D, θ) = {0}, hence d = (x + s)t’1 for some x ∈ F , s ∈ Skew(D, θ)

and t ∈ Sym(D, θ). For u = (x + s)(x ’ s)’1 , show that u ∈ U(D, θ) and

’1

NrdD (u) = NrdD (d)ι NrdD (d) .

(b) (Suresh [?, Lemma 2.6]) If deg(B, „ ) is odd, show that

NrdB U(B, „ ) = NrdB (B — ) © K 1 .

Hint: Let deg(B, „ ) = 2r + 1. If Nrd(b) = ι(y)y ’1 , then

y = Nrd(ybr )NK/F (y)’r ∈ F — · Nrd(B — ).

13. Let (A, σ) be a central simple algebra with involution (of any type) over a ¬eld F

and let L/F be a ¬eld extension. Suppose •, ψ : L ’ A are two embeddings

such that •(L), ψ(L) ‚ (A, σ)+ . The Skolem-Noether theorem shows that

there exists a ∈ A— such that • = Int(a) —¦ ψ. Show that σ(a)a ∈ CA ψ(L) and

¬nd a necessary and su¬cient condition on this element for the existence of a

similitude g ∈ Sim(A, σ) such that • = Int(g) —¦ ψ.

NOTES 203

Notes

§??. The Dickson invariant ∆ of (??) was originally de¬ned by Dickson [?,

Theorem 205, p. 206] by means of an explicit formula involving the entries of

the matrix. Subsequently, Dieudonn´ [?] showed that it can also be de¬ned by

e

considering the action of the similitude on the center of the even Cli¬ord algebra

(see (??)). The presentation given here is new.

A functor M : Bn ’ Bn such that End —¦ M ∼ IdBn and M —¦ End ∼ IdBn

= =

(thus providing an alternate proof of (??)) can be made explicit as follows. Recall

the canonical direct sum decomposition of Cli¬ord algebras (see Wonenburger [?,

Theorem 1]): if (V, q) is a quadratic space of dimension d,

C(V, q) = M0 • M1 • · · · • Md

where M0 = F , M1 = V and, for k ≥ 2, the space Mk is the linear span of the

elements v · m ’ (’1)k m · v with v ∈ V and m ∈ Mk’1 . For k = 1, . . . , d the vector

space Mk is also spanned by the products of k vectors in any orthogonal basis of V .

In particular, the dimension of Mk is given by the binomial coe¬cient:

d

dimF Mk = .

k

Clearly, Mk ‚ C0 (V, q) if and only if k is even; hence

C0 (V, q) = Mi .

i even

Suppose d is odd and disc q = 1. We then have

Md’1 = ζ · V ‚ C0 (V, q)

for any orientation ζ of (V, q), hence x2 ∈ F for all x ∈ Md’1 . Therefore, we may

de¬ne a quadratic map

s : Md’1 ’ F

by s(x) = x2 . The embedding Md’1 ’ C0 (V, q) induces an F -algebra homo-

morphism e : C(Md’1 , s) ’ C0 (V, q), which shows that disc s = 1. A canonical

orientation · on (Md’1 , s) can be characterized by the condition e(·) = 1.

Since the decomposition of C0 (V, q) is canonical, it can be de¬ned for the Clif-

ford algebra of any central simple algebra with orthogonal involution (A, σ), as

Jacobson shows in [?, p. 294]. If the degree of the algebra A is odd: deg A =

d = 2n + 1, the construction above associates to (A, σ) an oriented quadratic space

(M, s, ·) of dimension 2n + 1 (where M ‚ C(A, σ) and s(x) = x2 for x ∈ M )

and de¬nes a functor M : Bn ’ Bn . We leave it to the reader to check that

End —¦ M ∼ IdBn and M —¦ End ∼ IdBn .

= =

§??. If char F = 0 and deg A ≥ 10, Lie algebra techniques can be used to

prove that the Lie-automorphism ψ of Alt(A, σ) de¬ned in (??) extends to an

automorphism of (A, σ): see Jacobson [?, p. 307].

The extended Cli¬ord group „¦(A, σ) was ¬rst considered by Jacobson [?] in

characteristic di¬erent from 2. (Jacobson uses the term “even Cli¬ord group”.)

In the split case, this group has been investigated by Wonenburger [?]. The spin

groups Spin(A, σ, f ) were de¬ned by Tits [?] in arbitrary characteristic.

204 III. SIMILITUDES

The original proof of Dieudonn´™s theorem on multipliers of similitudes (??)

e

appears in [?, Th´or`me 2]. The easy argument presented here is due to Elman-

ee

Lam [?, Lemma 4]. The generalization in (??) is due to Merkurjev-Tignol [?]. An-

other proof of (??), using Galois cohomology, has been found by Bayer-Fluckiger [?]

assuming that char F = 2. (This assumption is also made in [?].)

From (??), it follows that every central simple algebra with orthogonal in-

volution which contains an improper isometry is split. In characteristic di¬erent

from 2, this statement can be proved directly by elementary arguments; it was ¬rst

observed by Kneser [?, Lemma 1.b, p. 42]. (See also Exercise ??; the proof in [?] is

di¬erent.)

™

§??. The canonical Lie homomorphism »k : L(A) ’ L(»k A) is de¬ned as the

™

di¬erential of the polynomial map »k . It is of course possible to de¬ne »k indepen-

k

dently of » : it su¬ces to mimic (??), substituting in the proof a — 1 — · · · — 1 +

™

1 — a — · · · — 1 + · · · + 1 — 1 — · · · — a for a—k . The properties of »k may also be

™ ™

proved directly (by mimicking (??) and (??)), but the proof that „ §k —¦ »k = »k —¦ „

involves rather tedious computations.

CHAPTER IV

Algebras of Degree Four

Among groups of automorphisms of central simple algebras with involution,

there are certain isomorphisms, known as exceptional isomorphisms, relating alge-

bras of low degree. (The reason why these isomorphisms are indeed exceptional

comes from the fact that in some special low rank cases Dynkin diagrams coincide,

see §??.) Algebras of degree 4 play a central rˆle from this viewpoint: their three

o

types of involutions (orthogonal, symplectic, unitary) are involved with three of the

exceptional isomorphisms, which relate them to quaternion algebras, 5-dimensional

quadratic spaces and orthogonal involutions on algebras of degree 6 respectively.

A correspondence, ¬rst considered by Albert [?], between tensor products of two

quaternion algebras and quadratic forms of dimension 6 arises as a special case of

the last isomorphism.

The exceptional isomorphisms provide the motivation for, and can be obtained

as a consequence of, equivalences between certain categories of algebras with invo-

lution which are investigated in the ¬rst section. In the second section, we focus on

tensor products of two quaternion algebras, called biquaternion algebras, and their

Albert quadratic forms. The third section yields a quadratic form description of the

reduced Whitehead group of a biquaternion algebra, making use of symplectic in-

volutions. Analogues of the reduced Whitehead group for algebras with involution

are also discussed.

§15. Exceptional Isomorphisms

The exceptional isomorphisms between groups of similitudes of central simple

algebras with involution in characteristic di¬erent from 2 are easily derived from

Wonenburger™s theorem (??), as the following proposition shows:

(15.1) Proposition. Let (A, σ) be a central simple algebra with orthogonal invo-

lution over a ¬eld F of characteristic di¬erent from 2. If 2 < deg A ¤ 6, the

canonical homomorphism of (??):

C : PGO(A, σ) ’ AutF C(A, σ), σ

is an isomorphism.

Proof : Proposition (??) shows that if deg A > 2, then C is injective and its image

consists of the automorphisms of C(A, σ), σ which preserve the image c(A) of A

under the canonical map c. Moreover, (??) shows that c(A) = F • c(A)0 where

c(A)0 = c(A) © Skew C(A, σ), σ .

Therefore, it su¬ces to show that every automorphism of C(A, σ), σ preserves

c(A)0 if deg A ¤ 6.

205

206 IV. ALGEBRAS OF DEGREE FOUR

From (??) (or (??)), it follows that dim c(A)0 = dim Skew(A, σ). Direct com-

putations show that

dimF Skew(A, σ) = dimF Skew C(A, σ), σ

if deg A = 3, 4, 5; thus c(A)0 = Skew C(A, σ), σ in these cases, and every auto-

morphism of C(A, σ), σ preserves c(A)0 .

If deg A = 6 we get dimF Skew C(A, σ), σ = 16 while dimF c(A)0 = 15.

However, the involution σ is unitary; if Z is the center of C(A, σ), there is a

canonical decomposition

0

Skew C(A, σ), σ = Skew(Z, σ) • Skew C(A, σ), σ

where

0

Skew C(A, σ), σ = { u ∈ Skew(C(A, σ), σ) | TrdC(A,σ) (u) = 0 }.

Inspection of the split case shows that TrdC(A,σ) (x) = 0 for all x ∈ c(A)0 . Therefore,

by dimension count,

0

c(A)0 = Skew C(A, σ), σ .

0

Since Skew C(A, σ), σ is preserved under every automorphism of C(A, σ), σ ,

the proof is complete.

This proposition relates central simple algebras with orthogonal involutions of

degree n = 3, 4, 5, 6 with their Cli¬ord algebra. We thus get relations between:

central simple F -algebras

quaternion F -algebras

of degree 3 ←’

with symplectic involution

with orthogonal involution

central simple F -algebras quaternion algebras with

of degree 4 symplectic involution over an

←’

with orthogonal involution ´tale quadratic extension of F

e

central simple F -algebras central simple F -algebras

of degree 5 of degree 4

←’

with orthogonal involution with symplectic involution

central simple algebras

central simple F -algebras

of degree 4

of degree 6 ←’

with unitary involution over an

with orthogonal involution

´tale quadratic extension of F

e

In order to formalize these relations,23 we introduce various groupoids whose objects

are the algebras considered above. The groupoid of central simple F -algebras of

degree 2n + 1 with orthogonal involution has already been considered in §??, where

23 Inthe cases n = 4 and n = 6, the relation also holds if the ´tale quadratic extension is

e

F — F ; central simple algebras of degree d over F — F should be understood as products B 1 — B2

of central simple F -algebras of degree d. Similarly, quaternion algebras over F — F are de¬ned as

products Q1 — Q2 of quaternion F -algebras.

§15. EXCEPTIONAL ISOMORPHISMS 207

it is denoted Bn . In order to extend the relations above to the case where char F = 2,

we replace it by the category Bn of oriented quadratic spaces of dimension 2n + 1:

see (??). If char F = 2, we de¬ne an orientation on an odd-dimensional nonsingular

quadratic space (V, q) of trivial discriminant as in the case where char F = 2:

an orientation of (V, q) is a central element ζ ∈ C1 (V, q) such that ζ 2 = 1. If

char F = 2, the orientation is unique, hence the category of oriented quadratic

spaces is isomorphic to the category of quadratic spaces of trivial discriminant.

We thus consider the following categories, for an arbitrary ¬eld F :

- A1 is the category of quaternion F -algebras, where the morphisms are the

F -algebra isomorphisms;

- A2 is the category of quaternion algebras over an ´tale quadratic extension

e

1

of F , where the morphisms are the F -algebra isomorphisms;

- An , for an arbitrary integer n ≥ 2, is the category of central simple algebras

of degree n + 1 over an ´tale quadratic extension of F with involution of

e

the second kind leaving F elementwise invariant, where the morphisms are

the F -algebra isomorphisms which preserve the involutions;

- Bn , for an arbitrary integer n ≥ 1, is the category of oriented quadratic

spaces of dimension 2n + 1, where the morphisms are the isometries which

preserve the orientation (if char F = 2, every isometry preserves the orien-

tation, since the orientation is unique);

- Cn , for an arbitrary integer n ≥ 1, is the category of central simple F -

algebras of degree 2n with symplectic involution, where the morphisms are

the F -algebra isomorphisms which preserve the involutions;

- Dn , for an arbitrary integer n ≥ 2, is the category of central simple F -

algebras of degree 2n with quadratic pair, where the morphisms are the

F -algebra isomorphisms which preserve the quadratic pairs.

In each case, maps are isomorphisms, hence these categories are groupoids.

Note that there is an isomorphism of groupoids:

A1 = C 1

which follows from the fact that each quaternion algebra has a canonical symplectic

involution.

In the next sections, we shall successively establish equivalences of groupoids:

B1 ≡ C 1

D2 ≡ A 2

1

B2 ≡ C 2

D3 ≡ A 3 .

In each case, it is the Cli¬ord algebra construction which provides the functors

de¬ning these equivalences from the left-hand side to the right-hand side. Not

surprisingly, one will notice deep analogies between the ¬rst and the third cases, as

well as between the second and the fourth cases.

Our proofs do not rely on (??), and indeed provide an alternative proof of that

proposition.

15.A. B1 ≡ C1 . In view of the isomorphism A1 = C1 , it is equivalent to prove

A1 ≡ B 1 .

208 IV. ALGEBRAS OF DEGREE FOUR

For every quaternion algebra Q ∈ A1 , the vector space

Q0 = { x ∈ Q | TrdQ (x) = 0 }

has dimension 3, and the squaring map s : Q0 ’ F de¬ned by

s(x) = x2 for x ∈ Q0

is a canonical quadratic form of discriminant 1. Moreover, the inclusion Q0 ’ Q

induces an orientation π on the quadratic space (Q0 , s), as follows: by the universal

property of Cli¬ord algebras, this inclusion induces a homomorphism of F -algebras

hQ : C(Q0 , s) ’ Q.

Since dim C(Q0 , s) = 8, this homomorphism has a nontrivial kernel; in the center of

C(Q0 , s), there is a unique element π ∈ C1 (Q0 , s) such that π 2 = 1 which is mapped

to 1 ∈ Q. This element π is an orientation on (Q0 , s). Explicitly, if (1, i, j, k) is a

quaternion basis of Q in characteristic di¬erent from 2, the orientation π ∈ C(Q0 , s)

is given by π = i · j · k ’1 . If (1, u, v, w) is a quaternion basis of Q in characteristic 2,

the orientation is π = 1 (the image of 1 ∈ Q0 in C(Q0 , s), not the unit of C(Q0 , s)).

We de¬ne a functor

P : A1 ’ B1

by mapping Q ∈ A1 to the oriented quadratic space (Q0 , s, π) ∈ B1 .

A functor C in the opposite direction is provided by the even Cli¬ord algebra

construction: we de¬ne

C : B1 ’ A 1

by mapping every oriented quadratic space (V, q, ζ) ∈ B1 to C0 (V, q) ∈ A1 .

(15.2) Theorem. The functors P and C de¬ne an equivalence of groupoids

A1 ≡ B 1 .

Proof : For any quaternion algebra Q, the homomorphism hQ : C(Q0 , s) ’ Q in-

duced by the inclusion Q0 ’ Q restricts to a canonical isomorphism

∼

hQ : C0 (Q0 , s) ’ Q.

’

We thus have a natural transformation: C —¦ P ∼ IdA1 .

=

For (V, q, ζ) ∈ B1 , we de¬ne a bijective linear map mζ : V ’ C0 (V, q)0 by

mζ (v) = vζ for v ∈ V .

Since s(vζ) = (vζ)2 = v 2 = q(v), this map is an isometry:

∼

mζ : (V, q) ’ C0 (V, q)0 , s .

’

We claim that mζ carries ζ to the canonical orientation π on C0 (V, q)0 , s ; this

map therefore yields a natural transformation P —¦ C ∼ IdB1 which completes the

=

proof.

To prove the claim, it su¬ces to consider the case where char F = 2, since the

orientation is unique if char F = 2. Therefore, for the rest of the proof we assume

char F = 2. The isometry mζ induces an isomorphism

∼

mζ : C(V, q) ’ C C0 (V, q)0 , s

’

§15. EXCEPTIONAL ISOMORPHISMS 209

which maps ζ to π or ’π. Composing this isomorphism with the homomorphism

hC0 (V,q) : C C0 (V, q)0 , s ’ C0 (V, q) induced by the inclusion C0 (V, q)0 ’ C0 (V, q),

we get a homomorphism

hC0 (V,q) —¦ mζ : C(V, q) ’ C0 (V, q)

which carries v ∈ V to vζ ∈ C0 (V, q). Since ζ has the form ζ = v1 · v2 · v3 for a

suitable orthogonal basis (v1 , v2 , v3 ) of V , we have

hC0 (V,q) —¦ mζ (ζ) = (v1 ζ)(v2 ζ)(v3 ζ) = ζ 4 = 1.

Since the orientation π on C0 (V, q)0 , s is characterized by the condition

hC0 (V,q) (π) = 1,

it follows that mζ (ζ) = π.

(15.3) Corollary. For every oriented quadratic space (V, q, ζ) of dimension 3, the

Cli¬ord algebra construction yields a group isomorphism

∼

O+ (V, q) = Aut(V, q, ζ) ’ AutF C0 (V, q) = PGSp C0 (V, q), „

’

where „ is the canonical involution on C0 (V, q).

Proof : Since the functor C de¬nes an equivalence of groupoids, it induces isomor-

phisms between the automorphism groups of corresponding objects.

By combining Theorem (??) with (??) (in characteristic di¬erent from 2), we

obtain an equivalence between A1 and B1 :