The embedding ν of ι C(A, σ, f )op —Z C(A, σ, f ) into the endomorphism algebra of

the Cli¬ord bimodule B(A, σ, f ) (see (??)) yields an embedding

νg : EndZ C(A, σ, f ) ’ EndA B(A, σ, f )

de¬ned by

op

xνg (c1 —c2 )

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

Let γg ∈ EndF C(A, σ, f ) be the endomorphism C(g ’1 ), i.e.,

cγg = C(g ’1 )(c) for c ∈ C(A, σ, f ).

Since g is improper, γg is not Z-linear, but ι-semilinear. Thus Int(γg ), which maps

’1

f ∈ EndZ C(A, σ, f ) to γg —¦ f —¦ γg , is an automorphism of EndZ C(A, σ, f ).

De¬ne an F -algebra Eg as follows:

Eg = EndZ C(A, σ, f ) • EndZ C(A, σ, f ) · y

where y is subject to the following relations:

’1

yf = (γg —¦ f —¦ γg )y for f ∈ EndZ C(A, σ, f ),

y 2 = µ(g)γg .

2

The algebra Eg is thus a generalized cyclic algebra (see Albert [?, Theorem 11.11],

Jacobson [?, § 1.4]); the same arguments as for the usual cyclic algebras show E g

is central simple over F .

(13.39) Proposition. The homomorphism νg extends to an isomorphism of F -

algebras:

∼

νg : Eg ’ EndA B(A, σ, f )

’

by mapping y to the endomorphism β(g) of (??).

Proof : Since Eg and EndA B(A, σ, f ) are central simple F -algebras of the same

dimension, it su¬ces to show that the map de¬ned above is a homomorphism, i.e.,

that β(g) satis¬es the same relations as y:

’1

β(g) —¦ νg (f ) = νg (γg —¦ f —¦ γg ) —¦ β(g) for f ∈ EndZ C(A, σ, f ),

(13.40)

β(g)2 = µ(g)νg (γg ).

2

It su¬ces to check these relations over an extension of the base ¬eld. We may

thus assume that F is algebraically closed and (A, σ, f ) = EndF (V ), σq , fq for

some nonsingular quadratic space (V, q). Choose » ∈ F — satisfying »2 = µ(g).

Then q »’1 g(v) = q(v) for all v ∈ V , hence there is an automorphism of C(V, q)

which maps v to »’1 g(v) for all v ∈ V . By the Skolem-Noether theorem we may

thus ¬nd b ∈ C(V, q)— such that

b · v · b’1 = »’1 g(v) for v ∈ V .

Then C(g) is the restriction of Int(b) to C0 (V, q) = C(A, σ, f ), hence γg = Int(b’1 )

and

γg = (b2 )op — b’2 ∈ C(A, σ, f )op —Z C(A, σ, f ).

2

On the other hand, for v, w1 , . . . , w2r’1 ∈ V ,

β(g)

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

v — (w1 · · · w2r’1 )

= » v — b’1 · (w1 · · · w2r’1 ) · b .

192 III. SIMILITUDES

The equations (??) then follow by explicit computation.

To complete the proof of (??) we have to show that the algebra Eg is Brauer-

equivalent to the quaternion algebra Z, µ(g) F . As pointed out by A. Wadsworth,

this is a consequence of the following proposition:

(13.41) Proposition. Let S be a central simple F -algebra, let Z be a quadratic

Galois ¬eld extension of F contained in S, with nontrivial automorphism ι. Let

s ∈ S be such that Int(s)|Z = ι. Let E = CS (Z) and ¬x t ∈ F — . Let T be the

F -algebra with presentation

where yey ’1 = ses’1 for all e ∈ E, and y 2 = ts2 .

T = E • Ey,

Then M2 (T ) (Z, t) —F S.

Proof : Let j be the standard generator of (Z, t) with jzj ’1 = ι(z) for all z ∈ Z

and j 2 = t. Let R = (Z, t) —F S, and let

T = (1 — E) + (1 — E)y ‚ R, where y = j — s.

Then T is isomorphic to T , since y satis¬es the same relations as y. (That is, for

’1 2

= 1 — (ses’1 ) and y = 1 — ts2 .) By the double centralizer

any e ∈ E, y (1 — e)y

theorem (see (??)) R T —F Q where Q = CR (T ), and Q is a quaternion algebra

over F by a dimension count. It su¬ces to show that Q is split. For, then

R T —F Q T —F M2 (F ) M2 (T ).

Consider Z —F Z ‚ R; Z —F Z centralizes 1 — E and Int(y ) restricts to ι — ι on

Z—F Z. Now Z—F Z has two primitive idempotents e1 and e2 , since Z—F Z Z•Z.

The automorphisms Id — ι and ι — Id permute them, so ι — ι maps each ei to itself.

Hence e1 and e2 lie in Q since they centralize 1 — E and also y . Because e1 e2 = 0,

Q is not a division algebra, so Q is split, as desired.

(13.42) Remark. We have assumed in the above proposition that Z is a ¬eld.

The argument still works, with slight modi¬cation, if Z F — F .

As a consequence of Theorem (??), we may compare the group G(A, σ, f ) of

multipliers of similitudes with the subgroup G+ (A, σ, f ) of multipliers of proper

similitudes. Since the index of GO+ (A, σ, f ) in GO(A, σ, f ) is 1 or 2, it is clear

that either G(A, σ, f ) = G+ (A, σ, f ) or G+ (A, σ, f ) is a subgroup of index 2 in

G(A, σ, f ). If A is split, then

[GO(A, σ, f ) : GO+ (A, σ, f )] = [O(A, σ, f ) : O+ (A, σ, f )] = 2

and

G(A, σ, f ) = G+ (A, σ, f )

since hyperplane re¬‚ections are improper isometries (see (??)). If A is not split, we

deduce from (??):

(13.43) Corollary. Suppose that (σ, f ) is a quadratic pair on a central simple

F -algebra A of even degree. If A is not split, then O(A, σ, f ) = O+ (A, σ, f ) and

[GO(A, σ, f ) : GO+ (A, σ, f )] = [G(A, σ, f ) : G+ (A, σ, f )].

If G(A, σ, f ) = G+ (A, σ, f ), then A is split by Z = Z(A, σ, f ).

§14. UNITARY INVOLUTIONS 193

Proof : If g ∈ O(A, σ, f ) is an improper isometry, then (??) shows that A is Brauer-

equivalent to Z, µ(g) F , which is split since µ(g) = 1. This contradiction shows

that O(A, σ, f ) = O+ (A, σ, f ).

If [GO(A, σ, f ) : GO+ (A, σ, f )] = [G(A, σ, f ) : G+ (A, σ, f )], then necessarily

GO(A, σ, f ) = GO+ (A, σ, f ) and G(A, σ, f ) = G+ (A, σ, f ).

Therefore, A contains an improper similitude g, and µ(g) = µ(g ) for some proper

similitude g . It follows that g ’1 g is an improper isometry, contrary to the equality

O(A, σ, f ) = O+ (A, σ, f ).

Finally, if µ is the multiplier of an improper similitude, then (??) shows that

A is Brauer-equivalent to (Z, µ)F , hence it is split by Z.

(13.44) Corollary. If disc(σ, f ) is trivial, then G(A, σ, f ) = G+ (A, σ, f ).

Proof : It su¬ces to consider the case where A is not split. Then, if G(A, σ, f ) =

G+ (A, σ, f ), the preceding corollary shows that A is split by Z; this is impossible

if disc(σ, f ) is trivial, for then Z F — F .

§14. Unitary Involutions

In this section, we let (B, „ ) be a central simple algebra with involution of the

second kind over an arbitrary ¬eld F . Let K be the center of B and ι the nontrivial

automorphism of K/F .

We will investigate the group GU(B, „ ) of similitudes of (B, „ ) and the unitary

group U(B, „ ), which is the kernel of the multiplier map µ (see §??). The group

GU(B, „ ) has di¬erent properties depending on the parity of the degree of B. When

deg B is even, we relate this group to the group of similitudes of the discriminant

algebra D(B, „ ).

14.A. Odd degree.

(14.1) Proposition. If deg B is odd, the group G(B, „ ) of multipliers of simili-

tudes of (B, „ ) is the group of norms of K/F :

G(B, „ ) = NK/F (K — ).

Moreover, GU(B, „ ) = K — · U(B, „ ).

Proof : The inclusion NK/F (K — ) ‚ G(B, „ ) is clear, since K — ‚ GU(B, „ ) and

µ(±) = NK/F (±) for ± ∈ K — . In order to prove the reverse inclusion, let deg B =

2m + 1 and let g ∈ GU(B, „ ). By applying the reduced norm to the equation

„ (g)g = µ(g) we obtain

NK/F NrdB (g) = µ(g)2m+1 .

Therefore,

µ(g) = NK/F µ(g)’m NrdB (g) ∈ NK/F (K — ),

hence G(B, „ ) ‚ NK/F (K — ). This proves the ¬rst assertion.

The preceding equation shows moreover that µ(g)m NrdB (g)’1 g ∈ U(B, „ ).

Therefore, letting u = µ(g)m NrdB (g)’1 g and ± = µ(g)’m NrdB (g) ∈ K — , we get

g = ±u. Thus, GU(B, „ ) = K — · U(B, „ ).

Note that in the decomposition g = ±u above, the elements ± ∈ K — and

u ∈ U(B, „ ) are uniquely determined up to a factor in the group K 1 of norm 1

elements, since K — © U(B, „ ) = K 1 .

194 III. SIMILITUDES

14.B. Even degree. Suppose now that deg B = 2m and let g ∈ GU(B, „ ).

By applying the reduced norm to the equation „ (g)g = µ(g), we obtain

NK/F NrdB (g) = µ(g)2m ,

hence µ(g)m NrdB (g)’1 is in the group of elements of norm 1. By Hilbert™s The-

orem 90, there is an element ± ∈ K — , uniquely determined up to a factor in F — ,

such that

µ(g)’m NrdB (g) = ±ι(±)’1 .

We may therefore de¬ne a homomorphism ν : GU(A, σ) ’ K — /F — by

ν(g) = ± · F — .

(14.2)

Let SGU(B, „ ) be the kernel of ν, and let SU(B, „ ) be the intersection SGU(B, „ ) ©

U(B, „ ):

SGU(B, „ ) = { g ∈ GU(B, „ ) | NrdB (g) = µ(g)m }

SU(B, „ ) = { u ∈ GU(B, „ ) | NrdB (u) = µ(u) = 1 }.

We thus have the following diagram, where all the maps are inclusions:

SU(B, „ ) ’ ’ ’ SGU(B, „ )

’’

¦ ¦

¦ ¦

U(B, „ ) ’ ’ ’ GU(B, „ ).

’’

Consider for example the case where K = F —F ; we may then assume B = E —E op

for some central simple F -algebra E of degree 2m, and „ = µ is the exchange

involution. We then have

GU(B, „ ) = { x, ±(x’1 )op | ± ∈ F — , x ∈ E — } E— — F —

and the maps µ and ν are de¬ned by

µ x, ±(x’1 )op = ±, ν x, ±(x’1 )op = NrdE (x), ±m · F — .

Therefore,

{ (x, ±) ∈ E — — F — | NrdE (x) = ±m },

SGU(B, „ )

E—

U(B, „ )

and the group SU(B, „ ) is isomorphic to the group of elements of reduced norm 1

in E, which we write SL(E):

{ x ∈ E — | NrdE (x) = 1 } = SL(E).

SU(B, „ )

14.C. Relation with the discriminant algebra. Our ¬rst results in this

direction do not assume the existence of an involution; we formulate them for an

arbitrary central simple F -algebra A:

§14. UNITARY INVOLUTIONS 195

The canonical map »k .

(14.3) Proposition. Let A be any central simple algebra over a ¬eld F . For all

integers k such that 1 ¤ k ¤ deg A, there is a homogeneous polynomial map of

degree k:

»k : A ’ » k A

which restricts to a group homomorphism A— ’ (»k A)— . If the algebra A is split,

let A = EndF (V ), then under the identi¬cation »k A = EndF ( k V ) the map »k is

de¬ned by

k

»k (f ) = f = f § ···§ f for f ∈ EndF (V ).

Proof : Let gk : Sk ’ (A—k )— be the homomorphism of (??). By (??), it is clear

that for all a ∈ A— the element —k a = a — · · · — a commutes with gk (π) for all

π ∈ Sk , hence also with sk = π∈Sk sgn(π)gk (π). Multiplication on the right by

—k a is therefore an endomorphism of the left A—k -module A—k sk . We denote this

endomorphism by »k a; thus »k a ∈ EndA—k (A—k sk ) = »k A is de¬ned by

»k a

= (a1 — · · · — ak ) · sk · —k a = (a1 a — · · · — ak a) · sk .

(a1 — · · · — ak ) · sk

If A = EndF (V ), there is a natural isomorphism (see (??)):

k

A—k sk = HomF ( V , V —k ),

k

V ’ V —k de¬ned by

under which sk is identi¬ed with the map sk :

sk (v1 § · · · § vk ) = sk (v1 — · · · — vk ) for v1 , . . . , vk ∈ V .

»k f

(v1 § · · · § vk ) = sk —k f (v1 — · · · — vk ) , hence

For f ∈ EndF (V ) we have sk

»k f

sk (v1 § · · · § vk ) = sk f (v1 ) § · · · § f (vk ) .

»k f k

f , which means that »k (f ) ∈ »k EndF (V ) is identi¬ed

Therefore, sk = sk —¦

k k

f ∈ EndF ( V ). It is then clear that »k is a homogeneous polynomial map

with

of degree k, and that its restriction to A— is a group homomorphism to (»k A)— .

For the following result, we assume deg A = 2m, so that »m A has a canonical

involution γ of the ¬rst kind (see (??)).

(14.4) Proposition. If deg A = 2m, then γ(»m a)»m a = NrdA (a) for all a ∈ A.

In particular, if a ∈ A— , then »m a ∈ Sim(»m A, γ) and µ(»m a) = NrdA (a).

Proof : It su¬ces to check the split case. We may thus assume A = EndF (V ),

m

hence »m A = EndF ( V ) and the canonical involution γ is the adjoint involution

m m 2m

with respect to the canonical bilinear map § : V— V’ V . Moreover,

m

m m

» (f ) = f for f ∈ EndF (V ). The statement that » (f ) is a similitude for γ

therefore follows from the following identities

m m

f(v1 § · · · § vm ) § f (w1 § · · · § wm )

= f (v1 ) § · · · § f (vm ) § f (w1 ) § · · · § f (wm )

= det f · v1 § · · · § vm § w1 § · · · § wm

for v1 , . . . , vm , w1 , . . . , wm ∈ V .

196 III. SIMILITUDES

The canonical map D. We now return to the case of central simple algebras

with unitary involution (B, „ ). We postpone until after Proposition (??) the dis-

cussion of the case where the center K of B is isomorphic to F — F ; we thus assume

for now that K is a ¬eld.

(14.5) Lemma. For k = 1, . . . , deg B, let „ §k be the involution on »k B induced

by „ (see (??)). For all k, the canonical map »k : B ’ »k B satis¬es

„ §k —¦ »k = »k —¦ „.

Proof : By extending scalars to a splitting ¬eld of B, we reduce to considering

the split case. We may thus assume B = EndK (V ) and „ = σh is the adjoint

involution with respect to some nonsingular hermitian form h on V . According to

(??), the involution „ §k is the adjoint involution with respect to h§k . Therefore, for

f ∈ EndK (V ), the element „ §k —¦ »k (f ) ∈ EndK ( k V ) is de¬ned by the condition:

h§k „ §k —¦ »k (f )(v1 § · · · § vk ), w1 § · · · § wk =

h§k v1 § · · · § vk , »k (f )(w1 § · · · § wk )

k

for v1 , . . . , vk , w1 , . . . , wk ∈ V . Since »k (f ) = f , the right-hand expression

equals

det h vi , f (wj ) = det h „ (f )(vi ), wj

1¤i,j¤k 1¤i,j¤k

§k k

=h » „ (f ) (v1 § · · · § vk ), w1 § · · · § wk .

Assume now deg B = 2m; we may then de¬ne the discriminant algebra D(B, „ )

as the subalgebra of »m B of elements ¬xed by „ §m —¦ γ, see (??).

(14.6) Lemma. For g ∈ GU(B, „ ), let ± ∈ K — be such that ν(g) = ± · F — ; then

±’1 »m g ∈ D(B, „ ) and „ (±’1 »m g) · ±’1 »m g = NK/F (±)’1 µ(g)m . In particular,

»m g ∈ Sim D(B, „ ), „ for all g ∈ SGU(B, „ ).

Proof : By (??) we have

γ(»m g) = NrdB (g)»m g ’1 = NrdB (g)µ(g)’m »m „ (g) ,

hence, by (??),

„ §m —¦ γ(»m g) = ι NrdB (g) µ(g)’m »m g = ±’1 ι(±)»m g.

Therefore, ±’1 »m g ∈ D(B, „ ). Since „ is the restriction of γ to D(B, „ ), we have

„ (±’1 »m g) · ±’1 »m g = ±’2 γ(»m g)»m g,

and (??) completes the proof.

The lemma shows that the inner automorphism Int(»m g) = Int(±’1 »m g)

of »m B preserves D(B, „ ) and induces an automorphism of D(B, „ ), „ . Since

this automorphism is also induced by the automorphism Int(g) of (B, „ ), by func-

toriality of the discriminant algebra construction, we denote it by D(g). Alternately,

under the identi¬cation Aut D(B, „ ), „ = PSim D(B, „ ), „ of (??), we may set

D(g) = ±’1 »m g · F — , where ± ∈ K — is a representative of ν(g) as above.

The next proposition follows from the de¬nitions and from (??):

§14. UNITARY INVOLUTIONS 197

(14.7) Proposition. The following diagram commutes:

SGU(B, „ ) ’’’

’’ GU(B, „ )

¦ ¦

¦ ¦

m

D

»

Int

Sim D(B, „ ), „ ’ ’ ’ AutF D(B, „ ), „ .

’’

Moreover, for g ∈ SGU(B, „ ), the multipliers of g and »m g are related by

µ(»m g) = µ(g)m = NrdB (g).

Therefore, »m restricts to a group homomorphism SU(B, „ ) ’ Iso D(B, „ ), „ .

We now turn to the case where K F — F , which was put aside for the

preceding discussion. In this case, we may assume B = E — E op for some central

simple F -algebra E of degree 2m and „ = µ is the exchange involution. As observed

in §?? and §??, we may then identify

D(B, „ ), „ = (»m E, γ) and GU(B, „ ) = E — — F — .

The discussion above remains valid without change if we set D(x, ±) = Int(»m x) for

(x, ±) ∈ E — — F — = GU(B, „ ), a de¬nition which is compatible with the de¬nitions

above (in the case where K is a ¬eld) under scalar extension.

™

The canonical Lie homomorphism »k . To conclude this section, we derive

from the map »m a Lie homomorphism from the Lie algebra Skew(B, „ )0 of skew-

symmetric elements of reduced trace zero to the Lie algebra Skew D(B, „ ), „ . This

Lie homomorphism plays a crucial rˆle in §?? (see (??)).

o

As above, we start with an arbitrary central simple F -algebra A. Let t be an

indeterminate over F . For k = 1, . . . , deg A, consider the canonical map

»k : A — F (t) ’ »k A — F (t).

Since this map is polynomial of degree k and »k (1) = 1, there is a linear map

™

»k : A ’ »k A such that for all a ∈ A,

™

»k (t + a) = tk + »k (a)tk’1 + · · · + »k (a).

™

(14.8) Proposition. The map »k is a Lie-algebra homomorphism

™

»k : L(A) ’ L(»k A).

k

If A = EndF (V ), then under the identi¬cation »k A = EndF ( V ) we have

™

»k (f )(v1 § · · · § vk ) =

f (v1 ) § v2 § · · · § vk + v1 § f (v2 ) § · · · § vk + · · · + v1 § v2 § · · · § f (vk )

for all f ∈ EndF (V ) and v1 , . . . , vk ∈ V .

™

Proof : The description of »k in the split case readily follows from that of »k in

™ k

(??). To prove that » is a Lie homomorphism, we may reduce to the split case by

a scalar extension. The property then follows from an explicit computation: for f ,

198 III. SIMILITUDES

g ∈ EndF (V ) and v1 , . . . , vk ∈ V we have

™ ™

»k (f ) —¦ »k (g)(v1 § · · · § vk ) = v1 § · · · § f (vi ) § · · · § g(vj ) § · · · § vk

1¤i<j¤k

+ v1 § · · · § f —¦ g(vi ) § · · · § vk

1¤i¤k

+ v1 § · · · § g(vj ) § · · · § f (vi ) § · · · § vk ,

1¤j<i¤k

™ ™ ™ ™

hence »k (f ) —¦ »k (g) ’ »k (g) —¦ »k (f ) maps v1 § · · · § vk to

™

v1 § · · · § (f —¦ g ’ g —¦ f )(vi ) § · · · § vk = »k [f, g] (v1 § · · · § vk ).

1¤i¤k

™ ™ ™

This shows »k (f ), »k (g) = »k [f, g] .

™

(14.9) Corollary. Suppose k ¤ deg A ’ 1. If a ∈ A satis¬es »k a ∈ F , then a ∈ F

™ ™

and »k a = ka. In particular, ker »k = { a ∈ F | ka = 0 }.

Proof : It su¬ces to consider the split case; we may thus assume that A = EndF (V )

™

for some vector space V . If »k a ∈ F , then for all x1 , . . . , xk ∈ V we have

™

x1 § »k a(x1 § · · · § xk ) = 0, hence x1 § a(x1 ) § x2 § · · · § xk = 0. Since k < dim V ,

this relation shows that a(x1 ) ∈ x1 · F for all x1 ∈ V , hence a ∈ F . The other

statements are then clear.

1

In the particular case where k = deg A, we have:

2

(14.10) Proposition. Suppose deg A = 2m, and let γ be the canonical involution

on »m A. For all a ∈ A,

™ ™

»m a + γ(»m a) = TrdA (a).

Proof : By (??), we have

γ »m (t + a) · »m (t + a) = Nrd(t + a).

The proposition follows by comparing the coe¬cients of t2m’1 on each side.

We now consider a central simple algebra with unitary involution (B, „ ) over F ,

and assume that the center K of B is a ¬eld. Suppose also that the degree of B is

even: deg B = 2m. Since (??) shows that

„ §m —¦ »m (t + b) = »m —¦ „ (t + b) for b ∈ B,

it follows that

™ ™

„ §m —¦ »m = »m —¦ „.

™

It is now easy to determine under which condition »m b ∈ D(B, „ ): this holds if and

™ ™

only if γ(»m b) = „ §m (»m b), which means that

™ ™

TrdB (b) ’ »m b = »m „ (b).

By (??), this equality holds if and only if b + „ (b) ∈ F and TrdB (b) = m b + „ (b) .

Let

(14.11) s(B, „ ) = { b ∈ B | b + „ (b) ∈ F and TrdB (b) = m b + „ (b) },

™

so s(B, „ ) = (»m )’1 D(B, „ ) . The algebra s(B, „ ) is contained in g(B, „ ) = { b ∈

B | b + „ (b) ∈ F } (see §??). If µB : g(B, „ ) ’ F is the map which carries

™

EXERCISES 199

b ∈ g(B, „ ) to b + „ (b), we may describe s(B, „ ) as the kernel of the F -linear

map TrdB ’mµB : g(B, „ ) ’ K. The image of this map lies in K 0 = { x ∈

™

K | TK/F (x) = 0 }, since taking the reduced trace of both sides of the relation

b + „ (b) = µ(b) yields TK/F TrdB (b) = 2mµB (b). On the other hand, this map is

™ ™

not trivial: since TrdB is surjective, we may ¬nd x ∈ B such that TrdB (x) ∈ F ;

/

then x ’ „ (x) ∈ g(B, „ ) is not mapped to 0. Therefore,

dimF s(B, „ ) = dimF g(B, „ ) ’ 1 = 4m2 .

™

(14.12) Proposition. The homomorphism »m restricts to a Lie algebra homo-

morphism

™

»m : s(B, „ ) ’ g D(B, „ ), „ .

By denoting by µD : g D(B, „ ), „ ’ F the map which carries x ∈ D(B, „ ) to

™

x + „ (x), we have

™™

µD (»m b) = mµB (b) = TrdB (b)

™ for b ∈ s(B, „ ).

™

Therefore, »m restricts to a Lie algebra homomorphism

™

»m : Skew(B, „ )0 ’ Skew D(B, „ ), „ ,

where Skew(B, „ )0 is the Lie algebra of skew-symmetric elements of reduced trace

zero in B.

™ ™

Proof : Since s(B, „ ) = (»m )’1 D(B, „ ) , it is clear that »m restricts to a homo-

morphism from s(B, „ ) to L D(B, „ ) . To prove that its image lies in g D(B, „ ), „ ,

it su¬ces to prove

™ ™

»m b + „ (»m b) = TrdB (b) for b ∈ s(B, „ ).

This follows from (??), since „ is the restriction of γ to D(B, „ ).

Suppose ¬nally K F — F ; we may then assume B = E — E op for some central

simple F -algebra E of degree 2m, and „ = µ is the exchange involution. We have

s(B, „ ) = { (x, ± ’ xop ) | x ∈ E, ± ∈ F , and TrdE (x) = m± },

and D(B, „ ), „ may be identi¬ed with (»m E, γ). The Lie algebra homomorphism

™ ™

»m : s(B, „ ) ’ g(»m E, γ) then maps (x, ± ’ xop ) to »m x. Identifying Skew(B, „ )0

with the Lie algebra E 0 of elements of reduced trace zero (see (??)), we may restrict

this homomorphism to a Lie algebra homomorphism:

™

»m : E 0 ’ Skew(»m E, γ).

Exercises

1. Let Q be a quaternion algebra with canonical involution γ over a ¬eld F of

arbitrary characteristic. On the algebra A = Q —F Q, consider the canonical

quadratic pair (γ — γ, f ) (see (??)). Prove that

GO+ (A, γ — γ, f ) = { q1 — q2 | q1 , q2 ∈ Q— }

and determine the group of multipliers µ GO+ (A, γ — γ, f ) .

2. Let (A, σ) be a central simple F -algebra with involution of any kind with cen-

ter K and let ± ∈ AutK (A). Prove that the following statements are equivalent:

200 III. SIMILITUDES

(a) ± ∈ AutK (A, σ).

(b) ± Sym(A, σ) = Sym(A, σ).

(c) ± Alt(A, σ) = Alt(A, σ).

3. Let (A, σ) be a central simple algebra with orthogonal involution and degree

a power of 2 over a ¬eld F of characteristic di¬erent from 2, and let B ‚ A

be a proper subalgebra with center F = B. Prove that every similitude f ∈

GO(A, σ) such that f Bf ’1 = B is proper.

4. (Wonenburger [?]) The aim of this exercise is to give a proof of Wonenburger™s

theorem on the image of GO(V, q) in Aut C0 (V, q) , see (??).

Let q be a nonsingular quadratic form on an even-dimensional vector space

V over a ¬eld F of arbitrary characteristic. Using the canonical identi¬cation

∼

•q : V — V ’ EndF (V ), we identify c EndF (V ) = V · V ‚ C0 (V, q) and

’

2

Alt EndF (V ), σq = V . An element x ∈ V · V is called a regular plane

element if x = v · w for some vectors v, w ∈ V which span a nonsingular 2-

dimensional subspace of V . The ¬rst goal is to show that the regular plane

elements are preserved by the automorphisms of C0 (V, q) which preserve V · V .

2 4

Let ρ : V’ V be the quadratic map which vanishes on elements of

the type v § w and whose polar is the exterior product (compare with (??)),

and let „ be the canonical involution on C(V, q) which is the identity on V .

2

(a) Show that x ∈ V has the form x = v § w for some v, w ∈ V if and only

if ρ(x) = 0.

(b) Show that the Lie homomorphism δ : V · V ’ 2 V maps v · w to v § w.

(c) Let W = { x + „ (x) | x ∈ V · V · V · V } ‚ C0 (V, q). Show that F ‚ W and

that there is a surjective map ω : 4 V ’ W/F such that

ω(v1 § v2 § v3 § v4 ) = v1 · v2 · v3 · v4 + v4 · v3 · v2 · v1 + F.

Show that for all x ∈ V · V ,

’ω —¦ ρ —¦ δ(x) = „ (x) · x + F.

(d) Assume that char F = 2. Show that ω is bijective and use the results

above to show that if x ∈ V · V satis¬es „ (x) · x ∈ F , then δ(x) = v § w

for some v, w ∈ V .

(e) Assume that (V, q) is a 6-dimensional hyperbolic space over a ¬eld F of

characteristic 2, and let (e1 , . . . , e6 ) be a symplectic basis of V consisting

of isotropic vectors. Show that x = e1 · e2 + e3 · e4 + e5 · e6 ∈ V · V satis¬es

„ (x) · x = 0, „ (x) + x = 1, but δ(x) cannot be written in the form v § w

with v, w ∈ V .

(f) For the rest of this exercise, assume that char F = 2. Show that x ∈ V · V

is a regular plane element if and only if „ (x) · x ∈ F , „ (x) + x ∈ F and

2

„ (x) + x = 4„ (x) · x. Conclude that every automorphism of C0 (V, q)

which commutes with „ and preserves V · V maps regular plane elements

to regular plane elements.

(g) Show that regular plane elements x, y ∈ V · V anticommute if and only if

x = u · v and y = u · w for some pairwise orthogonal anisotropic vectors u,

v, w ∈ V .

(h) Let (e1 , . . . , en ) be an orthogonal basis of V . Let θ ∈ Aut C0 (V, q), „ be

an automorphism which preserves V · V . Show that there is an orthogonal

basis (v1 , v2 , . . . , vn ) of V such that θ(e1 · ei ) = v1 · vi for i = 2, . . . , n. Let

EXERCISES 201

± = q(v1 )’1 q(e1 ). Show that the linear transformation of V which maps

e1 to ±v1 and ei to vi for i = 2, . . . , n is a similitude which induces θ.

5. Let D be a central division algebra with involution over a ¬eld F of charac-

teristic di¬erent from 2 and let V be a (¬nite dimensional) right vector space

over D with a nonsingular hermitian form h. Let v ∈ V be an anisotropic

vector and let d ∈ D — be such that

h(v, v) = dh(v, v)d (= h(vd, vd)).

De¬ne „v,d ∈ EndF (V ) by

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

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

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

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

Theorem 7.9.5].

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

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

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

1’d 1’e

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

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

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

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

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

e

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

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

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

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

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

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

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

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

U = { u ∈ A | σ(u)u = 1 }

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

U 0 = { u ∈ U | 1 + u ∈ A— }.

Let also

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

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

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

1’a

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

known as the Cayley parametrization of U .)

202 III. SIMILITUDES

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

let g ∈ GO(A, σ, f ). Show that

± m is odd and g is improper,

1 if

m is even and g is proper,

Sn µ(g)’1 g 2 =

m is odd and g is proper,

µ(g) · F —2 if

m is even and g is improper.

and that

µ GO+ (A, σ, f ) · F —2 if m is odd,

+

Sn O (A, σ, f ) ⊃

µ GO’ (A, σ, f ) · F —2 if m is even,

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

Hint: Use the arguments of (??).

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

quadratic pair. Show that

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

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

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

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

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

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

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

Brauer class of B.

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

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

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

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

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

phisms which have a diagonal matrix representation with respect to an

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

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

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

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

’1

NrdD (u) = NrdD (d)ι NrdD (d) .

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

NrdB U(B, „ ) = NrdB (B — ) © K 1 .

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

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

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

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

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

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

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

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

NOTES 203

Notes

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

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

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

e

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

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

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

= =

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

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

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

C(V, q) = M0 • M1 • · · · • Md

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

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

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

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

d

dimF Mk = .

k

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

C0 (V, q) = Mi .

i even

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

Md’1 = ζ · V ‚ C0 (V, q)

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

de¬ne a quadratic map

s : Md’1 ’ F

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

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

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

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

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

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

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

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

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

End —¦ M ∼ IdBn and M —¦ End ∼ IdBn .

= =

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

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

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

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

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

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

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

204 III. SIMILITUDES

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

e

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

ee

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

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

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

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

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

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

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

di¬erent.)

™

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

™

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

k

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

™

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

™ ™

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

involves rather tedious computations.

CHAPTER IV

Algebras of Degree Four

Among groups of automorphisms of central simple algebras with involution,

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

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

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

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

o

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

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

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

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

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

the last isomorphism.

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

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

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

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

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

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

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

are also discussed.

§15. Exceptional Isomorphisms

The exceptional isomorphisms between groups of similitudes of central simple

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

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

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

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

canonical homomorphism of (??):

C : PGO(A, σ) ’ AutF C(A, σ), σ

is an isomorphism.

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

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

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

c(A)0 = c(A) © Skew C(A, σ), σ .

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

c(A)0 if deg A ¤ 6.

205

206 IV. ALGEBRAS OF DEGREE FOUR

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

putations show that

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

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

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

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

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

canonical decomposition

0

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

where

0

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

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

by dimension count,

0

c(A)0 = Skew C(A, σ), σ .

0

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

the proof is complete.

This proposition relates central simple algebras with orthogonal involutions of

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

central simple F -algebras

quaternion F -algebras

of degree 3 ←’

with symplectic involution

with orthogonal involution

central simple F -algebras quaternion algebras with

of degree 4 symplectic involution over an

←’

with orthogonal involution ´tale quadratic extension of F

e

central simple F -algebras central simple F -algebras

of degree 5 of degree 4

←’

with orthogonal involution with symplectic involution

central simple algebras

central simple F -algebras

of degree 4

of degree 6 ←’

with unitary involution over an

with orthogonal involution

´tale quadratic extension of F

e

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

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

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

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

e

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

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

products Q1 — Q2 of quaternion F -algebras.

§15. EXCEPTIONAL ISOMORPHISMS 207

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

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

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

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

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

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

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

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

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

F -algebra isomorphisms;

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

e

1

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

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

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

e

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

the F -algebra isomorphisms which preserve the involutions;

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

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

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

tation, since the orientation is unique);

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

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

the F -algebra isomorphisms which preserve the involutions;

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

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

F -algebra isomorphisms which preserve the quadratic pairs.

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

Note that there is an isomorphism of groupoids:

A1 = C 1

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

involution.

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

B1 ≡ C 1

D2 ≡ A 2

1

B2 ≡ C 2

D3 ≡ A 3 .

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

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

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

well as between the second and the fourth cases.

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

proposition.

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

A1 ≡ B 1 .

208 IV. ALGEBRAS OF DEGREE FOUR

For every quaternion algebra Q ∈ A1 , the vector space

Q0 = { x ∈ Q | TrdQ (x) = 0 }

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

s(x) = x2 for x ∈ Q0

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

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

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

hQ : C(Q0 , s) ’ Q.

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

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

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

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

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

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

We de¬ne a functor

P : A1 ’ B1

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

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

construction: we de¬ne

C : B1 ’ A 1

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

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

A1 ≡ B 1 .

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

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

∼

hQ : C0 (Q0 , s) ’ Q.

’

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

=

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

mζ (v) = vζ for v ∈ V .

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

∼

mζ : (V, q) ’ C0 (V, q)0 , s .

’

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

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

=

proof.

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

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

char F = 2. The isometry mζ induces an isomorphism

∼

mζ : C(V, q) ’ C C0 (V, q)0 , s

’

§15. EXCEPTIONAL ISOMORPHISMS 209

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

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

we get a homomorphism

hC0 (V,q) —¦ mζ : C(V, q) ’ C0 (V, q)

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

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

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

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

hC0 (V,q) (π) = 1,

it follows that mζ (ζ) = π.

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

Cli¬ord algebra construction yields a group isomorphism

∼

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

’

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

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

phisms between the automorphism groups of corresponding objects.

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

obtain an equivalence between A1 and B1 :

(15.4) Corollary. Suppose char F = 2. The functor P : A1 ’ B1 , which maps

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 (??).

Since (??), (??), (??) and (??) exhaust all the possibilities for ind(A, σ, f ) and

for Q, the proof is complete.

15.C. B2 ≡ C2 . The arguments to prove the equivalence of B2 and C2 are

similar to those used in §?? to prove B1 ≡ C1 .

For any oriented quadratic space (V, q, ζ) ∈ B2 (of trivial discriminant), the even

Cli¬ord algebra C0 (V, q) is central simple of degree 4, and its canonical involution

„ (= σq ) is symplectic. We may therefore de¬ne a functor

C : B2 ’ C 2

by

C(V, q, ζ) = C0 (V, q), „ .

On the other hand, let (A, σ) be a central simple F -algebra of degree 4 with sym-

plectic involution. As observed in (??), the reduced characteristic polynomial of

every symmetrized element is a square; the pfa¬an trace Trpσ is a linear form on

V and the pfa¬an norm Nrpσ is a quadratic form on V such that

2

PrdA,a (X) = X 2 ’ Trpσ (a)X + Nrpσ (a) for a ∈ Symd(A, σ).

In particular, if a ∈ Symd(A, σ) is such that Trpσ (a) = 0, then a2 = ’ Nrpσ (a) ∈ F .

Let

Symd(A, σ)0 = { a ∈ Symd(A, σ) | Trpσ (a) = 0 }

= { a ∈ Sym(A, σ) | TrdA (a) = 0 } if char F = 2 .

This is a vector space of dimension 5 over F . The map sσ : Symd(A, σ)0 ’ F

de¬ned by

sσ (a) = a2

§15. EXCEPTIONAL ISOMORPHISMS 217

is a quadratic form on Symd(A, σ)0 . Inspection of the split case shows that this

form is nonsingular. By the universal property of Cli¬ord algebras, the inclusion

Symd(A, σ) ’ A induces an F -algebra homomorphism

hA : C Symd(A, σ)0 , sσ ’ A,

(15.15)

which is not injective since dimF C Symd(A, σ)0 , sσ = 25 while dimF A = 24 .

Therefore, the Cli¬ord algebra C Symd(A, σ)0 , sσ is not simple, hence the dis-

criminant of the quadratic space Symd(A, σ)0 , sσ is trivial. Moreover, there is a

unique central element · in C1 Symd(A, σ)0 , sσ such that · 2 = 1 and hA (·) = 1.

We may therefore de¬ne a functor

S : C2 ’ B 2

by

S(A, σ) = Symd(A, σ)0 , sσ , · .

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

B2 ≡ C 2 .

Proof : For any (A, σ) ∈ C2 , the F -algebra homomorphism hA of (??) restricts to

a canonical isomorphism

∼

hA : C0 Symd(A, σ)0 , sσ ’ A,

’

and yields a natural transformation C —¦ S ∼ IdC2 .

=

For (V, q, ζ) ∈ B2 , we de¬ne a linear map mζ : V ’ C0 (V, q) by

mζ (v) = vζ for v ∈ V .

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

0

∼

mζ : (V, q) ’ Symd C0 (V, q), „

’ , s„ .

The same argument as in the proof of (??) shows that this isometry carries ζ to the

0

canonical orientation · on Symd C0 (V, q), „ , s„ ; therefore, it de¬nes a natural

transformation S —¦ C ∼ IdB2 which completes the proof.

=

(15.17) Corollary. For every oriented quadratic space (V, q, ζ) of dimension 5,

the Cli¬ord algebra construction yields a group isomorphism

∼

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

’

Proof : The functor C de¬nes an isomorphism between automorphism groups of

corresponding objects.

Suppose now that (A, σ) ∈ C2 corresponds to (V, q, ζ) ∈ B2 under the equiv-

alence B2 ≡ C2 , so that we may identify (A, σ) = C0 (V, q), „ and (V, q, ζ) =

Symd(A, σ)0 , sσ , · .

(15.18) Proposition. The special Cli¬ord group of (V, q) is

“+ (V, q) = GSp(A, σ).

Under the identi¬cation V = Symd(A, σ)0 ‚ A, the vector representation

χ : GSp(A, σ) ’ O+ (V, q)

is given by χ(g)(v) = gvg ’1 for g ∈ GSp(A, σ) and v ∈ V .

218 IV. ALGEBRAS OF DEGREE FOUR

The spin group is Spin(V, q) = Sp(A, σ) and the group of spinor norms is

Sn(V, q) = G(A, σ).

Proof : By de¬nition, “+ (V, q) is a subgroup of A— , and it consists of similitudes

of (A, σ). We have a commutative diagram with exact rows:

χ

O+ (V, q)

1 ’ ’ ’ F — ’ ’ ’ “+ (V, q) ’ ’ ’

’’ ’’ ’’ ’’’ 1

’’

¦ ¦

¦ ¦

C

1 ’ ’ ’ F — ’ ’ ’ GSp(A, σ) ’ ’ ’ PGSp(A, σ) ’ ’ ’ 1

’’ ’’ ’’ ’’

(see (??)24 ). The corollary above shows that the right-hand vertical map is an

isomorphism, hence “+ (V, q) = GSp(A, σ).

For g ∈ “+ (V, q), we have χ(g)(v) = g · v · g ’1 in C(V, q), by the de¬nition of

the vector representation. Under the identi¬cation V = Symd(A, σ)0 , every vector

v ∈ V is mapped to vζ ∈ A, hence the action of χ(g) is by conjugation by g, since

ζ is central in C(V, q). The last assertions are clear.

An alternate proof is given in (??) below.

If char F = 2, we may combine (??) with the equivalence B2 ≡ B2 of (??) to

get the following relation between groupoids of algebras with involution:

(15.19) Corollary. Suppose char F = 2. The functor S : C2 ’ B2 , which maps

every central simple algebra of degree 4 with symplectic involution (A, σ) to the

algebra of degree 5 with orthogonal involution EndF Symd(A, σ)0 , σsσ where σsσ

is the adjoint involution with respect to sσ , and the functor C : B2 ’ C2 , which

maps every central simple algebra of degree 5 with orthogonal involution (A , σ ) to

its Cli¬ord algebra C(A , σ ), σ , de¬ne an equivalence of groupoids: