2 3

D C(A, σ, f ), σ , σ, fD = EndF U •( U — U ) , σq , f q .

In order to de¬ne an isomorphism of this algebra with (A, σ, f ), it su¬ces, by (??),

to de¬ne a similitude of quadratic spaces

2 3

g : (U — • U, qU ) ’ U •( U — U ), q .

224 IV. ALGEBRAS OF DEGREE FOUR

For • ∈ U — and u ∈ U , we set

g(• + u) = θ + µ — u,

2

where θ ∈ U is such that µ · •, x = θ § x for all x ∈ U . We then have

µ — µ · •, u = µ — (u § θ) = q(θ + µ — u),

hence g is an isometry of quadratic spaces. We claim that the inverse of the induced

isomorphism

2 3

∼

g— : EndF (U — • U ), σq ’ EndF

’ U •( U — U ) , σb

extends ». To prove the claim, we use the identi¬cations

2 3

D C(A, σ, f ), σ = EndF U •( U — U)

2 3 2 3

= U •( U — U) — U •( U — U)

and

A = EndF (U — • U ) = (U — • U ) — (U — • U ).

Since c(A) = (U — • U ) · (U — • U ) ‚ C0 H(U ), qU is spanned by elements of the

form (• + u) · (ψ + v) with •, ψ ∈ U — and u, v ∈ U , it su¬ces to show that

the corresponding element (d• + u ) —¦ (dψ + v ) ∈ EndF ( 0 U ) — EndF ( 1 U ) =

C0 H(U ), qU (under the isomorphism of (??)) is »— g(• + u) — g(ψ + v) . This

amounts to verifying that

™

= tr »2 h —¦ g(• + u) — g(ψ + v)

tr h —¦ (d• + u) —¦ (dψ + v)

for all h ∈ EndF ( U ) — EndF ( U ). Details are left to the reader.

0 1

(15.25) Remark. For (B, „ ) ∈ A3 , the Lie isomorphism »— : c D(B, „ ) ’ s(B, „ )

∼

restricts to a Lie isomorphism c D(B, „ ) 0 ’ Skew(B, „ )0 . The inverse of this

’

™

1

isomorphism is 2 c —¦ »2 if char F = 2. Similarly, for (A, σ, f ) ∈ D3 , the inverse of the

™

Lie isomorphism » used in the proof of the theorem above is »2 —¦ 1 c if char F = 2.

2

(15.26) Corollary. For every central simple algebra A of degree 6 with quadratic

pair (σ, f ), the functor C induces an isomorphism of groups

∼

PGO(A, σ, f ) = AutF (A, σ, f ) ’ AutF C(A, σ, f ), σ

’

which restricts into an isomorphism of groups:

∼

PGO+ (A, σ, f ) ’ AutZ(A,σ,f ) C(A, σ, f ), σ = PGU C(A, σ, f ), σ

’

where Z(A, σ, f ) is the center of C(A, σ, f ).

Proof : The ¬rst isomorphism follows from the fact that C de¬nes an equivalence

of groupoids D3 ’ A3 (see (??)). Under this isomorphism, the proper similitudes

correspond to automorphisms of C(A, σ, f ) which restrict to the identity on the

center Z(A, σ, f ), by (??).

We thus recover the fourth case (deg A = 6) of (??).

§15. EXCEPTIONAL ISOMORPHISMS 225

Cli¬ord groups. Let (A, σ, f ) ∈ D3 and (B, „ ) ∈ A3 . Let K be the center

of (B, „ ), and assume that (A, σ, f ) and (B, „ ) correspond to each other under the

groupoid equivalence A3 ≡ D3 , so that we may identify (B, „ ) = C(A, σ, f ), σ and

(A, σ, f ) = D(B, „ ), „ , fD . Our goal is to relate the Cli¬ord groups of (A, σ, f ) to

groups of similitudes of (B, „ ). We write µσ and µ„ for the multiplier maps for the

involutions σ and „ respectively.

(15.27) Proposition. The extended Cli¬ord group of (A, σ, f ) is the group of

similitudes of (B, „ ), i.e.,

„¦(A, σ, f ) = GU(B, „ ),

and the following diagram commutes:

χ

„¦(A, σ, f ) ’ ’ ’ PGO+ (A, σ, f )

’’

¦

¦

(15.28)

D

GU(B, „ ) ’ ’ ’

’’ Aut(A, σ, f )

where D is the canonical map of §??. The Cli¬ord group of (A, σ, f ) is

“(A, σ, f ) = SGU(B, „ ) = { g ∈ GU(B, „ ) | NrdB (g) = µ„ (g)2 }

and the following diagram commutes:

χ

“(A, σ, f ) ’ ’ ’ O+ (A, σ, f )

’’

¦

¦

(15.29)

»

SGU(B, „ ) ’ ’ ’ O(A, σ, f )

’’

where » (g) = µ„ (g)’1 »2 g ∈ D(B, „ ) = A for g ∈ SGU(B, „ ). Moreover,

Spin(A, σ, f ) = SU(B, „ ) = { g ∈ GU(B, „ ) | NrdB (g) = µ„ (g) = 1 }.

Proof : Since (??) shows that the canonical map

C : PGO+ (A, σ, f ) ’ AutK (B, „ )

is surjective, it follows from the de¬nition of the extended Cli¬ord group in (??) that

„¦(A, σ, f ) = GU(B, „ ). By the de¬nition of χ , the following diagram commutes:

χ

„¦(A, σ, f ) ’ ’ ’ PGO+ (A, σ, f )

’’

¦

¦

C

Int

GU(B, „ ) ’ ’ ’

’’ AutK (B, „ ).

The commutativity of (??) follows, since the inverse of

∼

C : PGO+ (A, σ, f ) ’ AutK (B, „ )

’

is given by the canonical map D; indeed, the groupoid equivalence A3 ≡ D3 is given

by the Cli¬ord and discriminant algebra constructions.

To identify “(A, σ, f ), it su¬ces to prove that the homomorphism

κ : „¦(A, σ, f ) ’ K — /F —

whose kernel is “(A, σ, f ) (see (??)) coincides with the homomorphism

ν : GU(B, „ ) ’ K — /F —

226 IV. ALGEBRAS OF DEGREE FOUR

whose kernel is SGU(B, „ ) (see (??)). The description of Spin(A, σ, f ) also follows,

since Spin(A, σ, f ) = “(A, σ, f )©U(B, „ ). The following lemma therefore completes

the proof:

(15.30) Lemma. Diagram (??) and the following diagram are commutative:

κ

„¦(A, σ, f ) ’ ’ ’ K — /F —

’’

ν

GU(B, „ ) ’ ’ ’ K — /F — .

’’

Proof : It su¬ces to prove commutativity of the diagrams over a scalar extension.

We may thus assume that the base ¬eld F is algebraically closed.

4

Let V be a 4-dimensional vector space over F . Pick a nonzero element e ∈ V

4 2 4

to identify V = F and view the canonical quadratic map q : V’ V

of (??) as a quadratic form. Since F is algebraically closed, we have (A, σ, f )

2

EndF ( V ), σq , fq where (σq , fq ) is the quadratic pair associated with q. We ¬x

such an isomorphism and use it to identify until the end of the proof

2

(A, σ, f ) = EndF ( V ), σq , fq .

2

V ’ EndF (V • V — ) de¬ned in (??) induces an isomorphism

The map i :

2 ∼

V, q) ’ EndF (V • V — )

i— : C( ’

2

which identi¬es the Cli¬ord algebra B = C(A, σ, f ) = C0 ( V, q) with EndF (V ) —

EndF (V — ). The involution „ is then given by

t t

„ (f1 , f2 ) = (f2 , f1 )

for f1 , f2 ∈ EndF (V ). Therefore,

GU(B, „ ) = { f, ρ(f ’1 )t | ρ ∈ F — , f ∈ GL(V ) }.

For g = f, ρ(f ’1 )t ∈ GU(B, „ ), we consider

2

V ) = A and γ = f 2 , det f (f ’2 )t ∈ GU(B, „ ).

f § f ∈ EndF (

A computation shows that

2 2

f § f ∈ GO+ ( V, q) = GO+ (A, σ, f ), γ ∈ “+ ( V, q) = “(A, σ, f )

and moreover

χ(γ) = µ(f § f )’1 (f § f )2 .

Int(g) = C(f § f ),

Therefore, κ(g) = z · F — where z = 1, ρ2 (det f )’1 ∈ K — = F — — F — is such that

g 2 = z · γ. On the other hand, we have

µ„ (g)’2 NrdB (g) = ρ’2 det f, ρ2 (det f )’1 = zι(z)’1 ,

hence ν(g) = z · F — = κ(g).

It remains only to prove the commutativity of (??). Since ν = κ, we have

“(A, σ, f ) = SGU(B, „ ). Therefore, every element in “(A, σ, f ) has the form

f, ρ(f ’1 )t for some ρ ∈ F — and some f ∈ GL(V ). A computation yields

f ’1

f 0 0 2

= ρ’1 i— f § f (ξ)

· i— (ξ) · for ξ ∈ V,

’1 t ’1 t

0 ρ(f ) 0 ρ f

hence χ f, ρ(f ’1 )t = ρ’1 f § f and (??) commutes.

§15. EXCEPTIONAL ISOMORPHISMS 227

(15.31) Corollary. For (A, σ, f ) ∈ D3 , the group of multipliers of proper simili-

tudes and the group of spinor norms of (A, σ, f ) are given by

G+ (A, σ, f ) = { NK/F (z) | zι(z)’1 = µ„ (g)’2 NrdB (g) for some g ∈ GU(B, „ ) }

and

Sn(A, σ, f ) = { µ„ (g) | g ∈ SGU(B, „ ) }.

Proof : In view of the description of „¦(A, σ, f ) and κ above, it follows from (??)

that µσ PGO+ (A, σ, f ) = NK/F —¦ ν GU(B, „ ) , proving the ¬rst relation. The

second relation follows from the description of “(A, σ, f ) in (??).

The case of trivial discriminant. If K/F is a given ´tale quadratic exten-

e

sion, the functors D and C of (??) relate algebras with involution (B, „ ) ∈ A3 with

center K and algebras with involutions (A, σ, f ) ∈ D3 whose Cli¬ord algebra has

center Z(A, σ, f ) K. In order to make explicit the special case where K = F — F ,

let 1A3 be the full subgroupoid of A3 whose objects are algebras of degree 4 over

F — F with involution of the second kind and let 1 D3 be the full subgroupoid of D3

whose objects are algebras of degree 6 with quadratic pair of trivial discriminant.

Every (B, „ ) ∈ 1A3 is isomorphic to an algebra of the form (E — E op , µ) where E is

a central simple F -algebra of degree 4 and µ is the exchange involution, hence 1A3

is also equivalent to the groupoid of algebras of the form (E — E op , µ). Since

D(E — E op , µ), µ, fD = (»2 E, γ, f )

where (γ, f ) is the canonical quadratic pair on »2 E (see (??) if char F = 2), the

following is a special case of (??):

(15.32) Corollary. The Cli¬ord algebra functor C : 1 D3 ’ 1A3 and the functor

D : 1A3 ’ 1 D3 , which maps E — E op , µ to (»2 E, γ, f ), de¬ne an equivalence of

groupoids

1

A3 ≡ 1 D 3 .

In particular, for every central simple F -algebra E of degree 4,

C(»2 E, γ, f ) (E — E op , µ).

Observe that the maps in 1A3 are isomorphisms of algebras over F , not over

F — F . In particular, E — E op , µ and E op — E, µ are isomorphic in 1A3 , under

the map which interchanges the two factors. Therefore, 1A3 is not equivalent to

the groupoid of central simple F -algebras of degree 4 where the maps are the F -

algebra isomorphisms. There is however a correspondence between isomorphism

classes which we now describe.

For (A, σ, f ) ∈ 1 D3 , the Cli¬ord algebra C(A, σ, f ) decomposes into a direct

product

C(A, σ, f ) = C + (A, σ, f ) — C ’ (A, σ, f )

for some central simple F -algebras C + (A, σ, f ), C ’ (A, σ, f ) of degree 4. The

fundamental relations (??) and (??) show that C + (A, σ, f ) C ’ (A, σ, f )op and

C + (A, σ, f )—2 C ’ (A, σ, f )—2 ∼ A.

If (V, q) is a quadratic space of dimension 6 and trivial discriminant, we also

let C ± (V, q) denote C ± EndF (V ), σq , fq . The algebras C + (V, q) and C ’ (V, q) are

isomorphic central simple F -algebras of degree 4 and exponent 2.

228 IV. ALGEBRAS OF DEGREE FOUR

(15.33) Corollary. Every central simple F -algebra of degree 4 and exponent 2 is

of the form C ± (V, q) for some quadratic space (V, q) of dimension 6 and trivial

discriminant, uniquely determined up to similarity.

Every central simple F -algebra of degree 4 and exponent 4 is of the form

C (A, σ, f ) for some (A, σ, f ) ∈ 1 D3 such that ind A = 2, uniquely determined

±

up to isomorphism.

Proof : For every central simple F -algebra E of degree 4, we have E — E op

C + (»2 E, γ, f ) — C ’ (»2 E, γ, f ) by (??), hence

C ± (»2 E, γ, f ).

E

Since »2 E is Brauer-equivalent to E —2 , it is split if E has exponent 2 and has

C ± (A, σ, f ) for some

index 2 if E has exponent 4, by (??). Moreover, if E

(A, σ, f ) ∈ 1 D3 , then (E — E op , µ) C(A, σ, f ), σ since all involutions on E — E op

are isomorphic to the exchange involution. Therefore, by (??), we have

D(E — E op , µ), µ, fD (»2 E, γ, f ).

(A, σ, f )

To complete the proof, observe that when A = EndF (V ) we have (σ, f ) = (σq , fq )

for some quadratic form q, and the quadratic space (V, q) is determined up to

similarity by the algebra with quadratic pair (A, σ, f ) by (??).

Corollaries (??) and (??), and Proposition (??), can also be specialized to the

case where the discriminant of (A, σ, f ) is trivial. In particular, (??) simpli¬es

remarkably:

(15.34) Corollary. Let (A, σ, f ) ∈ 1 D3 and let C(A, σ, f ) E — E op . The multi-

pliers of similitudes of (A, σ, f ) are given by

G(A, σ, f ) = G+ (A, σ, f ) = F —2 · NrdE (E — )

and the spinor norms of (A, σ, f ) by

Sn(A, σ, f ) = { ρ ∈ F — | ρ2 ∈ NrdE (E — ) }.

Proof : The equality G(A, σ, f ) = G+ (A, σ, f ) follows from the hypothesis that

disc(A, σ, f ) is trivial by (??). Since the canonical involution σ on C(A, σ, f ) is the

exchange involution, we have under the identi¬cation C(A, σ, f ) = E — E op that

x, ρ(x’1 )op ρ ∈ F —, x ∈ E— ,

GU C(A, σ, f ), σ =

and, for g = x, ρ(x’1 )op ,

µ(g)’2 NrdC(A,σ,f ) (g) = ρ’2 NrdE (x), ρ2 NrdE (x)’1 = zι(z)’1

with z = NrdE (x), ρ2 ∈ Z(A, σ, f ) = F — F . Since NZ(A,σ,f )/F (z) = ρ2 NrdE (x),

Corollary (??) yields the equality G+ (A, σ, f ) = F —2 · NrdE (E — ). Finally, we have

by (??):

“(A, σ, f ) = SGU C(A, σ, f ), σ

= { x, ρ(x’1 )op ∈ GU C(A, σ, f ), σ | ρ2 = NrdE (x) },

hence

= { ρ ∈ F — | ρ2 ∈ NrdE (E — ) }.

Sn(A, σ, f ) = µ SGU C(A, σ, f ), σ

§15. EXCEPTIONAL ISOMORPHISMS 229

Examples. In this subsection, we explicitly determine the algebra with invo-

lution (B, „ ) ∈ A3 corresponding to (A, σ, f ) ∈ D3 when the quadratic pair (σ, f ) is

isotropic. Since the correspondence is bijective, our computations also yield infor-

mation on the discriminant algebra of some (B, „ ) ∈ A3 , which will be crucial for

relating the indices of (A, σ, f ) and (B, „ ) in (??) and (??) below.

(15.35) Example. Let (A, σ, f ) = EndF (V ), σq , fq where (V, q) is a 6-dimen-

sional quadratic space over a ¬eld F of characteristic di¬erent from 2, and suppose

q is isotropic. Suppose that disc q = ± · F —2 , so that the center of C0 (V, q) is

isomorphic to F [X]/(X 2 ’ ±). Then multiplying q by a suitable scalar, we may

assume that q has a diagonalization of the form

q = 1, ’1, ±, ’β, ’γ, βγ

for some β, γ ∈ F — . Let (e1 , . . . , e6 ) be an orthogonal basis of V which yields

the diagonalization above. In C0 (V, q), the elements e1 · e4 and e1 · e5 generate a

quaternion algebra (β, γ)F . The elements e1 · e4 · e5 · e6 and e1 · e4 · e5 · e2 centralize

this algebra and generate a split quaternion algebra (βγ)2 , ’βγ F ; therefore,

M2 (β, γ)F — F [X]/(X 2 ’ ±)

C0 (V, q)

by the double centralizer theorem (see (??)), and Proposition (??) shows that the

canonical involution „0 on C0 (V, q) is hyperbolic.

There is a corresponding result in characteristic 2: if the nonsingular 6-dimen-

sional quadratic space (V, q) is isotropic, we may assume (after scaling) that

q = [0, 0] ⊥ [1, ± + β] ⊥ γ [1, β] = [0, 0] ⊥ [1, ± + β] ⊥ [γ, βγ ’1 ]

for some ±, β ∈ F , γ ∈ F — . Thus, disc q = ± + „˜(F ), hence the center of C0 (V, q) is

isomorphic to F [X]/(X 2 + X + ±). Let (e1 , . . . , e6 ) be a basis of V which yields the

decomposition above. In C0 (V, q), the elements r = (e1 +e2 )·e3 and s = (e1 +e2 )·e4

satisfy r2 = 1, s2 = ± + β and rs + sr = 1, hence they generate a split quaternion

algebra 1, ± + β F (see §??). The elements (e1 +e2 )·e4 and (e1 +e2 )·e5 centralize

this algebra and generate a quaternion algebra γ, βγ ’1 F [β, γ)F . Therefore,

M2 [β, γ)F — F [X]/(X 2 + X + ±).

C0 (V, q)

As above, Proposition (??) shows that the canonical involution „0 is hyperbolic.

(15.36) Corollary. Let (B, „ ) ∈ A3 , with „ hyperbolic.

(1) Suppose the center Z(B) is a ¬eld, hence B is Brauer-equivalent to a quaternion

algebra (so ind B = 1 or 2); then the discriminant algebra D(B, „ ) splits and its

canonical quadratic pair („ , fD ) is associated with an isotropic quadratic form q.

The Witt index of q is 1 if ind B = 2; it is 2 if ind B = 1.

(2) Suppose Z(B) F — F , so that B E — E op for some central simple F -algebra

E of degree 4. If E is Brauer-equivalent to a quaternion algebra, then D(B, „ ) splits

and its canonical quadratic pair („ , fD ) is associated with an isotropic quadratic

form q. The Witt index of q is 1 if ind E = 2; it is 2 if ind E = 1.

Proof : (??) Since B has an involution of the second kind, Proposition (??) shows

that the Brauer-equivalent quaternion algebra has a descent to F . We may thus

assume that

√

for some ±, β, γ ∈ F — if char F = 2,

M2 (β, γ)F — F ( ±)

B

M2 [β, γ)F — F „˜’1 (±) for some ±, β ∈ F , γ ∈ F — if char F = 2,

230 IV. ALGEBRAS OF DEGREE FOUR

hence, by (??), (B, „ ) C0 (V, q), „0 where

1, ’1, ±, ’β, ’γ, βγ if char F = 2;

q

[0, 0] ⊥ [1, ± + β] ⊥ γ [1, β] if char F = 2.

If w(V, q) = 2, then Corollary (??) shows that 1 ∈ ind C0 (V, q), „0 , hence B is

split. Conversely, if B is split, then we may assume that γ = 1, and it follows that

w(V, q) = 2.

(??) The hypothesis yields

M2 (β, γ)F — F [X]/(X 2 ’ 1) for some β, γ ∈ F — if char F = 2,

B

M2 [β, γ)F — F [X]/(X 2 ’ X) for some β ∈ F , γ ∈ F — if char F = 2,

hence (B, „ ) C0 (V, q), „0 where

1, ’1, 1, ’β, ’γ, βγ if char F = 2;

q

[0, 0] ⊥ 1, γ [1, β] if char F = 2.

Since q is the orthogonal sum of a hyperbolic plane and the norm form of the

quaternion algebra Brauer-equivalent to E, we have w(V, q) = 1 if and only if

ind E = 2.

We next consider the case where the algebra A is not split. Since deg A = 6,

we must have ind A = 2, by (??). We write Z(A, σ, f ) for the center of the Cli¬ord

algebra C(A, σ, f ).

(15.37) Proposition. Let (A, σ, f ) ∈ D3 with ind A = 2.

(1) If the quadratic pair (σ, f ) is isotropic, then Z(A, σ, f ) is a splitting ¬eld of A.

(2) For each separable quadratic splitting ¬eld Z of A, there is, up to conjuga-

Z. If d ∈ F — is

tion, a unique quadratic pair (σ, f ) on A such that Z(A, σ, f )

such that the quaternion algebra Brauer-equivalent to A has the form (Z, d) F , then

C(A, σ, f ) M4 (Z) and the canonical involution σ is the adjoint involution with re-

spect to the 4-dimensional hermitian form on Z with diagonalization 1, ’1, 1, ’d .

Proof : (??) Let I ‚ A be a nonzero isotropic right ideal. We have rdim I ≥

1

2 deg A = 3, hence rdim I = 2 since ind A divides the reduced dimension of every

right ideal. Let e be an idempotent such that I = eA. As in the proof of (??),

we may assume that eσ(e) = σ(e)e = 0, hence e + σ(e) is an idempotent. Let

e1 = e + σ(e) and e2 = 1 ’ e1 ; then e1 A = eA • σ(e)A, hence rdim e1 A = 4

and therefore rdim e2 A = 2. Let Ai = ei Aei and let (σi , fi ) be the restriction of

the quadratic pair (σ, f ) to Ai for i = 1, 2. By (??), we have deg A1 = 4 and

deg A2 = 2, hence A2 is a quaternion algebra Brauer-equivalent to A. Moreover,

by (??) (if char F = 2) or (??) (if char F = 2),

disc(σ1 , f1 ) disc(σ2 , f2 ) if char F = 2,

disc(σ, f ) =

disc(σ1 , f1 ) + disc(σ2 , f2 ) if char F = 2.

Since eAe1 is an isotropic right ideal of reduced dimension 2 in A1 , the quadratic

pair (σ1 , f1 ) is hyperbolic, and Proposition (??) shows that its discriminant is

trivial. Therefore, disc(σ, f ) = disc(σ2 , f2 ), hence Z(A, σ, f ) Z(A2 , σ2 , f2 ). If

char F = 2, it was observed in (??) that Z(A2 , σ2 , f2 ) splits A2 , hence Z(A, σ, f )

splits A. To see that the same property holds if char F = 2, pick ∈ A2 such

that f2 (s) = TrdA2 ( s) for all s ∈ Sym(A2 , σ2 ); then TrdA2 ( ) = 1 and SrdA2 ( ) =

§15. EXCEPTIONAL ISOMORPHISMS 231

NrdA2 ( ) represents disc(σ2 , f2 ) in F/„˜(F ), so Z(A2 , σ2 , f2 ) F ( ). This completes

the proof of (??).

(??) Let Z be a separable quadratic splitting ¬eld of A and let d ∈ F — be such

that A is Brauer-equivalent to the quaternion algebra (Z, d)F , which we denote

simply by Q. We then have A M3 (Q). To prove the existence of an isotropic

quadratic pair (σ, f ) on A such that Z(A, σ, f ) Z, start with a quadratic pair

(θ, f1 ) on Q such that Z(Q, θ, f1 ) Z, and let (σ, f ) = (θ — ρ, f1— ) on A = Q —

M3 (F ), where ρ is the adjoint involution with respect to an isotropic 3-dimensional

bilinear form. We may choose for instance ρ = Int(u) —¦ t where

«

010

u = 1 0 0 ;

001

the involution σ is then explicitly de¬ned by

«

θ(x22 ) θ(x12 ) θ(x32 )