every quaternion algebra Q to the algebra with involution EndF (Q0 ), σs where σs

is the adjoint involution with respect to s, and the functor C : B1 ’ A1 , which maps

every algebra with involution (A, σ) of degree 3 to the quaternion algebra C(A, σ),

de¬ne an equivalence of groupoids:

A1 ≡ B 1 .

In particular, for every central simple algebra with involution (A, σ) of degree 3,

the functor C induces an isomorphism of groups:

∼

O+ (A, σ) = PGO(A, σ) = AutF (A, σ) ’ AutF C(A, σ) = PGSp C(A, σ), σ .

’

We thus recover the ¬rst case (deg A = 3) of (??).

Indices. Let Q ∈ A1 and (V, q, ζ) ∈ B1 correspond to each other under the

(Q0 , s) and Q

equivalence A1 ≡ B1 , so that (V, q) C0 (V, q). Since Q is

a quaternion algebra, its (Schur) index may be either 1 or 2; for the canonical

involution γ on Q, the index ind(Q, γ) is thus (respectively) either {0, 1} or {0}.

On the other hand, since dim V = 3, the Witt index w(V, q) is 1 whenever q is

isotropic.

The following correspondence between the various cases is well-known:

(15.5) Proposition. The indices of Q, (Q, γ) and (V, q) are related as follows:

ind Q = 2 ⇐’ ind(Q, γ) = {0} ⇐’ w(V, q) = 0;

ind Q = 1 ⇐’ ind(Q, γ) = {0, 1} ⇐’ w(V, q) = 1.

In other words, Q is a division algebra if and only if q is anisotropic, and Q is split

if and only if q is isotropic.

210 IV. ALGEBRAS OF DEGREE FOUR

Proof : If Q is a division algebra, then Q0 does not contain any nonzero nilpotent

elements. Therefore, the quadratic form s, hence also q, is anisotropic. On the

other hand, q is isotropic if Q is split, since M2 (F ) contains nonzero matrices

whose square is 0.

15.B. A2 ≡ D2 . The Cli¬ord algebra construction yields a functor C : D2 ’

1

A2 . In order to show that this functor de¬nes an equivalence of groupoids, we ¬rst

1

describe a functor N: A2 ’ D2 which arises from the norm construction.

1

2

Let Q ∈ A1 be a quaternion algebra over some ´tale quadratic extension K/F .

e

(If K = F — F , then Q should be understood as a direct product of quaternion

F -algebras.) Let ι be the nontrivial automorphism of K/F . Recall from (??) that

NK/F (Q) is the F -subalgebra of ι Q —K Q consisting of elements ¬xed by the switch

map

s : ι Q —K Q ’ ι Q —K Q.

The tensor product ι γ — γ of the canonical involutions on ι Q and Q restricts to an

involution NK/F (γ) of the ¬rst kind on NK/F (Q). By (??), we have

ι

Q —K Q, ι γ — γ ,

NK/F (Q)K , NK/F (γ)K

hence NK/F (γ) has the same type as ι γ — γ. Proposition (??) thus shows that

NK/F (γ) is orthogonal if char F = 2 and symplectic if char F = 2. Corollary (??)

further yields a quadratic pair (ι γ — γ, f— ) on ι Q — Q, which is uniquely determined

by the condition that f— vanishes on Skew(ι Q, ι γ) —K Skew(Q, γ). It is readily seen

that (ι γ —γ, ι—¦f— —¦s) is a quadratic pair with the same property, hence ι—¦f— —¦s = f

and therefore

for all x ∈ Sym(ι Q — Q, ι γ — γ).

f— s(x) = ιf— (x)

It follows that f— (x) ∈ F for all x ∈ Sym NK/F (Q), NK/F (γ) , hence (ι γ — γ, f— )

restricts to a quadratic pair on NK/F (Q). We denote this quadratic pair by

NK/F (γ), fN . The norm thus de¬nes a functor

N : A2 ’ D 2

1

which maps Q ∈ A2 to N(Q) = NK/F (Q), NK/F (γ), fN where K is the center

1

of Q.

On the other hand, for (A, σ, f ) ∈ D2 the Cli¬ord algebra C(A, σ, f ) is a quater-

nion algebra over an ´tale quadratic extension, as the structure theorem (??) shows.

e

Therefore, the Cli¬ord algebra construction yields a functor

C : D2 ’ A 2 .

1

The key tool to show that N and C de¬ne an equivalence of categories is the Lie

algebra isomorphism which we de¬ne next. For Q ∈ A2 , consider the F -linear map

1

n : Q ’ NK/F (Q) de¬ned by

™

n(q) = ι q — 1 + ι 1 — q

™ for q ∈ Q.

This map is easily checked to be a Lie algebra homomorphism; it is in fact the

di¬erential of the group homomorphism n : Q— ’ NK/F (Q)— which maps q ∈ Q—

to ι q — q. We have the nonsingular F -bilinear form TNK/F (Q) on NK/F (Q) and the

nonsingular F -bilinear form on Q which is the transfer of TQ with respect to the

trace TK/F . Using these, we may form the adjoint linear map

n— : NK/F (Q) ’ Q

™

§15. EXCEPTIONAL ISOMORPHISMS 211

which is explicitly de¬ned as follows: for x ∈ NK/F (Q), the element n— (x) ∈ Q is

™

uniquely determined by the condition

TK/F TrdQ n— (x)y

™ = TrdNK/F (Q) xn(y)

™ for all y ∈ Q.

(15.6) Proposition. Let Q = { x ∈ Q | TrdQ (x) ∈ F }. The linear map n— ™

factors through the canonical map c : NK/F (Q) ’ c N(Q) and induces an isomor-

phism of Lie algebras

∼

n— : c N(Q) ’ Q .

™ ’

This isomorphism is the identity on F .

Proof : Suppose ¬rst that K F — F . We may then assume that Q = Q1 — Q2

for some quaternion F -algebras Q1 , Q2 , and NK/F (Q) = Q1 — Q2 . Under this

identi¬cation, the map n is de¬ned by

™

n(q1 , q2 ) = q1 — 1 + 1 — q2

™ for q1 ∈ Q1 and q2 ∈ Q2 .

It is readily veri¬ed that

n— (q1 — q2 ) = TrdQ2 (q2 )q1 , TrdQ1 (q1 )q2

™ for q1 ∈ Q1 and q2 ∈ Q2 ,

hence n— is the map ˜ of (??). From (??), it follows that n— factors through c and

™ ™

induces an isomorphism of Lie algebras

c N(Q) = c(Q1 — Q2 ) ’ Q = { (q1 , q2 ) ∈ Q1 — Q2 | TrdQ1 (q1 ) = TrdQ2 (q2 ) }.

For i = 1, 2, let i ∈ Qi be such that TrdQi ( i ) = 1. Then TrdQ1 —Q2 ( 1 — 2 ) = 1,

hence f 1 — 2 + γ1 ( 1 ) — γ2 ( 2 ) = 1, and therefore c 1 — 2 + γ1 ( 1 ) — γ2 ( 2 ) = 1.

On the other hand,

n—

™ — + γ1 ( 1 ) — γ2 ( 2 ) = ( 1 , 2) + γ1 ( 1 ), γ2 ( 2 ) = (1, 1),

1 2

hence n— maps 1 ∈ c N(Q) to 1 ∈ Q . The map n— thus restricts to the identity

™ ™

on F , completing the proof in the case where K F — F .

In the general case, it su¬ces to prove the proposition over an extension of the

base ¬eld F . Extending scalars to K, we are reduced to the special case considered

above, since K — K K — K.

(15.7) Theorem. The functors N and C de¬ne an equivalence of groupoids:

A2 ≡ D 2 .

1

Moreover, if Q ∈ A2 and (A, σ, f ) ∈ D2 correspond to each other under this equiv-

1

alence, then the center Z(Q) of Q satis¬es

F disc(A, σ, f ) if char F = 2;

Z(Q)

F „˜’1 disc(A, σ, f ) if char F = 2.

Proof : If the ¬rst assertion holds, then the quaternion algebra Q corresponding

to (A, σ, f ) ∈ D2 is the Cli¬ord algebra C(A, σ, f ), hence the description of Z(Q)

follows from the structure theorem for Cli¬ord algebras (??).

In order to prove the ¬rst statement, we establish natural transformations

N —¦ C ∼ IdD2 and C —¦ N ∼ IdA2 . Thus, for (A, σ, f ) ∈ D2 and for Q ∈ A2 , we

= = 1

1

have to describe canonical isomorphisms

(A, σ, f ) NZ(A,σ,f )/F C(A, σ, f ) , NZ(A,σ,f )/F (σ), fN

212 IV. ALGEBRAS OF DEGREE FOUR

and

Q C NZ(Q)/F (Q), NZ(Q)/F (γ), fN

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

Observe that the fundamental relation (??) between an algebra with invo-

lution and its Cli¬ord algebra already shows that there is an isomorphism A

NZ(A,σ,f )/F C(A, σ, f ) . However, we need a canonical isomorphism which takes

the quadratic pairs into account.

Our construction is based on (??): we use (??) to de¬ne isomorphisms of Lie

algebras and show that these isomorphisms extend to isomorphisms of associative

algebras over an algebraically closed extension, hence also over the base ¬eld.

Let (A, σ, f ) ∈ D2 and let

C(A, σ, f ) = { x ∈ C(A, σ, f ) | TrdC(A,σ,f ) (x) ∈ F }.

Lemma (??) shows that TrdC(A,σ,f ) c(a) = TrdA (a) for a ∈ A, hence c(A) ‚

C(A, σ, f ) , and dimension count shows that this inclusion is an equality. Propo-

sition (??) then yields a Lie algebra isomorphism n— : c N C(A, σ, f ) ’ c(A)

™

which is the identity on F . By (??), it follows that this isomorphism induces a Lie

algebra isomorphism

∼

n : Alt N C(A, σ, f ) ’ Alt(A, σ).

’

To prove that this isomorphism extends to an isomorphism of algebras with quad-

ratic pairs, it su¬ces by (??) to consider the split case. We may thus assume that A

is the endomorphism algebra of a hyperbolic quadratic space H(U ) of dimension 4.

Thus

A = EndF H(U ) = EndF (U — • U )

where U is a 2-dimensional vector space, U — is its dual, and (σ, f ) = (σqU , fqU ) is

the quadratic pair associated with the hyperbolic quadratic form on U — • U :

for • ∈ U — , u ∈ U .

qU (• + u) = •(u)

In that case, the Cli¬ord algebra C(A, σ, f ) can be described as

C(A, σ, f ) = C0 H(U ) = EndF ( U ) — EndF ( U ),

0 1

where 0 U (resp. 1 U ) is the 2-dimensional subspace of even- (resp. odd-) degree

elements in the exterior algebra of U (see (??)):

2

U =F • U, U = U.

0 1

Therefore,

NZ(A,σ,f )/F C(A, σ, f ) = EndF ( U— U ).

0 1

On the vector space 0 U — 1 U , we de¬ne a quadratic form q as follows: pick

2

a nonzero element (hence a basis) e ∈ U ; for x, y ∈ U , we may then de¬ne

q(1 — x + e — y) ∈ F by the equation

eq(1 — x + e — y) = x § y.

The associated quadratic pair (σq , fq ) on EndF ( 0 U — 1 U ) is the canonical

quadratic pair N (σ), fN (see Exercise ?? of Chapter ??). A computation shows

that the map g : H(U ) ’ 0 U — 1 U de¬ned by

g(• + u) = 1 — x + e — u,

§15. EXCEPTIONAL ISOMORPHISMS 213

where x ∈ U is such that x § y = e•(y) for all y ∈ U , is a similitude of quadratic

spaces

∼

g : H(U ) ’ (

’ U— U, q).

0 1

By (??), this similitude induces an isomorphism of algebras with quadratic pair

∼

g— : EndF H(U ) , σqU , fqU ’ EndF (

’ U— U ), σq , fq .

0 1

’1

We leave it to the reader to check that g— extends the Lie algebra homomorphism

n, completing the proof that n induces a natural transformation N —¦ C ∼ IdD2 .

=

We use the same technique to prove that C —¦ N ∼ IdA2 . For Q ∈ A1 , Proposi-

2

= 1

tion (??) yields a Lie algebra isomorphism

∼

n— : c N(Q) ’ Q .

™ ’

∼

To prove that n— extends to an isomorphism of F -algebras C N(Q) ’ Q, we

™ ’

may extend scalars, since N(Q) is generated as an associative algebra by c N(Q) .

Extending scalars to Z(Q) if this algebra is a ¬eld, we may therefore assume that

Z(Q) F — F . In that case Q Q1 — Q2 for some quaternion F -algebras Q1 , Q2 ,

hence NZ(Q)/F (Q) Q1 — Q2 , and n— is the map ˜ of (??), de¬ned by

™

˜ c(x1 — x2 ) = TrdQ2 (x2 )x1 , TrdQ1 (x1 )x2 for x1 ∈ Q1 , x2 ∈ Q2 .

Since it was proven in (??) that ˜ extends to an isomorphism of F -algebras

∼

C N(Q) = C(Q1 — Q2 , γ1 — γ2 , f— ) ’ Q1 — Q2 , the proof is complete.

’

(15.8) Remark. For Q ∈ A2 , the Lie isomorphism n— : c N(Q) ’ Q restricts to

™

1

∼ 0

an isomorphism c N(Q) 0 ’ Q . If char F = 2, the inverse of this isomorphism

’

1

is 2 c —¦ n (see Exercise ??). Similarly, for (A, σ, f ) ∈ D2 , the inverse of the Lie

™

™1

isomorphism n : Alt N C(A, σ, f ) ’ Alt(A, σ) is n —¦ 2 c if char F = 2.

(15.9) Corollary. For every central simple algebra A of degree 4 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 = AutF C(A, σ, f )

’

which restricts into an isomorphism of groups:

∼

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

’

PGSp C(A, σ, f ), σ = C(A, σ, f )— /Z(A, σ, f )— .

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

of groupoids D2 ’ A2 (see (??)). Under this isomorphism, the proper similitudes

1

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 second case (deg A = 4) of (??).

Cli¬ord groups. Let Q ∈ A2 and (A, σ, f ) ∈ D2 . Let Z be the center of Q,

1

and assume that Q and (A, σ, f ) correspond to each other under the groupoid

equivalence A2 ≡ D2 , so that we may identify Q = C(A, σ, f ) and (A, σ, f ) =

1

NZ/F (Q), NZ/F (γ), fN .

(15.10) Proposition. The extended Cli¬ord group of (A, σ, f ) is „¦(A, σ, f ) = Q —

and the canonical map χ0 : Q— ’ GO+ (A, σ, f ) of (??) is given by χ0 (q) = ι q —q ∈

NZ/F (Q) = A. For q ∈ Q— , the multiplier of χ0 (q) is µ χ0 (q) = NZ/F NrdQ (q) .

214 IV. ALGEBRAS OF DEGREE FOUR

The Cli¬ord group of (A, σ, f ) is

“(A, σ, f ) = { q ∈ Q— | NrdQ (q) ∈ F — },

and the vector representation map χ : “(A, σ, f ) ’ O+ (A, σ, f ) is given by

χ(q) = NrdQ (q)’1ι q — q = ι q — γ(q)’1 .

The spin group is

Spin(A, σ, f ) = SL1 (Q) = { q ∈ Q— | NrdQ (q) = 1 }.

Proof : We identify „¦(A, σ, f ) by means of (??): the canonical map b : A ’ B(A, σ, f )

maps A onto the subspace of invariant elements under the canonical involution ω.

Therefore, the condition σ(x) — Ab · x = Ab holds for all x ∈ Q— .

It su¬ces to check the description of χ0 in the split case, where it follows

from explicit computations. The Cli¬ord group is characterized in (??) by the

condition µ(q) ∈ F — , which here amounts to NrdQ (q) ∈ F — , and the description of

Spin(A, σ, f ) follows.

(15.11) Corollary. With the same notation as above, the group of multipliers of

proper similitudes of (A, σ, f ) is

G+ (A, σ, f ) = F —2 · NZ/F NrdQ (Q— )

and the group of spinor norms is

Sn(A, σ, f ) = F — © NrdQ (Q— ).

Moreover, G+ (A, σ, f ) = G(A, σ, f ) if and only if A is nonsplit and splits over Z.

Proof : The description of G+ (A, σ, f ) follows from (??) and the proposition above,

since χ (q) = χ0 (q) · F — for all q ∈ „¦(A, σ, f ). By de¬nition, the group of spinor

norms is Sn(A, σ, f ) = µ “(A, σ, f ) , and the preceding proposition shows that

µ “(A, σ, f ) = F — © NrdQ (Q— ).

If G(A, σ, f ) = G+ (A, σ, f ), then (??) shows that A is not split and splits

over Z. In order to prove the converse implication, we use the isomorphism

A NZ/F (Q) proved in (??) (and also in (??), see (??)). If A is split by Z,

scalar extension to Z shows that ι Q —Z Q is split, hence Q is isomorphic to

ι

Q as a Z-algebra. It follows that AutF (Q) = AutZ (Q), hence (??) shows that

PGO(A, σ, f ) = PGO+ (A, σ, f ). By (??), it follows that G(A, σ, f ) = G+ (A, σ, f )

if A is not split.

The case of trivial discriminant. If K is a given ´tale quadratic extension

e

2

of F , the equivalence A1 ≡ D2 set up in (??) associates quaternion algebras with

center K with algebras with quadratic pair (A, σ, f ) such that Z(A, σ, f ) = K. In

the particular case where K = F —F , we are led to consider the full subgroupoid 1A2

1

of A2 whose objects are F -algebras of the form Q1 —Q2 where Q1 , Q2 are quaternion

1

F -algebras, and the full subgroupoid 1 D2 of D2 whose objects are central simple

F -algebras with quadratic pair of trivial discriminant. Theorem (??) specializes to

the following statement:

(15.12) Corollary. The functor N : 1A2 ’ 1 D2 which maps the object Q1 — Q2

1

to (Q1 — Q2 , γ1 — γ2 , f— ) (where γ1 , γ2 are the canonical involutions on Q1 , Q2

respectively, and (γ1 — γ2 , f— ) is the quadratic pair of (??)) and the Cli¬ord algebra

functor C : 1 D2 ’ 1A2 de¬ne an equivalence of groupoids:

1

12

≡ 1 D2 .

A1

§15. EXCEPTIONAL ISOMORPHISMS 215

In particular, every central simple algebra A of degree 4 with quadratic pair (σ, f )

of trivial discriminant decomposes as a tensor product of quaternion algebras:

(A, σ, f ) = (Q1 — Q2 , γ1 — γ2 , f— ).

Proof : For (A, σ, f ) ∈ 1 D2 , we have C(A, σ, f ) = Q1 — Q2 for some quaternion

F -algebras Q1 , Q2 . The isomorphism (A, σ, f ) N —¦ C(A, σ, f ) yields:

(A, σ, f ) (Q1 — Q2 , γ1 — γ2 , f— ).

Note that the algebras Q1 , Q2 are uniquely determined by (A, σ, f ) up to

isomorphism since C(A, σ, f ) = Q1 — Q2 . Actually, they are uniquely determined

as subalgebras of A by the relation (A, σ, f ) = (Q1 — Q2 , γ1 — γ2 , f— ). If char F = 2,

this property follows from the observation that Skew(A, σ) = Skew(Q1 , γ1 ) — 1 +

1 — Skew(Q2 , γ2 ), since Skew(Q1 , γ1 ) — 1 and 1 — Skew(Q2 , γ2 ) are the only simple

Lie ideals of Skew(A, σ). See Exercise ?? for the case where char F = 2.

The results in (??), (??) and (??) can also be specialized to the case where the

discriminant of (σ, f ) is trivial. For instance, one has the following description of

the group of similitudes and their multipliers:

(15.13) Corollary. Let (A, σ, f ) = (Q1 — Q2 , γ1 — γ2 , f— ) ∈ 1 D2 . The functor C

induces isomorphisms of groups:

∼

PGO(A, σ, f ) ’ AutF (Q1 — Q2 )

’

and

∼

PGO+ (A, σ, f ) ’ AutF (Q1 ) — AutF (Q2 ) = PGL(Q1 ) — PGL(Q2 ).

’

Similarly, Spin(A, σ, f ) SL1 (Q1 ) — SL1 (Q2 ). Moreover,

G(A, σ, f ) = G+ (A, σ, f ) = NrdQ1 (Q— ) · NrdQ2 (Q— )

1 2

and

Sn(A, σ, f ) = NrdQ1 (Q— ) © NrdQ2 (Q— ).

1 2

Indices. Let Q ∈ A2 and (A, σ, f ) ∈ D2 correspond to each other under the

1

equivalence A2 ≡ D2 . Since deg A = 4, there are four possibilities for ind(A, σ, f ):

1

{0}, {0, 1}, {0, 2}, {0, 1, 2}.

The following proposition describes the corresponding possibilities for the algebra

Q. Let K be the center of Q, so K F disc(A, σ, f ) if char F = 2 and K

’1

F„˜ disc(A, σ, f ) if char F = 2.

(15.14) Proposition. With the notation above,

(1) ind(A, σ, f ) = {0} if and only if either Q is a division algebra (so K is a ¬eld )

or Q Q1 — Q2 for some quaternion division F -algebras Q1 , Q2 (so K F — F );

(2) ind(A, σ, f ) = {0, 1} if and only if K is a ¬eld and Q M2 (K);

(3) ind(A, σ, f ) = {0, 2} if and only if Q M2 (F ) — Q0 for some quaternion

division F -algebra Q0 ;

(4) ind(A, σ, f ) = {0, 1, 2} if and only if Q M2 (F ) — M2 (F ).

216 IV. ALGEBRAS OF DEGREE FOUR

Proof : If 1 ∈ ind(A, σ, f ), then A is split and the quadratic pair (σ, f ) is isotropic.

Thus, A EndF (V ) for some 4-dimensional F -vector space V , and (σ, f ) is

the quadratic pair associated with some isotropic quadratic form q on V . Since

dim V = 4, the quadratic space (V, q) is hyperbolic if and only if its discriminant

is trivial, i.e., K F — F . Therefore, if ind(A, σ, f ) = {0, 1}, then K is a ¬eld;

by (??), the canonical involution on C0 (V, q) Q is hyperbolic, hence Q is split.

If ind(A, σ, f ) = {0, 1, 2}, then (V, q) is hyperbolic and K F — F . By (??), it

follows that Q M2 (F ) — M2 (F ). Conversely, if Q M2 (K) (and K is either a

¬eld or isomorphic to F — F ), then Q contains a nonzero element q which is not

invertible. The element ι q — q ∈ NK/F (Q) A generates an isotropic right ideal of

reduced dimension 1, hence 1 ∈ ind(A, σ, f ). This proves (??) and (??).

If 2 ∈ ind(A, σ, f ), then (σ, f ) is hyperbolic, hence Proposition (??) shows

that Q M2 (F ) — Q0 for some quaternion F -algebra Q0 , since Q C(A, σ, f ).

Conversely, if Q M2 (F ) — Q0 for some quaternion F -algebra Q0 , then

(A, σ, f ) M2 (F ) — Q0 , γM — γ0 , f— ,

where γM and γ0 are the canonical (symplectic) involutions on M2 (F ) and Q0

respectively. If x ∈ M2 (F ) is a nonzero singular matrix, then x — 1 generates an

isotropic right ideal of reduced dimension 2 in A, hence 2 ∈ ind(A, σ, f ). This

proves (??) and yields an alternate proof of (??).