for the trace bilinear form (see (??)). Suppose that this condition holds and let

» = ’ g ’1 g ∈ F . We proceed to show that » = 0 or 1. Since TrdQ ( ) = 1 we

have NrdQ ( ) = 2 + and NrdQ ( + ») = 2 + + »2 + ». On the other hand, we

must have NrdQ ( ) = NrdQ ( + »), since + » = g ’1 g. Therefore, »2 + » = 0 and

» = 0 or 1. Therefore,

GO(Q, γ, f ) = { g ∈ Q— | g ’1 g = } ∪ { g ∈ Q— | g ’1 g = + 1 },

and the claim is proved.

162 III. SIMILITUDES

(12.19) Example. Let A = Q1 —F Q2 be a tensor product of two quaternion

algebras over a ¬eld F of characteristic di¬erent from 2, and σ = γ1 — γ2 , the

tensor product of the canonical involutions. A direct computation shows that the

Lie algebra Alt(A, σ) decomposes as a direct sum of the (Lie) algebras of pure

quaternions in Q1 and Q2 :

Alt(A, σ) = (Q0 — 1) • (1 — Q0 ).

1 2

Since Lie algebras of pure quaternions are simple and since the decomposition of

a semisimple Lie algebra into a direct product of simple subalgebras is unique,

it follows that every automorphism θ ∈ AutF (A, σ) preserves the decomposition

above, hence also the pair of subalgebras {Q1 , Q2 }. If Q1 Q2 , then θ must

preserve separately Q1 and Q2 ; therefore, it restricts to automorphisms of Q1 and

of Q2 . Let q1 ∈ Q— , q2 ∈ Q— be such that

1 2

θ|Q1 = Int(q1 ), θ|Q2 = Int(q2 ).

Then θ = Int(q1 — q2 ); so

GO(A, σ) = { q1 — q2 | q1 ∈ Q— , q2 ∈ Q— }

1 2

and the map which carries (q1 ·F — , q2 ·F — ) to (q1 —q2 )·F — induces an isomorphism

∼

(Q— /F — ) — (Q— /F — ) ’ PGO(A, σ).

’

1 2

If Q1 Q2 , then we may assume for notational convenience that A = Q—F Q where

Q is a quaternion algebra isomorphic to Q1 and to Q2 . Under the isomorphism

γ— : A ’ EndF (Q) such that γ— (q1 — q2 )(x) = q1 xγ(q2 ) for q1 , q2 , x ∈ Q, the

involution σ = γ — γ corresponds to the adjoint involution with respect to the

reduced norm quadratic form nQ ; therefore GO(A, σ) is the group of similitudes of

the quadratic space (Q, nQ ):

GO(A, σ) GO(Q, nQ ).

(These results are generalized in §??).

Multipliers of similitudes. Let (A, σ) be a central simple algebra with invo-

lution of any kind over an arbitrary ¬eld F . Let G(A, σ) be the group of multipliers

of similitudes of (A, σ):

G(A, σ) = { µ(g) | g ∈ Sim(A, σ) } ‚ F — .

If θ is an involution of the same kind as σ on a division algebra D Brauer-equivalent

to A, we may represent A as the endomorphism algebra of some vector space V

over D and σ as the adjoint involution with respect to some nonsingular hermitian

or skew-hermitian form h on V :

(A, σ) = EndD (V ), σh .

As in the split case (where D = F ), the similitudes of (A, σ) are the similitudes

of the hermitian or skew-hermitian space (V, h). It is clear from the de¬nition

that a similitude of (V, h) with multiplier ± ∈ F — may be regarded as an isometry

∼

(V, ±h) ’ (V, h). Therefore, multipliers of similitudes of (A, σ) can be character-

’

ized in terms of the Witt group W (D, θ) of hermitian spaces over D with respect

to θ (or of the group W ’1 (D, θ) of skew-hermitian spaces over D with respect to θ)

(see Scharlau [?, p. 239]). For the next proposition, note that the group W ±1 (D, θ)

is a module over the Witt ring W F .

§12. GENERAL PROPERTIES 163

(12.20) Proposition. For (A, σ) EndD (V ), σh as above,

G(A, σ) = { ± ∈ F — | (V, h) (V, ±h) }

= { ± ∈ F — | 1, ’± · h = 0 in W ±1 (D, θ) }.

In particular, if A is split and σ is symplectic, then G(A, σ) = F — .

Proof : The ¬rst part follows from the description above of similitudes of (A, σ).

The last statement then follows from the fact that W ’1 (F, IdF ) = 0.

As a sample of application, one can prove the following analogue of Scharlau™s

norm principle for algebras with involution by the same argument as in the classical

case (see Scharlau [?, Theorem 2.8.6]):

(12.21) Proposition. For any ¬nite extension L/F ,

NL/F G(AL , σL ) ‚ G(A, σ).

(12.22) Corollary. If σ is symplectic, then

F —2 · NrdA (A— ) ‚ G(A, σ).

If moreover deg A ≡ 2 mod 4, then this inclusion is an equality, and

G(A, σ) = F —2 · NrdA (A— ) = NrdA (A— ).

Proof : Let D be the division algebra (which is unique up to a F -isomorphism)

Brauer-equivalent to A. Then, NrdD (D— ) = NrdA (A— ) by Draxl [?, Theorem 1,

p. 146], hence it su¬ces to show that NrdD (d) ∈ G(A, σ) for all d ∈ D — to prove

the ¬rst part. Let L be a maximal sub¬eld in D containing d. The algebra AL is

then split, hence (??) shows:

G(AL , σL ) = L— .

From (??), it follows that

NL/F (d) ∈ G(A, σ).

This completes the proof of the ¬rst part, since NL/F (d) = NrdD (d).

Next, assume deg A = n = 2m, where m is odd. Since the index of A divides

its degree and its exponent, we have ind A = 1 or 2, hence D = F or D is a

quaternion algebra. In each case, NrdD (D— ) contains F —2 , hence F —2 ·NrdA (A— ) =

NrdA (A— ). On the other hand, taking the pfa¬an norm of each side of the equation

σ(g)g = µ(g), for g ∈ GSp(A, σ), we obtain NrdA (g) = µ(g)m by (??). Since m is

odd, it follows that

2

µ(g) = µ(g)’(m’1)/2 NrdA (g) ∈ F —2 · NrdA (A— ),

hence G(A, σ) ‚ F —2 · NrdA (A— ).

12.C. Proper similitudes. Suppose σ is an involution of the ¬rst kind on

a central simple F -algebra A of even degree n = 2m. For every similitude g ∈

Sim(A, σ) we have

NrdA (g) = ±µ(g)m ,

as can be seen by taking the reduced norm of both sides of the equation σ(g)g =

µ(g).

164 III. SIMILITUDES

(12.23) Proposition. If σ is a symplectic involution on a central simple F -algebra

A of degree n = 2m, then

NrdA (g) = µ(g)m for all g ∈ GSp(A, σ).

Proof : If A is split, the formula is a restatement of (??). The general case follows

by extending scalars to a splitting ¬eld of A.

By contrast, if σ is orthogonal, we may distinguish two types of similitudes

according to the sign of NrdA (g)µ(g)’m :

(12.24) De¬nition. Let σ be an orthogonal involution on a central simple alge-

bra A of even degree n = 2m over an arbitrary ¬eld F . A similitude g ∈ GO(A, σ)

is called proper (resp. improper ) if NrdA (g) = +µ(g)m (resp. NrdA (g) = ’µ(g)m ).

(Thus, if char F = 2, every similitude of (A, σ) is proper; however, see (??).)

It is clear that proper similitudes form a subgroup of index at most 2 in the

group of all similitudes; we write GO+ (A, σ) for this subgroup. The set of improper

similitudes is a coset of GO+ (A, σ), which may be empty.16 We also set:

PGO+ (A, σ) = GO+ (A, σ)/F — ,

and

O+ (A, σ) = GO+ (A, σ) © O(A, σ) = { g ∈ A | σ(g)g = NrdA (g) = 1 }.

The elements in O+ (A, σ) are the proper isometries.

(12.25) Example. Let Q be a quaternion algebra with canonical involution γ over

a ¬eld F of characteristic di¬erent from 2, and let σ = Int(u) —¦ γ for some invertible

pure quaternion u. Let v ∈ A be an invertible pure quaternion which anticommutes

with u. The group GO(A, σ) has been determined in (??); straightforward norm

computations show that the elements in F (u)— are proper similitudes, whereas

those in F (u)— · v are improper, hence

GO+ (Q, σ) = F (u)— .

However, no element in F (u)— · v has norm 1 unless Q is split, so

O+ (Q, σ) = O(Q, σ) = { z ∈ F (u) | NF (u)/F (z) = 1 } if Q is not split.

(12.26) Example. Let A = Q1 —F Q2 , a tensor product of two quaternion algebras

over a ¬eld F of characteristic di¬erent from 2, and σ = γ1 — γ2 where γ1 , γ2 are

the canonical involutions on Q1 and Q2 .

If Q1 Q2 , then we know from (??) that all the similitudes of (A, σ) are of

the form q1 — q2 for some q1 ∈ Q— , q2 ∈ Q— . We have

1 2

µ(q1 — q2 ) = γ1 (q1 )q1 — γ2 (q2 )q2 = NrdQ1 (q1 ) · NrdQ2 (q2 )

and

NrdA (q1 — q2 ) = NrdQ1 (q1 )deg Q2 · NrdQ2 (q2 )deg Q1 = µ(q1 — q2 )2 ,

so all the similitudes are proper:

GO(A, σ) = GO+ (A, σ) O(A, σ) = O+ (A, σ).

and

16 Fromthe viewpoint of linear algebraic groups, one would say rather that this coset may

have no rational point. It has a rational point over a splitting ¬eld of A however, since hyperplane

re¬‚ections are improper isometries.

§12. GENERAL PROPERTIES 165

On the other hand, if Q1 Q2 , then the algebra A is split, hence GO+ (A, σ) is a

subgroup of index 2 in GO(A, σ).

Proper similitudes of algebras with quadratic pair. A notion of proper

similitudes can also be de¬ned for quadratic pairs. We consider only the charac-

teristic 2 case, since if the characteristic is di¬erent from 2 the similitudes of a

quadratic pair (σ, f ) are the similitudes of the orthogonal involution σ.

Thus let (σ, f ) be a quadratic pair on a central simple algebra A of even degree

n = 2m over a ¬eld F of characteristic 2. Let ∈ A be an element such that

f (s) = TrdA ( s) for all s ∈ Sym(A, σ)

(see (??)). For g ∈ GO(A, σ, f ), we have f (gsg ’1 ) = f (s) for all s ∈ Sym(A, σ),

hence, as in (??),

TrdA (g ’1 g ’ )s = 0 for s ∈ Sym(A, σ).

By (??), it follows that

g ’1 g ’ ∈ Alt(A, σ).

Therefore, f (g ’1 g ’ ) ∈ F by property (??) of the de¬nition of a quadratic pair.

(12.27) Proposition. The element f (g ’1 g ’ ) depends only on the similitude g,

and not on the choice of . Moreover, f (g ’1 g ’ ) = 0 or 1.

Proof : As observed in (??), the element is uniquely determined by the quadratic

pair (σ, f ) up to the addition of an element in Alt(A, σ). If = + x + σ(x), then

g ’1 g ’ = (g ’1 g ’ ) + (g ’1 xg ’ x) + σ(g ’1 xg ’ x),

since σ(g) = µ(g)g ’1 . We have

f g ’1 xg ’ x + σ(g ’1 xg ’ x) = TrdA (g ’1 xg ’ x) = 0,

hence the preceding equation yields

f (g ’1 g ’ ) = f (g ’1 g ’ ),

proving that f (g ’1 g ’ ) does not depend on the choice of .

We next show that this element is either 0 or 1. By (??), we have σ( ) = + 1,

hence 2 + = σ( ) . It follows that

g ’1 2 g ’ 2

= µ(g)’1 σ(g)σ( ) g ’ σ( ) + (g ’1 g ’ ),

hence g ’1 2 g ’ 2

∈ Sym(A, σ). We shall show successively:

f (g ’1 g ’ )2 = f (g ’1 g ’ )2 ,

(12.28)

f (g ’1 g ’ )2 = f (g ’1 2 g ’ 2

(12.29) ),

f (g ’1 2 g ’ 2

) = f (g ’1 g ’ ).

(12.30)

By combining these equalities, we obtain

f (g ’1 g ’ )2 = f (g ’1 g ’ ),

hence f (g ’1 g ’ ) = 0 or 1.

We ¬rst show that f (x)2 = f (x2 ) for all x ∈ Alt(A, σ); equation (??) follows,

since g ’1 g ’ ∈ Alt(A, σ). Let x = y + σ(y) for some y ∈ A. Since σ( ) + = 1,

we have

σ(y)y = σ(y) y + σ σ(y) y ,

166 III. SIMILITUDES

hence

f σ(y)y = TrdA σ(y) y .

The right side also equals

TrdA yσ(y) = f yσ(y) ,

hence f σ(y)y + yσ(y) = 0. It follows that

f (x2 ) = f y 2 + σ(y 2 ) = TrdA (y 2 ).

On the other hand, (??) shows that TrdA (y 2 ) = TrdA (y)2 ; since f (x) = TrdA (y),

we thus have f (x)2 = f (x2 ).

To prove (??), it su¬ces to show

f (g ’1 g + g ’1 g) = 0,

since (g ’1 g ’ )2 = (g ’1 2 g ’ 2

) + (g ’1 g + g ’1 g). By the de¬nition of , we

have

f (g ’1 g + g ’1 g) = TrdA (g ’1 g + g ’1 g) ;

the right-hand expression vanishes, since TrdA ( g ’1 g ) = TrdA ( 2 g ’1 g).

To complete the proof, we show (??): since g is a similitude and 2 + =

σ( ) ∈ Sym(A, σ), we have f g ’1 ( 2 + )g = f ( 2 + ), hence

f (g ’1 2 g + g ’1 g + 2

+ )=0

and therefore

f (g ’1 2 g + 2

) = f (g ’1 g + ).

(12.31) Example. Suppose (V, q) is a nonsingular quadratic space of even dimen-

sion n = 2m and let (σq , fq ) be the associated quadratic pair on EndF (V ), so

that

GO EndF (V ), σq , fq = GO(V, q),

as observed in (??). If ∈ EndF (V ) is such that fq (s) = tr( s) for all s ∈

Sym EndF (V ), σq , we claim that for all g ∈ GO(V, q) the Dickson invariant ∆(g),

de¬ned in (??), satis¬es

∆(g) = f (g ’1 g ’ ).

Since the right-hand expression does not depend on the choice of , it su¬ces to

prove the claim for a particular . Pick a basis (e1 , . . . , en ) of V which is symplectic

for the alternating form bq , i.e.,

bq (e2i’1 , e2i ) = 1, bq (e2i , e2i+1 ) = 0 and bq (ei , ej ) = 0 if |i ’ j| > 1,

and identify every endomorphism of V with its matrix with respect to that basis.

An element ∈ EndF (V ) such that tr( s) = fq (s) for all s ∈ Sym EndF (V ), σq is

given in (??); the corresponding matrix is (see the proof of (??))

«

0

1

1 q(e2i )

¬ ·

..

= where i = q(e .

.

2i’1 ) 0

0 m

§12. GENERAL PROPERTIES 167

On the other hand, for a matrix M representing the quadratic form q, we may

choose

«

M1 0

q(e2i’1 ) 0

¬ ·

..

M = where Mi = .

. 1 q(e2i )

0 Mm

where W = M + M t . Therefore, for all

It is readily veri¬ed that M = W ·

invertible g ∈ EndF (V ),

g ’1 g + = W ’1 (W g ’1 W ’1 M g + M ).

Since σq (g) = W ’1 g t W , we have W g ’1 W ’1 = µ(g)’1 g t if g ∈ GO(V, q), hence

the preceding equation may be rewritten as

g ’1 g + = µ(g)’1 W ’1 g t M g + µ(g)M .

Let R be a matrix such that g t M g + µ(g)M = R + Rt , so that

∆(g) = tr µ(g)’1 W ’1 R .

We then have

g ’1 g + = µ(g)’1 W ’1 (R + Rt ) = µ(g)’1 W ’1 R + σq µ(g)’1 W ’1 R ,

hence

fq (g ’1 g + ) = tr µ(g)’1 W ’1 R ,

and the claim is proved.

Note that this result yields an alternate proof of the part of (??) saying that

’1

f (g g+ ) = 0 or 1 for all g ∈ GO(A, σ, f ). One invokes (??) after scalar extension

to a splitting ¬eld.

(12.32) De¬nition. Let (σ, f ) be a quadratic pair on a central simple algebra A

of even degree over a ¬eld F of characteristic 2. In view of (??), we may set

∆(g) = f (g ’1 g ’ ) ∈ {0, 1} for g ∈ GO(A, σ, f ),

where ∈ A is such that f (s) = TrdA ( s) for all s ∈ Sym(A, σ). We call ∆ the

Dickson invariant. By (??), this de¬nition is compatible with (??) when (A, σ, f ) =

EndF (V ), σq , fq .

It is easily veri¬ed that the map ∆ is a group homomorphism

∆ : GO(A, σ, f ) ’ Z/2Z.

We set GO+ (A, σ, f ) for its kernel; its elements are called proper similitudes. We

also set PGO+ (A, σ, f ) = GO+ (A, σ, f )/F — .

(12.33) Example. Let Q be a quaternion algebra with canonical involution γ over

a ¬eld F of characteristic 2, and let (γ, f ) be a quadratic pair on Q. Choose ∈ Q

satisfying f (s) = TrdQ ( s) for all s ∈ Sym(Q, γ). As observed in (??), we have

GO(Q, γ, f ) = F ( )— ∪ F ( )— · v

where v ∈ Q— satis¬es v ’1 v = + 1. For g ∈ F ( )— we have g ’1 g + = 0, hence

∆(g) = 0. On the other hand, if g ∈ F ( )— · v, then g ’1 g + = 1, hence ∆(g) = 1,

by (??). Therefore (compare with (??))

GO+ (Q, γ, f ) = F ( )— .

168 III. SIMILITUDES

12.D. Functorial properties. Elaborating on the observation that simili-

tudes of a given bilinear or hermitian space induce automorphisms of its endo-

morphism algebra with adjoint involution (see (??)), we now show how similitudes

between two hermitian spaces induce isomorphisms between their endomorphism

algebras. In the case of quadratic spaces of odd dimension in characteristic dif-

ferent from 2, the relationship with endomorphism algebras takes the form of an

equivalence of categories.

Let D be a division algebra with involution θ over an arbitrary ¬eld F . Let

K be the center of D, and assume F is the sub¬eld of θ-invariant elements in K

(so F = K if θ is of the ¬rst kind). Hermitian or skew-hermitian spaces (V, h),

(V , h ) over D with respect to θ are called similar if there exists a D-linear map

g : V ’ V and a nonzero element ± ∈ F — such that

h g(x), g(y) = ±h(x, y) for x, y ∈ V .

The map g is then called a similitude with multiplier ±.

Assuming (V, h), (V , h ) nonsingular, let σh , σh be their adjoint involutions

on EndD (V ), EndD (V ) respectively.

(12.34) Proposition. Every similitude g : (V, h) ’ (V , h ) induces a K-isomor-

phism of algebras with involution

g— : EndD (V ), σh ’ EndD (V ), σh

de¬ned by g— (f ) = g —¦ f —¦ g ’1 for f ∈ EndD (V ). Further, every K-isomorphism

of algebras with involution EndD (V ), σh ’ EndD (V ), σh has the form g— for

some similitude g : (V, h) ’ (V , h ), which is uniquely determined up to a factor

in K — .

Proof : It is straightforward to check that for every similitude g, the map g— is an

isomorphism of algebras with involution. On the other hand, suppose that

¦ : EndD (V ), σh ’ EndD (V ), σh

is a K-isomorphism of algebras with involution. We then have dimD V = dimD V ,

and we use ¦ to de¬ne on V the structure of a left EndD (V ) —K Dop -module, by

(f — dop ) — v = ¦(f )(v )d for f ∈ EndD (V ), d ∈ D, v ∈ V .

The space V also is a left EndD (V ) —K Dop -module, with the action de¬ned by

(f — dop ) — v = f (v)d for f ∈ EndD (V ), d ∈ D, v ∈ V .

Since dimK V = dimK V , it follows from (??) that V and V are isomorphic as

EndD (V ) —K Dop -modules. Hence, there exists a D-linear bijective map

g: V ’ V

such that f — g(v ) = g —¦ f (v ) for all f ∈ EndD (V ), v ∈ V ; this means

¦(f ) —¦ g = g —¦ f for f ∈ EndD (V ).

It remains to show that g is a similitude, and that it is uniquely determined up to

a factor in K — . To prove the ¬rst part, de¬ne a hermitian form h0 on V by

h0 (v, w) = h g(v), g(w) for v, w ∈ V .

For all f ∈ EndD (V ), we then have

h0 v, f (w) = h g(v), ¦(f ) —¦ g(w) .

§12. GENERAL PROPERTIES 169

Since ¦ is an isomorphism of algebras with involution, σh ¦(f ) = ¦ σh (f ) ,

hence the right-hand expression may be rewritten as

h ¦ σh (f ) —¦ g(v), g(w) = h0 σh (f )(v), w .

Therefore, σh is the adjoint involution with respect to h0 . By (??), it follows that

h0 = ±h for some ± ∈ F — , hence g is a similitude with multiplier ±, and ¦ = g— .

If g, g : (V, h) ’ (V , h ) are similitudes such that g— = g— , then g ’1 g ∈

EndD (V ) commutes with every f ∈ EndD (V ), hence g ≡ g mod K — .

(12.35) Corollary. All hyperbolic involutions of the same type on a central simple

algebra are conjugate. Similarly, all hyperbolic quadratic pairs on a central simple

algebra are conjugate.

Proof : Let A be a central simple algebra, which we represent as EndD (V ) for some

vector space V over a division algebra D, and let σ, σ be hyperbolic involutions

of the same type on A. These involutions are adjoint to hyperbolic hermitian or

skew-hermitian forms h, h on V , by (??). Since all the hyperbolic forms on V are

isometric, the preceding proposition shows that (A, σ) (A, σ ), hence σ and σ

are conjugate.

Consider next two hyperbolic quadratic pairs (σ, f ) and (σ , f ) on A. The

involutions σ and σ are hyperbolic, hence conjugate, by the ¬rst part of the proof.

After a suitable conjugation, we may thus assume σ = σ. By (??), there are

idempotents e, e ∈ A such that f (s) = TrdA (es) and f (s) = TrdA (e s) for all

s ∈ Sym(A, σ).

Claim. There exists x ∈ A— such that σ(x)x = 1 and e = xe x’1 .

The claim yields

f (s) = TrdA (x’1 exs) = TrdA (exsx’1 ) = f (xsx’1 ) for all s ∈ Sym(A, σ),

hence x conjugates (σ, f ) into (σ , f ).

To prove the claim, choose a representation of A:

(A, σ) = EndD (V ), σh

for some hyperbolic hermitian space (V, h) over a division algebra D. As in the

proof of (??), we may ¬nd a pair of complementary totally isotropic subspaces U ,

W (resp. U , W ) in V such that e is the projection on U parallel to W and e is the

projection on U parallel to W . It is easy to ¬nd an isometry of V which maps U

to U and W to W ; every such isometry x satis¬es σ(x)x = 1 and e = xe x’1 .

There is an analogue to (??) for quadratic pairs:

(12.36) Proposition. Let (V, q) and (V , q ) be even-dimensional and nonsingular

quadratic spaces over a ¬eld F . Every similitude g : (V, q) ’ (V , q ) induces an

F -isomorphism of algebras with quadratic pair

g— : EndF (V ), σq , fq ’ EndF (V ), σq , fq

de¬ned by

g— (h) = g —¦ h —¦ g ’1 for h ∈ EndF (V ).

Moreover, every F -isomorphism

EndF (V ), σq , fq ’ EndF (V ), σq , fq

170 III. SIMILITUDES

of algebras with quadratic pair is of the form g— for some similitude g : (V, q) ’

(V , q ), which is uniquely determined up to a factor in F — .

Proof : The same arguments as in the proof of (??) apply here. Details are left to

the reader.

Proposition (??) shows that mapping every hermitian or skew-hermitian space

(V, h) to the algebra EndD (V ), σh de¬nes a full functor from the category of

nonsingular hermitian or skew-hermitian spaces over D, where the morphisms are

the similitudes, to the category of central simple algebras with involution where

the morphisms are the isomorphisms. In the particular case where the degree is

odd and the characteristic is di¬erent from 2, this functor can be used to set up an

equivalence of categories, as we show in (??) below.

A particular feature of the categories we consider here (and in the next chapter)

is that all the morphisms are invertible (i.e., isomorphisms). A category which has

this property is called a groupoid . Equivalences of groupoids may be described in a

very elementary way, as the next proposition shows. For an arbitrary category X ,

let Isom(X ) be the class17 of isomorphism classes of objects in X . Every functor

S : X ’ Y induces a map Isom(X ) ’ Isom(Y ) which we also denote by S.

(12.37) Proposition. Let X , Y be groupoids. A covariant functor S : X ’ Y

de¬nes an equivalence of categories if and only if the following conditions hold :

(1) the induced map S : Isom(X ) ’ Isom(Y ) is a bijection;

(2) for each X ∈ X , the induced map AutX (X) ’ AutY S(X) is a bijection.

Proof : The conditions are clearly necessary. Suppose that the covariant functor S

satis¬es conditions (??) and (??) above. If X, X ∈ X and g : S(X) ’ S(X ) is a

morphism in Y , then S(X) and S(X ) are in the same isomorphism class of Y , hence

(??) implies that X and X are isomorphic. Let f : X ’ X be an isomorphism.

Then g —¦ S(f )’1 ∈ AutY S(X ) , hence g —¦ S(f )’1 = S(h) for some h ∈ AutX (X ).

It follows that g = S(h —¦ f ), showing that the functor S is full. It is also faithful: if

f , g : X ’ X are morphisms in X , then S(f ) = S(g) implies S(f ’1 —¦ g) = IdS(X) ,

hence f = g by (??). Since every object in Y is isomorphic to an object of the form

S(X) with X ∈ X , it follows that S is an equivalence of categories (see Mac Lane [?,

p. 91]).

(12.38) Remarks. (1) The proof above also applies mutatis mutandis to con-

travariant functors, showing that the same conditions as in (??) characterize the

contravariant functors which de¬ne anti-equivalence of categories.

∼

(2) The bijection AutX (X) ’ AutY S(X) induced by an equivalence of cat-

’

egories is a group isomorphism if the same convention is used in X and Y for

mapping composition. It is an anti-isomorphism if opposite conventions are used

in X and Y . By contrast, the bijection induced by an anti-equivalence of categories

is an anti-isomorphism if the same convention is used in X and Y , and it is an

isomorphism if opposite conventions are used.

For the rest of this section F is a ¬eld of characteristic di¬erent from 2. For

any integer n ≥ 1, let Bn denote the groupoid of central simple F -algebras of

17 For all the categories we consider in the sequel, this class is a set.

§12. GENERAL PROPERTIES 171

degree 2n + 1 with involution of the ¬rst kind,18 where the morphisms are the F -

algebra isomorphisms which preserve the involutions. Note that these algebras are

necessarily split, and the involution is necessarily of orthogonal type, by (??).

For any integer n ≥ 1, let Qn be the groupoid of all nonsingular quadratic

spaces of dimension n over the ¬eld F , where the morphisms are the isometries,

1

and let Qn be the full subcategory of quadratic spaces with trivial discriminant. For

(V, q) ∈ Qn , let σq denote the adjoint involution on EndF (V ) with respect to (the

polar of) q. If (V, q) ∈ Q2n+1 , then EndF (V ), σq ∈ Bn , and we have a functor

End : Q2n+1 ’ Bn

given by mapping (V, q) to EndF (V ), σq , as observed in (??).

(12.39) Proposition. The functor End de¬nes a bijection between the sets of

isomorphism classes:

∼

1

End : Isom(Q2n+1 ) ’ Isom(Bn ).

’

Proof : By (??), every algebra with involution in Bn is isomorphic to an algebra

with involution of the form EndF (V ), σq for some quadratic space (V, q) of dimen-

sion 2n+1. Since the adjoint involution does not change when the quadratic form is

multiplied by a scalar, we may substitute (disc q)q for q and thus assume disc q = 1.

Therefore, the map induced by End on isomorphism classes is surjective.

On the other hand, suppose

¦ : EndF (V ), σq ’ EndF (V ), σq

1

is an isomorphism, for some quadratic spaces (V, q), (V , q ) ∈ Q2n+1 . By (??), we

may ¬nd a similitude g : (V, q) ’ (V , q ) such that ¦ = g— . This similitude may be

∼

regarded as an isometry (V, ±q) ’ (V , q ) , where ± is the multiplier of g. Since

’

disc q = disc q and dim V = dim V is odd, we must have ± = 1, hence g is an

∼

isometry (V, q) ’ (V , q ).

’

Even though it de¬nes a bijection between the sets of isomorphism classes, the

1

functor End is not an equivalence between Q2n+1 and Bn : this is because the group

of automorphisms of the algebra with involution EndF (V ), σq is

AutF EndF (V ), σq = PGO(V, q) = O+ (V, q)

(the second equality follows from (??)), whereas the group of automorphisms of

(V, q) is O(V, q). However, we may de¬ne some additional structure on quadratic

spaces to restrict the automorphism group and thereby obtain an equivalence of

categories.

(12.40) De¬nition. Let (V, q) be a quadratic space of odd dimension and trivial

discriminant over a ¬eld F of characteristic di¬erent from 2. The center Z of the

Cli¬ord algebra C(V, q) is then an ´tale quadratic extension of F isomorphic to

e

F — F . An orientation of (V, q) is an element ζ ∈ Z F such that ζ 2 = 1. Thus,

each quadratic space (V, q) as above has two possible orientations which di¬er by

a sign. Triples (V, q, ζ) are called oriented quadratic spaces.

18 This notation is motivated by the fact that the automorphism group of each object in this

groupoid is a classical group of type Bn : see Chapter ??. However this groupoid is only de¬ned

for ¬elds of characteristic di¬erent from 2.

172 III. SIMILITUDES

∼

Every isometry g : (V, q) ’ (V , q ) induces an isomorphism g— : C(V, q) ’ ’

C(V , q ) which carries an orientation of (V, q) to an orientation of (V , q ). The

isometries g : (V, q) ’ (V, q) which preserve a given orientation form the group

O+ (V, q).

Let Bn be the groupoid of oriented quadratic spaces of dimension 2n+1 over F .

The objects of Bn are triples (V, q, ζ) where (V, q) is a quadratic space of dimen-

sion 2n + 1 over F with trivial discriminant and ζ is an orientation of (V, q), and

the morphisms are the orientation-preserving isometries. For each (V, q, ζ) ∈ Bn ,

the map ’IdV : V ’ V de¬nes an isomorphism (V, q, ζ) ’ (V, q, ’ζ), hence two

oriented quadratic spaces are isomorphic if and only if the quadratic spaces are iso-

metric. In other words, the functor which forgets the orientation de¬nes a bijection

∼ 1

Isom(Bn ) ’ Isom(Q2n+1 ).

’

(12.41) Theorem. The functor End which maps every oriented quadratic space

(V, q, ζ) in Bn to the algebra with involution EndF (V ), σq ∈ Bn de¬nes an equiv-

alence of categories:

Bn ≡ B n .

1

Proof : Since the isomorphism classes of Bn and Q2n+1 coincide, (??) shows that the

∼

functor End de¬nes a bijection Isom(Bn ) ’ Isom(Bn ). Moreover, as we observed

’

above, for every oriented quadratic space (V, q, ζ) of dimension 2n + 1 we have

Aut EndF (V ), σq = O+ (V, q) = Aut(V, q, ζ).

(12.42)

Therefore, (??) shows that End is an equivalence of categories.

§13. Quadratic Pairs

In this section, (σ, f ) is a quadratic pair on a central simple algebra A over

an arbitrary ¬eld F . If the degree of A is odd, then A is split, char F = 2, and

the group of similitudes of (A, σ, f ) reduces to the orthogonal group of an F -vector

space (see (??)). We therefore assume throughout this section that the degree is

even, and we set

deg A = n = 2m.

Our goal is to obtain additional information on the group GO(A, σ, f ) by relating

similitudes of (A, σ, f ) to the Cli¬ord algebra C(A, σ, f ) and the Cli¬ord bimodule

B(A, σ, f ). We use this to de¬ne a Cli¬ord group “(A, σ, f ), which is a twisted ana-

logue of the special Cli¬ord group of a quadratic space, and also de¬ne an extended

Cli¬ord group „¦(A, σ, f ). These constructions are used to prove an analogue of a

classical theorem of Dieudonn´ on the multipliers of similitudes.

e

13.A. Relation with the Cli¬ord structures. Since the Cli¬ord alge-

bra C(A, σ, f ) and the Cli¬ord bimodule B(A, σ, f ) are canonically associated to

(A, σ, f ), every automorphism in AutF (A, σ, f ) induces automorphisms of C(A, σ, f )

and B(A, σ, f ). Our purpose in this section is to investigate these automorphisms.

The Cli¬ord algebra. Every automorphism θ ∈ Aut F (A, σ, f ) induces an

automorphism

C(θ) ∈ AutF C(A, σ), σ .

Explicitly, C(θ) can be de¬ned as the unique automorphism of C(A, σ, f ) such that

C(θ) c(a) = c θ(a) for a ∈ A,

§13. QUADRATIC PAIRS 173

where c : A ’ C(A, σ, f ) is the canonical map (??). We thereby obtain a canonical

group homomorphism

C : AutF (A, σ, f ) ’ AutF C(A, σ, f ), σ .

Slightly abusing notation, we also call C the homomorphism

C : GO(A, σ, f ) ’ AutF C(A, σ, f ), σ

obtained by composing the preceding map with the epimorphism

Int : GO(A, σ, f ) ’ AutF (A, σ, f )

of (??). Thus, for g ∈ GO(A, σ, f ) and a ∈ A,

C(g) c(a) = c(gag ’1 ).

(13.1) Proposition. Suppose A is split; let (A, σ, f ) = EndF (V ), σq , fq for

some nonsingular quadratic space (V, q). Then, under the standard identi¬cations

GO(A, σ, f ) = GO(V, q) (see (??)) and C(A, σ, f ) = C0 (V, q) (see (??)), the canon-

ical map C : GO(V, q) ’ AutF C0 (V, q) is de¬ned by

C(g)(v1 · · · v2r ) = µ(g)’r g(v1 ) · · · g(v2r )

for g ∈ GO(V, q) and v1 , . . . , v2r ∈ V .

Proof : It su¬ces to check the formula above on generators v · w of C0 (V, q). For

v, w ∈ V , the product v · w in C(V, q) is the image of v — w under the canonical

map c: we thus have v · w = c(v — w), hence

C(g)(v · w) = c g —¦ (v — w) —¦ g ’1 .

Let ± = µ(g) be the multiplier of g; then σ(g)’1 = ±’1 g, hence, for x ∈ V ,

g —¦ (v — w) —¦ g ’1 (x) = g(v)bq w, g ’1 (x) = g(v)bq ±’1 g(w), x .

Therefore, g —¦ (v — w) —¦ g ’1 (x) = ±’1 g(v) — g(w) (x), which shows

g —¦ (v — w) —¦ g ’1 = ±’1 g(v) — g(w),

hence c g —¦ (v — w) —¦ g ’1 = ±’1 g(v) · g(w).

Note that, for g ∈ GO(A, σ, f ), the automorphism C(g) of C(A, σ, f ) is F -linear

but is not necessarily the identity on the center of C(A, σ, f ). The behavior of C(g)

on the center in fact determines whether g is proper, as the next proposition shows.

(13.2) Proposition. A similitude g ∈ GO(A, σ, f ) is proper if and only if C(g)

restricts to the identity map on the center Z of C(A, σ, f ).

Proof : Suppose ¬rst that char F = 2. Choose ∈ A satisfying f (s) = TrdA ( s) for

all s ∈ Sym(A, σ) (see (??)). By (??), we have Z = F c( ) , hence it su¬ces to

show

C(g) c( ) = c( ) + ∆(g) for g ∈ GO(A, σ, f ).

For g ∈ GO(A, σ, f ) we have

∆(g) = f (g ’1 g ’ );

since g is a similitude, the right side also equals

f g(g ’1 g ’ )g ’1 = f (g g ’1 ’ ).

174 III. SIMILITUDES

On the other hand, since g g ’1 ’ ∈ Sym(A, σ), we have

f (g g ’1 ’ ) = c(g g ’1 ’ ),

hence

∆(g) = c(g g ’1 ) ’ c( ) = C(g) c( ) ’ c( ),

proving the proposition when char F = 2.

Suppose now that char F = 2. It su¬ces to check the split case; we may

thus assume (A, σ, f ) = EndF (V ), σq , fq for some nonsingular quadratic space

(V, q), and use the standard identi¬cations and the preceding proposition. Let

(e1 , . . . , e2m ) be an orthogonal basis of (V, q). Recall that e1 · · · e2m ∈ Z F . For

g ∈ GO(A, σ, f ) = GO(V, q), we have

C(g)(e1 · · · e2m ) = µ(g)’m g(e1 ) · · · g(e2m ).

On the other hand, a calculation in the Cli¬ord algebra shows that

g(e1 ) · · · g(e2m ) = det(g)e1 · · · e2m ;

hence e1 · · · e2m is ¬xed by C(g) if and only if det(g) = µ(g)m . This proves the

proposition in the case where char F = 2.

In view of this proposition, the Dickson invariant ∆ : GO(A, σ, f ) ’ Z/2Z

de¬ned in (??) may alternately be de¬ned by

0 if C(g)|Z = IdZ ,

(13.3) ∆(g) =

1 if C(g)|Z = IdZ .

The image of the canonical map C has been determined by Wonenburger in

characteristic di¬erent from 2:

(13.4) Proposition. If deg A > 2, the canonical homomorphism

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

is injective. If char F = 2, the image of C is the group of those automorphisms

which preserve the image c(A) of A under the canonical map c : A ’ C(A, σ, f ).

Proof : If θ ∈ AutF (A, σ) lies in the kernel of C, then

c θ(a) = c(a) for a ∈ A,

since the left side is the image of c(a) under C(θ). By applying the map δ of (??),

we obtain

θ a ’ σ(a) = a ’ σ(a) for a ∈ A,

hence θ is the identity on Alt(A, σ). Since deg A > 2, (??) shows that Alt(A, σ)

generates A, hence θ = IdA , proving the injectivity of C.

It follows from the de¬nition that every automorphism of the form C(θ) maps

c(A) to itself. Conversely, suppose ψ is an automorphism of C(A, σ, f ) which pre-

serves c(A), and suppose char F = 2. The map f is then uniquely determined by

σ, so we may denote C(A, σ, f ) simply by C(A, σ). The restriction of ψ to

c(A)0 = c(A) © Skew C(A, σ), σ = { x ∈ c(A) | Trd(x) = 0 }

§13. QUADRATIC PAIRS 175

is a Lie algebra automorphism. By (??), the Lie algebra c(A)0 is isomorphic to

Alt(A, σ) via δ, with inverse isomorphism 1 c; therefore, there is a corresponding

2

Lie automorphism ψ of Alt(A, σ) such that

c ψ (a) = ψ c(a) for a ∈ Alt(A, σ).

Let L be a splitting ¬eld of A. A theorem19 of Wonenburger [?, Theorem 4] shows

that the automorphism ψL = ψ — IdL of C(A, σ)L = C(AL , σL ) is induced by a

similitude g of (AL , σL ), hence

ψL (a) = gag ’1 for a ∈ Alt(AL , σL ).

Therefore, the automorphism ψL of Alt(AL , σL ) extends to an automorphism of

(AL , σL ). By (??), ψ extends to an automorphism θ of (A, σ), and this au-

tomorphism satis¬es C(θ) = ψ since c θ(a) = c ψ (a) = ψ c(a) for all a ∈

Alt(A, σ).

If deg A = 2, then C(A, σ) = Z, so AutF C(A, σ), σ = {Id, ι} and the canon-

ical homomorphism C maps PGO+ (A, σ) to Id, so C is not injective.

The Cli¬ord bimodule. The bimodule B(A, σ, f ) is canonically associated to

(A, σ, f ), just as the Cli¬ord algebra C(A, σ, f ) is. Therefore, every automorphism

θ ∈ AutF (A, σ, f ) induces a bijective linear map

B(θ) : B(A, σ, f ) ’ B(A, σ, f ).

This map satis¬es

B(θ)(ab ) = θ(a)b for a ∈ A

(where b : A ’ B(A, σ, f ) is the canonical map of (??)) and

(13.5) B(θ) a · (c1 — x · c2 ) = θ(a) · C(θ)(c1 ) — B(θ)(x) · C(θ)(c2 )

for a ∈ A, c1 , c2 ∈ C(A, σ, f ) and x ∈ B(A, σ, f ). Explicitly, B(θ) is induced by

the map θ : T + (A) ’ T + (A) such that

θ(a1 — · · · — ar ) = θ(a1 ) — · · · — θ(ar ).

As in the previous case, we modify the domain of de¬nition of B to be the group

GO(A, σ, f ), by letting B(g) = B Int(g) for g ∈ GO(A, σ, f ). We thus obtain a

canonical homomorphism

B : GO(A, σ, f ) ’ GLF B(A, σ, f ).

For g ∈ GO(A, σ, f ), we also de¬ne a map

(13.6) β(g) : B(A, σ, f ) ’ B(A, σ, f )

by

xβ(g) = g · B(g ’1 )(x) for x ∈ B(A, σ, f ).

The map β(g) is a homomorphism of left A-modules, since for a ∈ A and x ∈

B(A, σ, f ),

(a · x)β(g) = g · (g ’1 ag) · B(g ’1 )(x) = a · xβ(g) .

Moreover, the following equation is a straightforward consequence of the de¬nitions:

(1b )β(g) = g b .

(13.7)

19 This theorem is proved under the assumption that char F = 2. See Exercise ?? for a sketch

of proof.

176 III. SIMILITUDES

Since b is injective, it follows that the map

β : GO(A, σ, f ) ’ AutA B(A, σ, f )

is injective. This map also is a homomorphism of groups; to check this, we compute,

for g, h ∈ GO(A, σ, f ) and x ∈ B(A, σ, f ):

β(h)

xβ(g)—¦β(h) = g · B(g ’1 )(x) .

Since β(h) is a homomorphism of left A-modules, the right-hand expression equals

β(h)

g · B(g ’1 )(x) = g · h · B(h’1 ) —¦ B(g ’1 )(x)

= gh · B(h’1 g ’1 )(x)

= xβ(gh) ,

proving the claim.

Let Z be the center of C(A, σ, f ). Recall the right ι C(A, σ, f )op —Z C(A, σ, f )-

module structure on B(A, σ, f ), which yields the canonical map

ν : ι C(A, σ, f ) —Z C(A, σ, f ) ’ EndA—Z B(A, σ, f )

of (??). It follows from (??) that the following equation holds in EndA B(A, σ, f ):

β(g) —¦ ν(ιcop — c2 ) = ν ι C(g)(c1 )op — C(g)(c2 ) —¦ β(g)

(13.8) 1

for all g ∈ GO(A, σ, f ), c1 , c2 ∈ C(A, σ, f ). In particular, it follows by (??) that

β(g) is Z-linear if and only if g is proper.

The following result describes the maps B(g) and β(g) in the split case; it

follows by the same arguments as in (??).

(13.9) Proposition. Suppose A is split; let

(A, σ, f ) = EndF (V ), σq , fq

for some nonsingular quadratic space (V, q). Under the standard identi¬cations

GO(A, σ, f ) = GO(V, q) and B(A, σ, f ) = V — C1 (V, q) (see (??)), the maps B

and β are given by

B(g)(v — w1 · · · w2r’1 ) = µ(g)’r g(v) — g(w1 ) · · · g(w2r’1 )

and

(v — w1 · · · w2r’1 )β(g) = µ(g)r v — g ’1 (w1 ) · · · g ’1 (w2r’1 )

for g ∈ GO(A, σ, f ) and v, w1 , . . . , w2r’1 ∈ V .

13.B. Cli¬ord groups. For a nonsingular quadratic space (V, q) over an ar-

bitrary ¬eld F , the special Cli¬ord group “+ (V, q) is de¬ned by

“+ (V, q) = { c ∈ C0 (V, q)— | c · V · c’1 ‚ V }

where the product c · V · c’1 is computed in the Cli¬ord algebra C(V, q) (see for

instance Knus [?, Ch. 4, §6], or Scharlau [?, §9.3] for the case where char F = 2).

Although there is no analogue of the (full) Cli¬ord algebra for an algebra with

quadratic pair, we show in this section that the Cli¬ord bimodule may be used to

de¬ne an analogue of the special Cli¬ord group. We also show that an extended

Cli¬ord group can be de¬ned by substituting the Cli¬ord algebra for the Cli¬ord

bimodule. These constructions are used to de¬ne spinor norm homomorphisms on

the groups O+ (A, σ, f ) and PGO+ (A, σ, f ).

§13. QUADRATIC PAIRS 177

The special Cli¬ord group in the split case. Let (V, q) be a nonsingular

quadratic space of even20 dimension over an arbitrary ¬eld F , and let “+ (V, q) be

the special Cli¬ord group de¬ned above. Conjugation by c ∈ “+ (V, q) in C(V, q)

induces an isometry of (V, q), since

q(c · v · c’1 ) = (c · v · c’1 )2 = v 2 = q(v) for v ∈ V .

We set χ(c) for this isometry:

χ(c)(v) = c · v · c’1 for v ∈ V .

The map χ : “+ (V, q) ’ O(V, q) is known as the vector representation of the special

Cli¬ord group. The next proposition shows that its image is in O+ (V, q).

(13.10) Proposition. The elements in “+ (V, q) are similitudes of the even Clif-

ford algebra C0 = C0 (V, q) for the canonical involution „0 (see (??)). More pre-

cisely, „0 (c) · c ∈ F — for all c ∈ “+ (V, q). The vector representation χ and the

canonical homomorphism C of (??) ¬t into the following commutative diagram

with exact rows:

χ

O+ (V, q)

1 ’ ’ ’ F— ’ ’ ’ “+ (V, q)

’’ ’’ ’’’

’’ ’’’ 1

’’

¦ ¦ ¦

¦ ¦ ¦

C

Int

1 ’ ’ ’ Z — ’ ’ ’ Sim(C0 , „0 ) ’ ’ ’ AutZ (C0 , „0 ) ’ ’ ’ 1

’’ ’’ ’’ ’’

where Z denotes the center of C0 .

Proof : Let „ be the canonical involution on C(V, q), whose restriction to (the image

of) V is the identity. For c ∈ “+ (V, q) and v ∈ V , we have c · v · c’1 ∈ V , hence

c · v · c’1 = „ (c · v · c’1 ) = „0 (c)’1 · v · „0 (c).

This shows that the element „0 (c) · c centralizes V ; since V generates C(V, q),

it follows that „0 (c) · c is central in C(V, q), hence „0 (c) · c ∈ F — . This proves

“+ (V, q) ‚ Sim(C0 , „0 ).

The elements in ker χ centralize V , hence the same argument as above shows

ker χ = F — .

Let c ∈ “+ (V, q). By (??), the automorphism C χ(c) of C0 maps v1 · · · v2r to

χ(c)(v1 ) · · · χ(c)(v2r ) = c · (v1 · · · v2r ) · c’1

for v1 , . . . , v2r ∈ V , hence

C χ(c) = Int(c).

This automorphism is the identity on Z, hence (??) shows that χ(c) ∈ O+ (V, q).

Moreover, the last equation proves that the diagram commutes. Therefore, it re-

mains only to prove surjectivity of χ onto O+ (V, q).

To prove that every proper isometry is in the image of χ, observe that for every

v, x ∈ V with q(v) = 0,

v · x · v ’1 = v ’1 bq (v, x) ’ x = vq(v)’1 bq (v, x) ’ x,

hence the hyperplane re¬‚ection ρv : V ’ V satis¬es

ρv (x) = ’v · x · v ’1 for all x ∈ V .

20 Cli¬ord groups are also de¬ned in the odd-dimensional case, where results similar to those

of this section can be established. Since we are interested in the generalization to the nonsplit

case, given below, we consider only the even-dimensional case.

178 III. SIMILITUDES

Therefore, for anisotropic v1 , v2 ∈ V , we have v1 · v2 ∈ “+ (V, q) and

χ(v1 · v2 ) = ρv1 —¦ ρv2 .

The Cartan-Dieudonn´ theorem (see Dieudonn´ [?, pp. 20, 42], or Scharlau [?,

e

e

p. 15] for the case where char F = 2) shows that the group O(V, q) is generated by

hyperplane re¬‚ections, except in the case where F is the ¬eld with two elements,

dim V = 4 and q is hyperbolic. Since hyperplane re¬‚ections are improper isometries

(see (??)), it follows that every proper isometry has the form

ρv1 —¦ · · · —¦ ρv2r = χ(v1 · · · v2r )

for some anisotropic vectors v1 , . . . , v2r ∈ V , in the nonexceptional case.

Direct computations, which we omit, prove that χ is surjective in the excep-

tional case as well.

The proof shows that every element in “+ (V, q) is a product of an even number

of anisotropic vectors in V , except when (V, q) is the 4-dimensional hyperbolic space

over the ¬eld with two elements.

The Cli¬ord group of an algebra with quadratic pair. Let (σ, f ) be a

quadratic pair on a central simple algebra A of even degree over an arbitrary ¬eld F .

The Cli¬ord group consists of elements in C(A, σ, f ) which preserve the image Ab

of A in the bimodule B(A, σ, f ) under the canonical map b : A ’ B(A, σ, f ) of (??):

(13.11) De¬nition. The Cli¬ord group “(A, σ, f ) is de¬ned by

“(A, σ, f ) = { c ∈ C(A, σ, f )— | c’1 — Ab · c ‚ Ab }.

Since the C(A, σ, f )-bimodule actions on B(A, σ, f ) commute with the left A-

module action and since the canonical map b is a homomorphism of left A-modules,

the condition de¬ning the Cli¬ord group is equivalent to

c’1 — 1b · c ∈ Ab .

For c ∈ “(A, σ, f ), de¬ne χ(c) ∈ A by the equation

c’1 — 1b · c = χ(c)b .

(The element χ(c) is uniquely determined by this equation, since the canonical

map b is injective: see (??)).

(13.12) Proposition. In the split case (A, σ, f ) = EndF (V ), σq , fq , the standard

identi¬cations C(A, σ, f ) = C0 (V, q), B(A, σ, f ) = V — C1 (V, q) of (??), (??),

induce an identi¬cation “(A, σ, f ) = “+ (V, q), and the map χ de¬ned above is the

vector representation.

Proof : Under the standard identi¬cations, we have A = V — V and Ab = V — V ‚

V — C1 (V, q). Moreover, for c ∈ C(A, σ, f ) = C0 (V, q) and v, w ∈ V ,

c’1 — (v — w)b · c = v — (c’1 · w · c).

(13.13)

Therefore, the condition c’1 — Ab · c ‚ Ab amounts to:

v — (c’1 · w · c) ∈ V — V for v, w ∈ V ,

or c’1 · V · c ‚ V . This proves the ¬rst assertion.

Suppose now that c’1 —1b ·c = g b . Since b is a homomorphism of left A-modules,

we then get for all v, w ∈ V :

b

c’1 — (v — w)b · c = (v — w) · (c’1 — 1b · c) = (v — w) —¦ g .

(13.14)

§13. QUADRATIC PAIRS 179

By evaluating (v — w) —¦ g at an arbitrary x ∈ V , we obtain

vbq w, g(x) = vbq σ(g)(w), x = v — σ(g)(w) (x),

hence (v — w) —¦ g = v — σ(g)(w). Therefore, (??) yields

b

c’1 — (v — w)b · c = v — σ(g)(w) .

In view of (??), this shows: σ(g)(w) = c’1 · w · c, so

χ(c) = σ(g)’1 = g.

By extending scalars to a splitting ¬eld of A, it follows from the proposition

above and (??) that χ(c) ∈ O+ (A, σ, f ) for all c ∈ “(A, σ, f ). The commutative

diagram of (??) has an analogue for algebras with quadratic pairs:

(13.15) Proposition. For brevity, set C(A) = C(A, σ, f ). Let Z be the center

of C(A) and let σ denote the canonical involution on C(A). For all c ∈ “(A, σ, f ) we

have σ(c)c ∈ F — , hence “(A, σ, f ) ‚ Sim C(A), σ . The map χ and the restriction

to O+ (A, σ, f ) of the canonical map

C : GO(A, σ, f ) ’ AutF C(A), σ

¬t into a commutative diagram with exact rows:

χ

O+ (A, σ, f )

1 ’ ’ ’ F— ’ ’ ’

’’ ’’ “(A, σ, f ) ’’’

’’ ’’’ 1

’’

¦ ¦ ¦

¦ ¦ ¦

C

Int

1 ’ ’ ’ Z — ’ ’ ’ Sim C(A), σ

’’ ’’ ’ ’ ’ AutZ C(A), σ

’’ ’’’ 1

’’

Proof : All the statements follow by scalar extension to a splitting ¬eld of A and

comparison with (??), except for the surjectivity of χ onto O+ (A, σ, f ).

To prove this last point, recall the isomorphism ν of (??) induced by the C(A)-

bimodule structure on B(A, σ, f ):

∼

ν : ι C(A)op —Z C(A) ’ EndA—Z B(A, σ, f ) .

’

This isomorphism satis¬es

ι op

xν( c1 —c2 )

= c1 — x · c2 for x ∈ B(A, σ, f ), c1 , c2 ∈ C(A).

For g ∈ O+ (A, σ, f ), it follows from (??) that C(g) is the identity on Z, hence β(g)

is an A — Z-endomorphism of B(A, σ, f ), by (??). Therefore, there exists a unique

element ξ ∈ ι C(A)op —Z C(A) such that ν(ξ) = β(g).

In the split case, (??) shows that ξ = ι (c’1 )op — c, where c ∈ “+ (V, q) is such

that χ(c) = g. Since the minimal number of terms in a decomposition of an element

of a tensor product is invariant under scalar extension, it follows that ξ = ιcop — c2

1

for some c1 , c2 ∈ C(A). Moreover, if s is the switch map on ι C(A)op —Z C(A),

de¬ned by

op

s(ιcop — c ) = ιc — c for c, c ∈ C(A),

then s(ξ)ξ = 1, since ξ = ι (c’1 )op — c over a splitting ¬eld. Therefore, the elements

c1 , c2 ∈ C(A)— satisfy

c1 c2 = » ∈ Z with NZ/F (») = 1.

180 III. SIMILITUDES

By Hilbert™s Theorem 90 (??), there exists »1 ∈ Z such that » = »1 ι(»1 )’1 .

(Explicitly, we may take »1 = z + »ι(z), where z ∈ Z is such that z + »ι(z) is

invertible.) Then ξ = ι (c’1 )op — c3 for c3 = »’1 c2 , hence

3 1

xβ(g) = c’1 — x · c3 for x ∈ B(A, σ, f ).

3

In particular, for x = 1b we obtain by (??):

c’1 — 1b · c3 = g b .

3

This shows that c3 ∈ “(A, σ, f ) and χ(c3 ) = g.

(13.16) Corollary. Suppose deg A ≥ 4; then

“(A, σ, f ) © Z = { z ∈ Z — | z 2 ∈ F — };

thus “(A, σ, f ) © Z = F — if char F = 2, and “(A, σ, f ) © Z = F — ∪ z · F — if

char F = 2 and Z = F [z] with z 2 ∈ F — .

Proof : Since χ is surjective, it maps “(A, σ, f ) © Z to the center of O+ (A, σ, f ). It

follows by scalar extension to a splitting ¬eld of A that the center of O+ (A, σ, f )

is trivial if char F = 2 and is {1, ’1} if char F = 2 (see Dieudonn´ [?, pp. 25, 45]).

e

—

Therefore, the factor group “(A, σ, f ) © Z /F is trivial if char F = 2 and has two

elements if char F = 2, proving the corollary.

(13.17) Example. Suppose Q is a quaternion F -algebra with canonical involution

γ. If char F = 2, let σ = Int(u) —¦ γ for some invertible pure quaternion u. Let F (u)1

be the group of elements of norm 1 in F (u):

F (u)1 = { z ∈ F (u) | zγ(z) = 1 }.

As observed in (??), we have O+ (Q, σ) = F (u)1 . On the other hand, it follows

from the structure theorem for Cli¬ord algebras (??) that C(Q, σ) is a commutative

algebra isomorphic to F (u). The Cli¬ord group “(Q, σ) is the group of invertible

elements in C(Q, σ). It can be identi¬ed with F (u)— in such a way that the vector

representation χ maps x ∈ F (u)— to xγ(x)’1 ∈ F (u)1 . The upper exact sequence

of the commutative diagram in (??) thus takes the form

1’γ

1 ’ F — ’’ F (u)— ’’ F (u)1 ’ 1.

’

Similar results hold if char F = 2. Using the same notation as in (??) and (??),

we have O+ (Q, γ, f ) = F ( )1 , “(Q, γ, f ) F ( )— , and the upper exact sequence of

the commutative diagram in (??) becomes

1’γ

1 ’ F — ’’ F ( )— ’’ F ( )1 ’ 1.

’

The extended Cli¬ord group. In this subsection, we de¬ne an intermediate

group „¦(A, σ, f ) between “(A, σ, f ) and Sim C(A, σ, f ), σ , which covers the group

PGO+ (A, σ, f ) in the same way as “(A, σ, f ) covers O+ (A, σ, f ) by the vector rep-

resentation χ. This construction will enable us to de¬ne an analogue of the spinor

norm for the group PGO+ (A, σ).

The notation is as above: (σ, f ) is a quadratic pair on a central simple algebra A

of even degree over an arbitrary ¬eld F . Let Z be the center of the Cli¬ord algebra

C(A, σ, f ). Since (??) plays a central rˆle almost from the start (we need injectivity

o

of the canonical map C for the de¬nition of χ in (??) below), we exclude the case

of quaternion algebras from our discussion. We thus assume

deg A = n = 2m ≥ 4.

§13. QUADRATIC PAIRS 181

We identify PGO(A, σ, f ) with AutF (A, σ, f ) by mapping g · F — to Int(g), for

g ∈ GO(A, σ, f ). By (??) and (??), the canonical map C induces an injective

homomorphism C : PGO+ (A, σ, f ) ’ AutZ C(A, σ, f ), σ . Consider the following

diagram:

PGO+ (A, σ, f )

¦

¦

C

Int

Sim C(A, σ, f ), σ ’ ’ ’ AutZ C(A, σ, f ), σ .

’’

(13.18) De¬nition. The extended Cli¬ord group of (A, σ, f ) is the inverse image

under Int of the image of the canonical map C:

„¦(A, σ, f ) = { c ∈ Sim C(A, σ, f ), σ | Int(c) ∈ C PGO+ (A, σ, f ) }.

Thus, „¦(A, σ, f ) ‚ Sim C(A, σ, f ), σ , and there is an exact sequence

χ

1 ’ Z — ’ „¦(A, σ, f ) ’ PGO+ (A, σ, f ) ’ 1

(13.19) ’

where the map χ is de¬ned by

if Int(c) = C(g), with g ∈ GO+ (A, σ, f ).

χ (c) = g · F —

If char F = 2, the group „¦(A, σ, f ) = „¦(A, σ) may alternately be de¬ned by

„¦(A, σ) = { c ∈ C(A, σ)— | c · c(A) · c’1 = c(A) },

since the Z-automorphisms of C(A, σ) which preserve c(A) are exactly those which

are of the form C(g) for some g ∈ GO+ (A, σ), by (??). We shall not use this

alternate de¬nition, since we want to keep the characteristic arbitrary.

For c ∈ “(A, σ, f ), we have Int(c) = C(g) for some g ∈ O+ (A, σ, f ), by (??),

hence “(A, σ, f ) ‚ „¦(A, σ, f ). Our ¬rst objective in this subsection is to describe

“(A, σ, f ) as the kernel of a map κ : „¦(A, σ, f ) ’ Z — /F — .

The multiplier map µ : GO(A, σ, f ) ’ F — induces a map

µ : PGO+ (A, σ, f ) ’ F — /F —2 ,

since µ(±) = ±2 for all ± ∈ F — . This map ¬ts into an exact sequence:

µ

π

O+ (A, σ, f ) ’ PGO+ (A, σ, f ) ’ F — /F —2 .

’ ’

(13.20) Lemma. The kernel of the map µ —¦ χ : „¦(A, σ, f ) ’ F — /F —2 is the sub-

group Z — · “(A, σ, f ). In particular, if F is algebraically closed, then, since µ is

trivial, „¦(A, σ, f ) = Z — · “(A, σ, f ).

Proof : For c ∈ “(A, σ, f ), we have χ (c) = g · F — for some g ∈ O+ (A, σ, f ), hence

µ—¦χ (c) = 1. Since ker χ = Z — , the inclusion Z — ·“(A, σ, f ) ‚ ker(µ—¦χ ) follows. In

order to prove the reverse inclusion, pick c ∈ ker(µ—¦χ ); then χ (c) = g·F — for some

g ∈ GO+ (A, σ, f ) such that µ(g) ∈ F —2 . Let µ(g) = ±2 for some ± ∈ F — . Then

±’1 g ∈ O+ (A, σ, f ), so there is an element γ ∈ “(A, σ, f ) such that χ(γ) = ±’1 g.

We then have

Int(γ) = C(±’1 g) = C(g) = Int(c),

hence c = z · γ for some z ∈ Z — .

182 III. SIMILITUDES

We now de¬ne a map

κ : „¦(A, σ, f ) ’ Z — /F —

as follows: for ω ∈ „¦(A, σ, f ), we pick g ∈ GO+ (A, σ, f ) such that χ (ω) = g · F — ;

then µ(g)’1 g 2 ∈ O+ (A, σ, f ), hence there exists γ ∈ “(A, σ, f ) such that χ(γ) =

µ(g)’1 g 2 . By (??) it follows that Int(γ) = C(g 2 ) = Int(ω 2 ), hence

ω2 = z · γ for some z ∈ Z — .

We then set

κ(ω) = z · F — ∈ Z — /F — .

To check that κ is well-de¬ned, suppose g ∈ GO+ (A, σ, f ) also satis¬es χ (ω) =

2

C(g ). We then have g ≡ g mod F — , hence µ(g)’1 g 2 = µ(g )’1 g . On the other

hand, the element γ ∈ “(A, σ, f ) such that χ(γ) = µ(g)’1 g 2 is uniquely determined

up to a factor in F — , by (??), hence the element z ∈ Z — is uniquely determined

modulo F — .

Note that, if char F = 2, the element z is not uniquely determined by the

condition that ω 2 = z · γ for some γ ∈ “(A, σ, f ), since “(A, σ, f ) © Z — = F — (see

(??)).

(13.21) Proposition. The map κ : „¦(A, σ, f ) ’ Z — /F — is a group homomor-

phism and ker κ = “(A, σ, f ).

Proof : It su¬ces to prove the proposition over an algebraic closure. We may thus

assume F is algebraically closed; (??) then shows that

„¦(A, σ, f ) = Z — · “(A, σ, f ).

For z ∈ Z — and γ ∈ “(A, σ, f ) we have κ(z · γ) = z 2 · F — , hence κ is a group

homomorphism. Moreover, ker κ consists of the elements z · γ such that z 2 ∈

F — . In view of (??), this condition implies that z ∈ “(A, σ, f ), hence ker κ =

“(A, σ, f ).

Our next objective is to relate κ(ω) to µ —¦ χ (ω), for ω ∈ „¦(A, σ, f ). We need

the following classical result of Dieudonn´, which will be generalized in the next

e

section:

(13.22) Lemma (Dieudonn´). Let (V, q) be a nonsingular even-dimensional quad-

e

ratic space over an arbitrary ¬eld F . Let Z be the center of the even Cli¬ord al-

gebra C0 (V, q). For every similitude g ∈ GO(V, q), the multiplier µ(g) is a norm

from Z/F .

Proof : The similitude g may be viewed as an isometry µ(g) · q q. Therefore,

the quadratic form q ⊥ ’ µ(g) · q is hyperbolic. The Cli¬ord algebra of any

form 1, ’± · q is Brauer-equivalent to the quaternion algebra21 (Z, ±)F (see for

instance [?, (3.22), p. 47]), hence Z, µ(g) F splits, proving that µ(g) is a norm

from Z/F .

Continuing with the same notation, and assuming dim V ≥ 4, consider ω ∈

„¦ EndF (V ), σq , fq ‚ C0 (V, q) and g ∈ GO+ (V, q) such that χ (ω) = g · F — . The

preceding lemma yields an element z ∈ Z — such that µ(g) = NZ/F (z) = zι(z),

where ι is the nontrivial automorphism of Z/F .

21 We use the same notation as in § ??.

§13. QUADRATIC PAIRS 183

(13.23) Lemma. There exists an element z0 ∈ Z — such that κ(ω) = (zz0 )’1 · F —

2

and, in C(V, q),

’1

ω · v · ω ’1 = ι(z0 )z0 z ’1 g(v) for v ∈ V .

Proof : For all v ∈ V , we have in C(V, q)

2

z ’1 g(v) = z ’1 ι(z)’1 g(v)2 = µ(g)’1 q g(v) = q(v),

hence the map v ’ z ’1 g(v) extends to an automorphism of C(V, q), by the universal

property of Cli¬ord algebras. By the Skolem-Noether theorem, we may represent

this automorphism as Int(c) for some c ∈ C(V, q)— . For v, ∈ V , we then have

Int(c)(v · w) = z ’1 g(v) · z ’1 g(w) = µ(g)’1 g(v) · g(w),

hence (??) shows that the restriction of Int(c) to C0 (V, q) is C(g). Since g is a proper

similitude, it follows from (??) that Int(c) is the identity on Z, hence c ∈ C0 (V, q)— .

Moreover, Int(c)|C0 (V,q) = Int(ω)|C0 (V,q) since χ (ω) = g · F — , hence c = z0 ω for

some z0 ∈ Z — . It follows that for all v ∈ V ,

’1 ’1

ω · v · ω ’1 = z0 c · v · c’1 z0 = ι(z0 )z0 z ’1 g(v).

Observe next that

Int(c2 )(v) = z ’2 g 2 (v) for v ∈ V ,

since c commutes with z. If γ ∈ “+ (V, q) satis¬es χ(γ) = µ(g)’1 g 2 , then

γ · v · γ ’1 = µ(g)’1 g 2 (v) for v ∈ V ,

hence γ ’1 zc2 centralizes V . Since V generates C(V, q), it follows that

γ ≡ zc2 ≡ zz0 ω

2

mod F — .

By de¬nition of κ, these congruences yield κ(ω) = (zz0 )’1 · F — .

2

The main result of this subsection is the following:

(13.24) Proposition. The following diagram is commutative with exact rows and

columns:

χ

O+ (A, σ, f )

1 ’ ’ ’ F — ’ ’ ’ “(A, σ, f ) ’ ’ ’

’’ ’’ ’’ ’’’ 1

’’

¦ ¦ ¦

¦ ¦ ¦π

χ

1 ’ ’ ’ Z — ’ ’ ’ „¦(A, σ, f ) ’ ’ ’ PGO+ (A, σ, f ) ’ ’ ’ 1

’’ ’’ ’’ ’’

¦ ¦

¦κ ¦µ

NZ/F

Z — /F — F — /F —2 .

’’’

’’

Proof : In view of (??) and (??), it su¬ces to prove commutativity of the lower

square. By extending scalars to a splitting ¬eld of A in which F is algebraically

closed, we may assume that A is split. Let (V, q) be a nonsingular quadratic space

such that (A, σ, f ) = EndF (V ), σq , fq .

Fix some ω ∈ „¦(A, σ, f ) and g ∈ GO+ (V, q) such that χ (ω) = g · F — . Let

z ∈ Z — satisfy µ(g) = NZ/F (z). The preceding lemma yields z0 ∈ Z — such that

κ(ω) = (zz0 )’1 · F — . Then

2

NZ/F κ(ω) = NZ/F (zz0 )’1 · F —2 = µ(g) · F —2 .

2

184 III. SIMILITUDES

To conclude our discussion of the extended Cli¬ord group, we examine more

closely the case where deg A is divisible by 4. In this case, the map κ factors through

the multiplier map, and the homomorphism χ factors through a homomorphism

χ0 : „¦(A, σ, f ) ’ GO+ (A, σ, f ).

We denote by µ : „¦(A, σ, f ) ’ Z — the multiplier map, de¬ned by

µ(ω) = σ(ω)ω for ω ∈ „¦(A, σ, f ).

The element µ(ω) thus de¬ned belongs to Z — since „¦(A, σ, f ) ‚ Sim C(A, σ, f ), σ .

(13.25) Proposition. Suppose deg A ≡ 0 mod 4. For all ω ∈ „¦(A, σ, f ),

κ(ω) = µ(ω) · F — .

The Cli¬ord group can be characterized as

“(A, σ, f ) = { ω ∈ „¦(A, σ, f ) | µ(ω) ∈ F — }.

Proof : The second part follows from the ¬rst, since (??) shows that “(A, σ, f ) =

ker κ.

To prove the ¬rst part, we may extend scalars to an algebraic closure. For

ω ∈ „¦(A, σ, f ) we may thus assume, in view of (??), that there exist z ∈ Z — and

γ ∈ “(A, σ, f ) such that ω = z · γ. We then have κ(ω) = z 2 · F — . On the other

hand, since deg A ≡ 0 mod 4 the involution σ is of the ¬rst kind, by (??), hence

µ(ω) = z 2 µ(γ). Now, (??) shows that µ(γ) ∈ F — , hence

µ(ω) · F — = z 2 · F — = κ(ω).

Another interesting feature of the case where deg A ≡ 0 mod 4 is that the

extended Cli¬ord group has an alternate description similar to the de¬nition of the

Cli¬ord group. In the following proposition, we consider the image Ab of A in the

Cli¬ord bimodule B(A, σ, f ) under the canonical map b : A ’ B(A, σ, f ).

(13.26) Proposition. If deg A ≡ 0 mod 4, then

„¦(A, σ, f ) = { ω ∈ Sim C(A, σ, f ), σ | σ(ω) — Ab · ω = Ab }.

Proof : Let ω ∈ Sim C(A, σ, f ), σ . We have to show that Int(ω) = C(g) for some

g ∈ GO+ (A, σ, f ) if and only if σ(ω) — Ab · ω = Ab .

Assume ¬rst that ω ∈ „¦(A, σ, f ), i.e., Int(ω) = C(g) for some g ∈ GO+ (A, σ, f ).

To prove the latter equality, we may reduce by scalar extension to the split case.

Thus, suppose that (V, q) is a nonsingular quadratic space of dimension divisible

by 4 and (A, σ, f ) = EndF (V ), σq , fq . Under the standard identi¬cations asso-

ciated to q we have Ab = V — V ‚ V — C1 (V, q) (see (??)), hence it su¬ces to

show

σ(ω) · v · ω ∈ V for v ∈ V .

Let ι be, as usual, the nontrivial automorphism of the center Z of C(A, σ, f ) =

C0 (V, q) over F , and let z ∈ Z — be such that µ(g) = zι(z). By (??), there is an

element z0 ∈ Z — such that κ(ω) = (zz0 )’1 · F — and

2

’1

ω · v · ω ’1 = ι(z0 )z0 z ’1 g(v) for v ∈ V .

§13. QUADRATIC PAIRS 185

The canonical involution „ of C(V, q) restricts to σ = „0 on C0 (V, q); by (??)

this involution is the identity on Z. Therefore, by applying „ to each side of the

preceding equation, we obtain

’1

σ(ω)’1 · v · σ(ω) = g(v)ι(z0 )z0 z ’1 = z0 ι(z0 )’1 ι(z)’1 g(v) for v ∈ V ,

hence

’1

σ(ω) · v · ω = z0 ι(z0 )ι(z)g ’1 (v)µ(ω) for v ∈ V .

By (??), we have µ(ω) · F — = κ(ω), hence µ(ω) = ±(zz0 )’1 for some ± ∈ F — . By

2

substituting this in the preceding relation, we get for all v ∈ V

’1

σ(ω) · v · ω = ±z0 ι(z0 )ι(z)ι(zz0 )’1 g ’1 (v) = ±NZ/F (z0 )’1 g ’1 (v).

2

Since the right-hand term lies in V , we have thus shown σ(ω) — Ab · ω = Ab .

Suppose conversely that ω ∈ Sim C(A, σ, f ), σ satis¬es σ(ω) — Ab · ω = Ab .

Since b is injective, there is a unique element g ∈ A such that

σ(ω) — 1b · ω = g b .

(13.27)

We claim that g ∈ GO+ (A, σ, f ) and that Int(ω) = C(g). To prove the claim,

we may extend scalars to a splitting ¬eld of A; we may thus assume again that

(A, σ, f ) = EndF (V ), σq , fq for some nonsingular quadratic space (V, q) of dimen-

sion divisible by 4, and use the standard identi¬cations A = V — V , Ab = V — V ‚

B(A, σ, f ) = V — C1 (V, q). Since B(A, σ, f ) is a left A-module, we may multiply

each side of (??) by v — w ∈ V — V = A; we thus obtain

b

σ(ω) — (v — w)b · ω = (v — w) —¦ g for v, w ∈ V .

Since (v — w) —¦ g = v — σ(g)(w), it follows that, in C(V, q),

σ(ω) · w · ω = σ(g)(w) for w ∈ V .

Since ω is a similitude of C(A, σ, f ), by squaring this equation we obtain

NZ/F µ(ω) q(w) = q σ(g)(w) for w ∈ V .

This shows that σ(g) is a similitude, hence g ∈ GO(V, q), and

(13.28) µ σ(g) = µ(g) = NZ/F µ(ω) .

Moreover, for v, ∈ V ,

ω ’1 · (v · w) · ω =µ(ω)’1 σ(ω) · v · ω µ(ω)’1 σ(ω) · w · ω

’1

=NZ/F µ(ω) σ(g)(v) · σ(g)(w).

By (??) it follows that

ω ’1 · (v · w) · ω = µ(g)’1 σ(g)(v) · σ(g)(w) for v, w ∈ V ,

hence, by (??),

ω · (v · w) · ω ’1 = µ(g)’1 g(v) · g(w) = C(g)(v · w) for v, w ∈ V .

This equality shows that Int(ω)|C0 (V,q) = C(g), hence g is proper, by (??), and the

proof is complete.

186 III. SIMILITUDES

(13.29) De¬nition. Suppose deg A ≡ 0 mod 4. By using the description of the

extended Cli¬ord group „¦(A, σ, f ) in the proposition above, we may de¬ne a ho-

momorphism

χ0 : „¦(A, σ, f ) ’ GO+ (A, σ, f )

mapping ω ∈ „¦(A, σ, f ) to the element g ∈ GO+ (A, σ, f ) satisfying (??). Thus, for

ω ∈ „¦(A, σ, f ) the similitude χ0 (ω) is de¬ned by the relation

σ(ω) — 1b · ω = χ0 (ω)b .

The proof above shows that

χ (ω) = χ0 (ω) · F — for ω ∈ „¦(A, σ, f );

moreover, by (??), the following diagram commutes:

χ0

„¦(A, σ, f ) ’ ’ ’ GO+ (A, σ, f )

’’

¦ ¦

¦ ¦µ

µ

NZ/F

Z— F —.

’’’

’’

Spinor norms. Let (σ, f ) be a quadratic pair on a central simple algebra A

of even degree over an arbitrary ¬eld F .

(13.30) De¬nition. In view of (??), we may de¬ne a homomorphism

Sn : O+ (A, σ, f ) ’ F — /F —2

as follows: for g ∈ O+ (A, σ, f ), pick γ ∈ “(A, σ, f ) such that χ(γ) = g and let

Sn(g) = σ(γ)γ · F —2 = µ(γ) · F —2 .

This square class depends only on g, since γ is uniquely determined up to a factor

in F — . In other words, Sn is the map which makes the following diagram commute:

χ

1 ’ ’ ’ F — ’ ’ ’ “(A, σ, f ) ’ ’ ’ O+ (A, σ, f ) ’ ’ ’ 1

’’ ’’ ’’ ’’

¦ ¦ ¦

¦ ¦µ ¦

2 Sn

1 ’ ’ ’ F —2 ’ ’ ’ F— F — /F —2

’’ ’’ ’’’

’’ ’ ’ ’ 1.

’’

We also de¬ne the group of spinor norms:

Sn(A, σ, f ) = { µ(γ) | γ ∈ “(A, σ, f ) } ‚ F — ,

so Sn O+ (A, σ, f ) = Sn(A, σ, f )/F —2 , and the spin group:

Spin(A, σ, f ) = { γ ∈ “(A, σ, f ) | µ(γ) = 1 } ‚ “(A, σ, f ).

In the split case, if (A, σ, f ) = EndF (V ), σq , fq for some nonsingular quadratic

space (V, q) of even dimension, the standard identi¬cations associated to q yield

Spin(A, σ, f ) = Spin(V, q) = { c ∈ “+ (V, q) | „ (c) · c = 1 },

where „ is the canonical involution of C(V, q) which is the identity on V . From

the description of the spinor norm in Scharlau [?, Chap. 9, §3], it follows that the

group of spinor norms Sn(V, q) = Sn(A, σ, f ) consists of the products of any even

number of represented values of q.

§13. QUADRATIC PAIRS 187

The vector representation χ induces by restriction a homomorphism

Spin(A, σ, f ) ’ O+ (A, σ, f )

which we also denote χ.

(13.31) Proposition. The vector representation χ ¬ts into an exact sequence:

χ Sn

1 ’ {±1} ’’ Spin(A, σ, f ) ’’ O+ (A, σ, f ) ’’ F — /F —2 .

Proof : This follows from the exactness of the top sequence in (??) and the de¬nition

of Sn.

Assume now deg A = 2m ≥ 4. We may then use the extended Cli¬ord group

„¦(A, σ, f ) to de¬ne an analogue of the spinor norm on the group PGO+ (A, σ, f ), as

we proceed to show. The map S de¬ned below may be obtained as a connecting map

in a cohomology sequence, see §??. Its target group is the ¬rst cohomology group of

the absolute Galois group of F with coe¬cients in the center of the algebraic group

Spin(A, σ, f ). The approach we follow in this subsection does not use cohomology,

but since the structure of the center of Spin(A, σ, f ) depends on the parity of m,

we divide the construction into two parts, starting with the case where the degree

of A is divisible by 4. As above, we let Z = Z(A, σ, f ) be the center of the Cli¬ord

algebra C(A, σ, f ).

(13.32) De¬nition. Assume deg A ≡ 0 mod 4. We de¬ne a homomorphism

S : PGO+ (A, σ, f ) ’ Z — /Z —2

as follows: for g · F — ∈ PGO+ (A, σ, f ), pick ω ∈ „¦(A, σ, f ) such that χ (ω) = g · F —

and set

S(g · F — ) = σ(ω)ω · Z —2 = µ(ω) · Z —2 .

Since ω is determined by g · F — up to a factor in Z — and σ is of the ¬rst kind, by

(??), the element µ(ω)ω · Z —2 depends only on g · F — . The map S thus makes the

following diagram commute:

χ

1 ’ ’ ’ Z — ’ ’ ’ „¦(A, σ, f ) ’ ’ ’ PGO+ (A, σ, f ) ’ ’ ’ 1

’’ ’’ ’’ ’’

¦ ¦ ¦

¦ ¦µ ¦

2 S

1 ’ ’ ’ Z —2 ’ ’ ’ Z— Z — /Z —2

’’ ’’ ’’’

’’ ’ ’ ’ 1.

’’

Besides the formal analogy between the de¬nition of S and that of the spinor

norm Sn, there is also an explicit relationship demonstrated in the following propo-

sition:

(13.33) Proposition. Assume deg A ≡ 0 mod 4. Let

π : O+ (A, σ, f ) ’ PGO+ (A, σ, f )

be the canonical map. Then, the following diagram is commutative with exact rows:

µ

π

O+ (A, σ, f ) ’ ’ ’ PGO+ (A, σ, f ) ’ ’ ’ F — /F —2

’’ ’’

¦ ¦

¦ ¦

Sn S

NZ/F

F — /F —2 Z — /Z —2 ’ ’ ’ F — /F —2 .

’’’

’’ ’’

188 III. SIMILITUDES

Proof : Consider g ∈ O+ (A, σ, f ) and γ ∈ “(A, σ, f ) such that χ(γ) = g. We then

have Sn(g) = µ(γ) · F —2 . On the other hand, we also have χ (γ) = g · F — , hence

S(g · F — ) = µ(γ) · Z —2 . This proves that the left square is commutative.

Consider next g · F — ∈ PGO+ (A, σ, f ) and ω ∈ „¦(A, σ, f ) such that χ (ω) =

g · F — , so that S(g · F — ) = µ(ω) · Z —2 . By (??) we have µ(ω) · F — = κ(ω) and, by

(??), NZ/F κ(ω) = µ(g) · F —2 . Therefore, NZ/F S(g · F — ) = µ(g) · F —2 and the

right square is commutative.

Exactness of the lower sequence is a consequence of Hilbert™s Theorem 90 (??):

if z ∈ Z — is such that zι(z) = x2 for some x ∈ F — , then NZ/F (zx’1 ) = 1, hence

by (??) there exists some y ∈ Z — such that zx’1 = ι(y)y ’1 . (Explicitly, we may

take y = t + xz ’1 ι(t), where t ∈ Z is such that t + xz ’1 ι(t) is invertible.) Then

zy 2 = xNZ/F (y), hence z · Z —2 lies in the image of F — /F —2 .

Note that the spin group may also be de¬ned as a subgroup of the extended

Cli¬ord group: for deg A ≡ 0 mod 4,

Spin(A, σ, f ) = { ω ∈ „¦(A, σ, f ) | µ(ω) = 1 }.

Indeed, (??) shows that the right side is contained in “(A, σ, f ).

The restriction of the homomorphism

χ : „¦(A, σ, f ) ’ PGO+ (A, σ, f )

to Spin(A, σ, f ), also denoted χ , ¬ts into an exact sequence:

(13.34) Proposition. Assume deg A ≡ 0 mod 4. The sequence

χ S

1 ’ µ2 (Z) ’ Spin(A, σ, f ) ’ PGO+ (A, σ, f ) ’ Z — /Z —2

’ ’

is exact, where µ2 (Z) = { z ∈ Z | z 2 = 1 }.

Proof : Let g · F — be in the kernel of S. Then there exists ω ∈ „¦(A, σ, f ), with

σ(ω)ω = z 2 for some z ∈ Z — , such that χ (ω) = g · F — . By replacing ω by ωz ’1 ,

we get ω ∈ Spin(A, σ, f ). Exactness at Spin(A, σ, f ) follows from the fact that

Z — © Spin(A, σ, f ) = µ2 (Z) in „¦(A, σ, f ).

In the case where deg A ≡ 2 mod 4, the involution σ is of the second kind,

hence µ(ω) ∈ F — for all ω ∈ „¦(A, σ, f ). The rˆle played by µ in the case where

o

deg A ≡ 0 mod 4 is now played by a map which combines µ and κ.

Consider the following subgroup U of F — — Z — :

U = { (±, z) | ±4 = NZ/F (z) } ‚ F — — Z —

and its subgroup U0 = { NZ/F (z), z 4 | z ∈ Z — }. Let22

H 1 (F, µ4 [Z] ) = U/U0 ,

and let [±, z] be the image of (±, z) ∈ U in H 1 (F, µ4 [Z] ).

For ω ∈ „¦(A, σ, f ), let k ∈ Z — be a representative of κ(ω) ∈ Z — /F — . The

element kι(k)’1 is independent of the choice of the representative k and we de¬ne

µ— (ω) = µ(ω), kι(k)’1 µ(ω)2 ∈ U.

22 It

will be seen in Chapter ?? (see (??)) that this factor group may indeed be regarded as

a Galois cohomology group if char F = 2. This viewpoint is not needed here, however, and this

de¬nition should be viewed purely as a convenient notation.

§13. QUADRATIC PAIRS 189

For z ∈ Z — , we have κ(z) = z 2 · F — and µ(z) = NZ/F (z), hence

µ— (z) = NZ/F (z), z 4 ∈ U0 .

(13.35) De¬nition. Assume deg A ≡ 2 mod 4. De¬ne a homomorphism

S : PGO+ (A, σ, f ) ’ H 1 (F, µ4 [Z] )

as follows: for g · F — ∈ PGO+ (A, σ, f ), pick ω ∈ „¦(A, σ, f ) such that χ (ω) = g · F —

and let S(g · F — ) be the image of µ— (ω) in H 1 (F, µ4 [Z] ). Since ω is determined up

to a factor in Z — and µ— (Z — ) ‚ U0 , the de¬nition of S(g · F — ) does not depend on

the choice of ω. In other words, S is the map which makes the following diagram

commute:

χ

1 ’ ’ ’ Z — ’ ’ ’ „¦(A, σ, f ) ’ ’ ’ PGO+ (A, σ, f ) ’ ’ ’ 1

’’ ’’ ’’ ’’

¦ ¦ ¦

¦ µ— ¦ µ— ¦

S

’ ’ ’ H 1 (F, µ4 [Z] ) ’ ’ ’ 1.

1 ’ ’ ’ U0 ’ ’ ’

’’ ’’ U ’’ ’’

In order to relate the map S to the spinor norm, we de¬ne maps i and j which

¬t into an exact sequence

j

i

F — /F —2 ’ H 1 (F, µ4 [Z] ) ’ F — /F —2 .

’ ’

For ± · F —2 ∈ F — /F —2 , we let i(± · F —2 ) = [±, ±2 ]. For [±, z] ∈ H 1 (F, µ4 [Z] ),

we pick z0 ∈ Z — such that ±’2 z = z0 ι(z0 )’1 , and let j[±, z] = NZ/F (z0 ) · F —2 ∈

F — /F —2 . If NZ/F (z0 ) = β 2 for some β ∈ F — , then we may ¬nd z1 ∈ Z — such that

z0 β ’1 = z1 ι(z1 )’1 . It follows that ±’2 z = z1 ι(z1 )’2 , hence

2

(±, z) = ±NZ/F (z1 )’1 , ±2 NZ/F (z1 )’2 · NZ/F (z1 ), z1 ,

4

and therefore [±, z] = i ±NZ/F (z1 )’1 · F —2 . This proves exactness of the sequence

above.

(13.36) Proposition. Assume deg A ≡ 2 mod 4. Let

π : O+ (A, σ, f ) ’ PGO+ (A, σ, f )

be the canonical map. Then, the following diagram is commutative with exact rows:

µ

π

O+ (A, σ, f ) ’ ’ ’ PGO+ (A, σ, f ) ’ ’ ’ F — /F —2

’’ ’’

¦ ¦

¦ ¦

Sn S

j

i

F — /F —2 ’ ’ ’ H 1 (F, µ4 [Z] ) ’ ’ ’ F — /F —2 .

’’ ’’

Proof : Let g ∈ O+ (A, σ, f ) and let γ ∈ “(A, σ, f ) be such that χ(γ) = g. We then

have Sn(g) = µ(γ) · F —2 . On the other hand, we also have κ(γ) = 1, by (??), hence

µ— (γ) = µ(γ), µ(γ)2 = i µ(γ) · F —2 .

Since χ (γ) = g · F — , this proves commutativity of the left square.

Consider next g · F — ∈ PGO+ (A, σ, f ) and ω ∈ „¦(A, σ, f ) such that χ (ω) =

g · F — . We have j —¦ S(g · F — ) = NZ/F (k) · F — , where k ∈ Z — is a representative of

κ(ω) ∈ Z — /F — . Proposition (??) shows that NZ/F (k) · F —2 = µ(g · F — ), hence the

right square is commutative. Exactness of the bottom row was proved above.

190 III. SIMILITUDES

As in the preceding case, the spin group may also be de¬ned as a subgroup of

„¦(A, σ, f ): we have for deg A ≡ 2 mod 4,

Spin(A, σ, f ) = { ω ∈ „¦(A, σ, f ) | µ— (ω) = (1, 1) },

since (??) shows that the right-hand group lies in “(A, σ, f ). Furthermore we have

a sequence corresponding to the sequence (??):

(13.37) Proposition. Assume deg A ≡ 2 mod 4 and deg A ≥ 4. The sequence

χ S

1 ’ µ4 [Z] (F ) ’ Spin(A, σ, f ) ’ PGO+ (A, σ, f ) ’ H 1 (F, µ4 [Z] ),

’ ’

is exact, where µ4 [Z] (F ) = { z ∈ Z — | z 4 = 1 and ι(z)z = 1 }.

Proof : As in the proof of (??) the kernel of S is the image of Spin(A, σ, f ) under

χ . Furthermore we have by (??)

Z — © Spin(A, σ, f ) = { z ∈ Z — | z 2 ∈ F — and σ(z)z = 1 } = µ4 [Z] (F )

in „¦(A, σ, f ).

13.C. Multipliers of similitudes. This section is devoted to a generalization

of Dieudonn´™s theorem on the multipliers of similitudes (??). As in the preceding

e

sections, let (σ, f ) be a quadratic pair on a central simple algebra A of even degree

over an arbitrary ¬eld F , and let Z = Z(A, σ, f ) be the center of the Cli¬ord algebra

C(A, σ, f ). The nontrivial automorphism of Z/F is denoted by ι.

For ± ∈ F — , let (Z, ±)F be the quaternion algebra Z • Zj where multiplication

is de¬ned by jz = ι(z)j for z ∈ Z and j 2 = ±. In other words,

F [X]/(X 2 ’ δ);

(δ, ±)F if char F = 2 and Z

(Z, ±)F =

F [X]/(X 2 + X + δ).

[δ, ±)F if char F = 2 and Z

(Compare with §??).

(13.38) Theorem. Let g ∈ GO(A, σ, f ) be a similitude of (A, σ, f ).

(1) If g is proper, then Z, µ(g) F splits.

(2) If g is improper, then Z, µ(g) F is Brauer-equivalent to A.

When A splits, the algebra Z, µ(g) F splits in each case, so µ(g) is a norm

from Z/F for every similitude g. We thus recover Dieudonn´™s theorem (??).

e

In the case where g is proper, the theorem follows from (??) (or, equivalently,

from (??) and (??)). For the rest of this section, we ¬x some improper similitude g.

According to (??), the automorphism C(g) of C(A, σ) then restricts to ι on Z,

∼

so C(g) induces a Z-algebra isomorphism C(A, σ, f ) ’ ι C(A, σ, f ) by mapping

’

c ∈ C(A, σ, f ) to ι C(g)(c) ∈ ι C(A, σ, f ). When we view C(A, σ, f ) as a left

Z-module, we have the canonical isomorphism

C(A, σ, f )op —Z C(A, σ, f ) = EndZ C(A, σ, f )

which identi¬es cop — c2 with the endomorphism de¬ned by

1

op

c c1 —c2

= c1 cc2 for c, c1 , c2 ∈ C(A, σ, f ).

We then have Z-algebra isomorphisms:

EndZ C(A, σ, f ) = C(A, σ, f )op —Z C(A, σ, f ) ι

C(A, σ, f )op —Z C(A, σ, f ).

§13. QUADRATIC PAIRS 191