and

(7.12) TrdA1 —A2 ( s1 — s2 ) = f1 (s1 ) TrdA2 (s2 )

for s1 ∈ Sym(A1 , σ1 ) and s2 ∈ Sym(A2 , σ2 ). Since σ1 — σ2 (x) and xσ1 — σ2 ( ) have

the same reduced trace, we have

TrdA1 —A2 x + σ1 — σ2 (x) = TrdA1 —A2 + σ 1 — σ2 ( ) x .

Now, σ1 — σ2 ( ) = σ1 ( 1 ) — 1, hence it follows by (??) that + σ1 — σ2 ( ) = 1,

proving (??). To prove (??), it su¬ces to observe

TrdA1 —A2 ( s1 — s2 ) = TrdA1 ( 1 s1 ) TrdA2 (s2 ) = f1 (s1 ) TrdA2 (s2 ),

hence the claim is proved.

The determinant of (σ1 —σ2 , f1— ) is thus represented by SrdA1 —A2 ( ) in F/„˜(F ).

Since

n2

= PrdA1 , 1 (X)n2 ,

PrdA1 —A2 , (X) = NrdA1 —A2 (X ’ — 1) = NrdA1 (X ’ 1)

1

we have SrdA1 —A2 ( ) = n2 SrdA1 ( 1 ) + n2 TrdA1 ( 1 ). The proposition follows,

2

since TrdA1 ( 1 ) = f1 (1) and (??) shows that f1 (1) = m1 .

As for orthogonal involutions (see (??)), quadratic pairs on a quaternion algebra

are classi¬ed by their discriminant:

(7.13) Proposition. Let (γ, f1 ) and (γ, f2 ) be quadratic pairs on a quaternion

algebra Q over a ¬eld F of characteristic 2. If disc(γ, f1 ) = disc(γ, f2 ), then there

exists x ∈ Q— such that f2 = f1 —¦ Int(x).

86 II. INVARIANTS OF INVOLUTIONS

Proof : For i = 1, 2, let i ∈ Q be such that fi (s) = TrdQ ( i s) for all s ∈ Sym(Q, γ).

We have TrdQ ( i ) = fi (1) = 1, by (??), and

disc(γ, fi ) = SrdQ ( i ) + „˜(F ) = NrdQ ( i ) + „˜(F ).

Therefore, the hypothesis yields

NrdQ ( 2 ) = NrdQ ( 1 ) + (±2 + ±) = NrdQ ( + ±)

1

for some ± ∈ F . Since 1 is determined up to the addition of an element in

Alt(Q, γ) = F , we may substitute 1 + ± for 1 , and assume NrdQ ( 2 ) = NrdQ ( 1 ).

The elements 1 , 2 then have the same reduced characteristic polynomial, hence

we may ¬nd x ∈ Q— such that 1 = x 2 x’1 . For s ∈ Sym(Q, γ), we then have

TrdQ ( 2 s) = TrdQ (x’1 1 xs) = TrdQ ( 1 xsx’1 ),

hence f2 (s) = f1 (xsx’1 ).

Our ¬nal result is an analogue of the formula for the discriminant of an orthog-

onal sum of quadratic spaces. Let (σ, f ) be a quadratic pair on a central simple

algebra A over a ¬eld F of characteristic 2, and let e1 , e2 ∈ A be symmetric idem-

potents such that e1 + e2 = 1. As in the preceding section, we let A1 = e1 Ae1 ,

A2 = e2 Ae2 and restrict σ to symplectic involutions σ1 and σ2 on A1 and A2 . The

degrees of A1 and A2 are therefore even. We have Sym(Ai , σi ) = Sym(A, σ) © Ai

for i = 1, 2, hence we may also restrict f to Sym(Ai , σi ) and get a quadratic pair

(σi , fi ) on Ai .

(7.14) Proposition. With the notation above,

disc(σ, f ) = disc(σ1 , f1 ) + disc(σ2 , f2 ).

Proof : For i = 1, 2, let i ∈ Ai be such that fi (s) = TrdAi ( i s) for all s ∈

Sym(Ai , σi ). For s ∈ Sym(A, σ), we have

s = e1 se1 + e1 se2 + σ(e1 se2 ) + e2 se2 ,

hence

f (s) = f1 (e1 se1 ) + TrdA (e1 se2 ) + f2 (e2 se2 ).

Since TrdA (e1 se2 ) = TrdA (se2 e1 ) and e2 e1 = 0, the middle term on the right side

vanishes. Therefore,

f (s) = TrdA1 ( 1 e1 se1 ) + TrdA2 ( 2 e2 se2 ) for all s ∈ Sym(A, σ).

Taking into account the fact that ei i ei = for i = 1, 2, we obtain

i

f (s) = TrdA1 ( 1 s) + TrdA2 ( 2 s) = TrdA ( + 2 )s for all s ∈ Sym(A, σ).

1

We may thus compute det(σ, f ):

det(σ, f ) = SrdA ( + 2) + „˜(F ).

1

Scalar extension to a splitting ¬eld of A shows that PrdA, = PrdA1 , 1 PrdA2 , 2 ,

1+ 2

hence

SrdA ( + 2) = SrdA1 ( 1 ) + SrdA2 ( 2 ) + TrdA1 ( 1 ) TrdA2 ( 2 ).

1

1

Since TrdAi ( i ) = fi ( i ) = deg Ai , by (??), the preceding relation yields

2

1

det(σ, f ) = det(σ1 , f1 ) + det(σ2 , f2 ) + deg A1 deg A2 .

4

The formula for disc(σ, f ) is then easily checked, using that deg A = deg A1 +

deg A2 .

§8. THE CLIFFORD ALGEBRA 87

§8. The Cli¬ord Algebra

Since the Cli¬ord algebra of a quadratic form is not invariant when the quad-

ratic form is multiplied by a scalar, it is not possible to de¬ne a corresponding

notion for involutions. However, the even Cli¬ord algebra is indeed an invariant for

quadratic forms up to similarity, and our aim in this section is to generalize its con-

struction to algebras with quadratic pairs. The ¬rst de¬nition of the (generalized,

even) Cli¬ord algebra of an algebra with orthogonal involution of characteristic

di¬erent from 2 was given by Jacobson [?], using Galois descent. Our approach is

based on Tits™ “rational” de¬nition [?] which includes the characteristic 2 case.

Since our main tool will be scalar extension to a splitting ¬eld, we ¬rst discuss

the case of a quadratic space.

8.A. The split case. Let (V, q) be a nonsingular quadratic space over a

¬eld F of arbitrary characteristic. The Cli¬ord algebra C(V, q) is the factor of

the tensor algebra T (V ) by the ideal I(q) generated by all the elements of the form

v — v ’ q(v) · 1 for v ∈ V . The natural gradation of T (V ) (by natural numbers)

induces a gradation by Z/2Z:

T (V ) = T0 (V ) • T1 (V ) = T (V — V ) • V — T (V — V ) .

Since generators of I(q) are in T0 (V ), the Z/2Z gradation of T (V ) induces a gra-

dation of C(V, q):

C(V, q) = C0 (V, q) • C1 (V, q).

We have dimF C(V, q) = 2dim V and dimF C0 (V, q) = 2(dim V )’1 : see Knus [?,

Ch. IV, (1.5.2)].

The even Cli¬ord algebra C0 (V, q) may also be de¬ned directly as a factor

algebra of T0 (V ) = T (V — V ):

(8.1) Lemma. In the tensor algebra T (V — V ), consider the following two-sided

ideals:

(1) I1 (q) is the ideal generated by all the elements of the form

v — v ’ q(v), for v ∈ V .

(2) I2 (q) is the ideal generated by all the elements of the form

u — v — v — w ’ q(v)u — w, for u, v, w ∈ V .

Then

T (V — V )

C0 (V, q) = .

I1 (q) + I2 (q)

Proof : The inclusion map T (V — V ) ’ T (V ) maps I1 (q) and I2 (q) into I(q); it

therefore induces a canonical epimorphism

T (V — V )

’ C0 (V, q).

I1 (q) + I2 (q)

The lemma follows if we show

T (V — V )

dimF ¤ dimF C0 (V, q).

I1 (q) + I2 (q)

This inequality is easily established by using an orthogonal decomposition of V

into subspaces of dimension 1 (if char F = 2) or of dimension 2 (if char F = 2 and

88 II. INVARIANTS OF INVOLUTIONS

dim V is even) or into one subspace of dimension 1 and subspaces of dimension 2

(if char F = 2 and dim V is odd).

Structure of even Cli¬ord algebras. We recall the structure theorem for

even Cli¬ord algebras:

(8.2) Theorem. Let (V, q) be a nonsingular quadratic space over a ¬eld F of ar-

bitrary characteristic.

(1) If dim V is odd : dim V = 2m + 1, then C0 (V, q) is central simple F -algebra of

degree 2m .

(2) If dim V is even: dim V = 2m, the center of C0 (V, q) is an ´tale quadratic F -

e

algebra Z. If Z is a ¬eld, then C0 (V, q) is a central simple Z-algebra of degree 2m’1 ;

if Z F — F , then C0 (V, q) is the direct product of two central simple F -algebras

of degree 2m’1 . Moreover, the center Z can be described as follows:

(a) If char F = 2, Z F [X]/(X 2 ’ δ) where δ ∈ F — is a representative of the

discriminant: disc q = δ · F —2 ∈ F — /F —2 .

(b) If char F = 2, Z F [X]/(X 2 + X + δ) where δ ∈ F is a representative of

the discriminant: disc q = δ + „˜(F ) ∈ F/„˜(F ).

A proof can be found in Knus [?, Ch. IV] or Lam [?, Ch. 5] (for the case where

char F = 2) or Scharlau [?, Ch. 9] (for the cases where char F = 2 or char F = 2

and dim V even).

For future reference, we recall an explicit description of the Cli¬ord algebra

of hyperbolic quadratic spaces, from which a proof of the theorem above can be

derived by scalar extension.

Let U be an arbitrary ¬nite dimensional vector space over F and let H(U ) =

—

(U • U, qU ) be the hyperbolic quadratic space de¬ned by

qU (• + u) = •(u)

for • ∈ U — and u ∈ U , as in §??.

In order to give an explicit description of the Cli¬ord algebra of H(U ), consider

the exterior algebra U . Collecting separately the even and odd exterior powers

of U , we get a Z/2Z-gradation

U= U• U,

0 1

where

2i 2i+1

U= U and U= U.

0 i≥0 1 i≥0

For u ∈ U , let ∈ EndF ( U ) denote (exterior) multiplication on the left by u:

u

u (x1 § · · · § xr ) = u § x 1 § · · · § xr .

For • ∈ U — , let d• ∈ EndF ( U ) be the unique derivation of U extending •

which is explicitly de¬ned by

r

(’1)i+1 x1 § · · · § xi’1 § xi+1 § · · · § xr •(xi ).

d• (x1 § · · · § xr ) =

i=1

It is clear that and d• interchange the subspaces U and U for all u ∈ U ,

u 0 1

• ∈ U —.

§8. THE CLIFFORD ALGEBRA 89

(8.3) Proposition. The map which carries •+u ∈ U — •U to d• + ∈ EndF ( U)

u

induces an isomorphism

∼

˜ : C H(U ) ’ EndF (

’ U ).

The restriction of this isomorphism to the even Cli¬ord algebra is an isomorphism

∼

˜0 : C0 H(U ) ’ EndF (

’ U ) — EndF ( U ).

0 1

r s

U and • ∈ U — ,

Proof : A computation shows that for ξ ∈ U, · ∈

d• (ξ § ·) = d• (ξ) § · + (’1)r ξ § d• (·).

By applying this formula twice in the particular case where r = 1, we obtain

d2 (u § ·) = u § d2 (·);

• •

s

by induction on s we conclude that d2 = 0. Therefore, for · ∈ U , u ∈ U and

•

—

•∈U ,

2

(d• + u) (·) = d• (u § ·) + u § d• (·) = ·•(u).

By the universal property of Cli¬ord algebras, it follows that the map U — • U ’

EndF ( U ) which carries • + u to d• + u induces an algebra homomorphism

˜ : C H(U ) ’ EndF ( U ). The fact that ˜ is an isomorphism is established

by induction on dim U (see Knus [?, Ch. IV, (2.1.1)]). (Alternately, assuming the

structure theorem for Cli¬ord algebras, injectivity of ˜ follows from the fact that

C H(U ) is simple, and surjectivity follows by dimension count).

Let ˜0 be the restriction of ˜ to C0 H(U ) . Since d• + u exchanges 0 U and

—

1 U for all • ∈ U and u ∈ U , the elements in the image of ˜0 preserve 0 U and

1 U . Therefore, ˜0 maps C0 H(U ) into EndF ( 0 U ) — EndF ( 1 U ). This map

is onto by dimension count.

The canonical involution. For every quadratic space (V, q), the identity map

on V extends to an involution on the tensor algebra T (V ) which preserves the ideal

I(q). It therefore induces an involution „ on the Cli¬ord algebra C(V, q). This

involution is called the canonical involution of C(V, q); it is the unique involution

which is the identity on (the image of) V . The involution „ clearly restricts to an

involution on C0 (V, q) which we denote by „0 and call the canonical involution of

C0 (V, q). The type of this canonical involution is determined as follows:

(8.4) Proposition. (1) If dim V ≡ 2 mod 4, then „0 is unitary.

(2) If dim V ≡ 0 mod 4, then „0 is the identity on the center Z of C0 (V, q). It

is orthogonal if dim V ≡ 0 mod 8 and char F = 2, and symplectic if dim V ≡ 4

mod 8 or char F = 2. (In the case where Z F — F , this means that „0 is of

orthogonal or symplectic type on each factor of C0 (V, q).)

(3) If dim V ≡ 1, 7 mod 8, then „0 is orthogonal if char F = 2 and symplectic if

char F = 2.

(4) If dim V ≡ 3, 5 mod 8, then „0 is symplectic.

Proof : Consider ¬rst the case where dim V is even: dim V = 2m. By extending

scalars, we may assume that (V, q) is a hyperbolic quadratic space. Let (V, q) =

H(U ) for some m-dimensional vector space U , hence C(V, q) EndF ( U ) by (??).

Under this isomorphism, the canonical involution „ on C(V, q) corresponds to the

adjoint involution with respect to some bilinear form on U which we now describe.

90 II. INVARIANTS OF INVOLUTIONS

1

Let : U ’ U be the involution such that u = u for all u ∈ U = U

r

and let s : U ’ F be a nonzero linear map which vanishes on U for r < m.

De¬ne a bilinear form b : U — U ’ F by

b(ξ, ·) = s(ξ § ·) for ξ, · ∈ U.

We have b(·, ξ) = s(ξ § ·) for ξ, · ∈ U . Since ζ = (’1)m(m’1)/2 ζ for ζ ∈ m U ,

it follows that b is symmetric if m ≡ 0, 1 mod 4, and it is skew-symmetric if

m ≡ 2, 3 mod 4. If char F = 2, then U is commutative, is the identity on U

and ξ § ξ = 0 for all ξ ∈ U , hence b is alternating. In all cases, the form b is

nonsingular.

For u ∈ U and ξ, · ∈ U we have u = u, hence

b(u § ξ, ·) = s(ξ § u § ·) = b(ξ, u § ·).

Similarly, for • ∈ U — , ξ, · ∈ U , a simple computation (using the fact that d• is

a derivation on U ) shows that

b d• (ξ), · = b ξ, d• (·) .

Therefore, the adjoint involution σb on EndF ( U ) is the identity on all the en-

domorphisms of the form d• + u . It follows that σb corresponds to the canonical

involution „ under the isomorphism ˜ of (??). In view of the type of b, the invo-

lution „ is orthogonal if m ≡ 0, 1 mod 4 and char F = 2, and it is symplectic in

the other cases.

If m is odd, the complementary subspaces 0 U and 1 U are totally isotropic

for b. Therefore, letting e0 ∈ EndF ( U ) (resp. e1 ∈ EndF ( U )) denote the

projection on 0 U (resp. 1 U ) parallel to 1 U (resp. 0 U ), we have σb (e0 ) = e1 .

It follows that σb exchanges EndF ( 0 U ) and EndF ( 1 U ), hence „0 is unitary.

If m is even, b restricts to nonsingular bilinear forms b0 on 0 U and b1 on 1 U ,

and the restriction of σb to EndF ( 0 U ) — EndF ( 1 U ) is σb0 — σb1 . Since b0 and b1

have the same type as b, the proof is complete in the case where dim V is even.

If dim V is odd: dim V = 2m + 1, we may extend scalars to assume (V, q)

decomposes as

(V, q) [’1] ⊥ (V , q )

for some nonsingular quadratic space (V , q ) of dimension 2m which may be as-

sumed hyperbolic. Considering this isometry as an identi¬cation, and letting e ∈ V

denote a basis element of the subspace [’1] such that q(e) = ’1, we get an isomor-

∼

phism C(V , q ) ’ C0 (V, q) by mapping x ∈ V to e · x ∈ C0 (V, q). If char F = 2,

’

the canonical involution „0 on C0 (V, q) corresponds to the canonical involution „

on C(V , q ) under this isomorphism. Therefore, „0 is symplectic. If char F = 2,

the canonical involution „0 corresponds to Int(ζ) —¦ „ where ζ ∈ C(V , q ) is the

product of the elements in an orthogonal basis of V . As observed above, „ is

orthogonal if m ≡ 0, 1 mod 4 and is symplectic if m ≡ 2, 3 mod 4. On the other

hand, „ (ζ) = (’1)m ζ, hence (??) shows that „0 is orthogonal if m ≡ 0, 3 mod 4

and symplectic if m ≡ 1, 2 mod 4.

(8.5) Proposition. The involutions „ and „0 are hyperbolic if the quadratic space

(V, q) is isotropic.

Proof : If (V, q) is isotropic, it contains a hyperbolic plane; we may thus ¬nd in V

vectors x, y such that q(x) = q(y) = 0 and bq (x, y) = 1. Let e = x · y ∈ C0 (V, q) ‚

§8. THE CLIFFORD ALGEBRA 91

C(V, q). The conditions on x and y imply e2 = e and „ (e) = „0 (e) = 1 ’ e, hence

„ and „0 are hyperbolic, by (??).

8.B. De¬nition of the Cli¬ord algebra. Let (σ, f ) be a quadratic pair on

a central simple algebra A over a ¬eld F of arbitrary characteristic. Our goal is

to de¬ne an algebra C(A, σ, f ) in such a way that for every nonsingular quadratic

space (V, q) (of even dimension if char F = 2),

C EndF (V ), σq , fq C0 (V, q)

where (σq , fq ) is the quadratic pair associated to q by (??). The idea behind the

de¬nition below (in (??)) is that EndF (V ) V — V under the standard iden-

ti¬cation •q of (??); since C0 (V, q) is a factor algebra of T (V — V ), we de¬ne

C EndF (V ), σq , fq as a factor algebra of T EndF (V ) .

Let A denote A viewed as an F -vector space. We recall the “sandwich” iso-

morphism

∼

Sand : A — A ’ EndF (A)

’

such that Sand(a — b)(x) = axb for a, b, x ∈ A (see (??)). We use this isomorphism

to de¬ne a map

σ2 : A — A ’ A — A

as follows: for ¬xed u ∈ A — A the map A ’ A de¬ned by x ’ Sand(u) σ(x)

is linear and therefore of the form Sand σ2 (u) for a certain σ2 (u) ∈ A. In other

words, the map σ2 is de¬ned by the condition

Sand σ2 (u) (x) = Sand(u) σ(x) for u ∈ A — A, x ∈ A.

(8.6) Lemma. Let (V, b) be a nonsingular symmetric bilinear space and let σ b be

its adjoint involution on EndF (V ). The map σ2 on EndF (V ) — EndF (V ) satis¬es

σ2 •b (x1 — x2 ) — •b (x3 — x4 ) = •b (x1 — x3 ) — •b (x2 — x4 )

∼

for x1 , x2 , x3 , x4 ∈ V where •b : V — V ’ EndF (V ) is the standard identi¬cation

’

of (??).

Proof : It su¬ces to see that, for x1 , x2 , x3 , x4 , v, w ∈ V ,

Sand •b (x1 — x3 ) — •b (x2 — x4 ) •b (v — w) =

Sand •b (x1 — x2 ) — •b (x3 — x4 ) •b (w — v) .

This follows from a straightforward computation: the left side equals

•b (x1 — x3 ) —¦ •b (v — w) —¦ •b (x2 — x4 ) = •b (x1 — x4 )b(x3 , v)b(w, x2 )

whereas the right side equals

•b (x1 — x2 ) —¦ •b (w — v) —¦ •b (x3 — x4 ) = •b (x1 — x4 )b(x2 , w)b(v, x3 ).

Let ∈ A be such that f (s) = TrdA ( s) for all s ∈ Sym(A, σ). The existence

of such an element is proved in (??), where it is also proved that is uniquely

determined up to the addition of an element in Alt(A, σ). If = + a ’ σ(a) for

some a ∈ A, then for all u ∈ A — A such that σ2 (u) = u we have

Sand(u)( ) = Sand(u)( ) + Sand(u)(a) ’ Sand(u) σ(a) .

92 II. INVARIANTS OF INVOLUTIONS

The last term on the right side is equal to Sand σ2 (u) (a) = Sand(u)(a), hence

Sand(u)( ) = Sand(u)( ).

Therefore, the following de¬nition does not depend on the choice of :

(8.7) De¬nition. The Cli¬ord algebra C(A, σ, f ) is de¬ned as a factor of the ten-

sor algebra T (A):

T (A)

C(A, σ, f ) =

J1 (σ, f ) + J2 (σ, f )

where

(1) J1 (σ, f ) is the ideal generated by all the elements of the form s ’ f (s) · 1, for

s ∈ A such that σ(s) = s;

(2) J2 (σ, f ) is the ideal generated by all the elements of the form u ’ Sand(u)( ),

for u ∈ A — A such that σ2 (u) = u and for ∈ A as above.

The following proposition shows that the de¬nition above ful¬lls our aim:

(8.8) Proposition. Let (V, q) be a nonsingular quadratic space (of even dimen-

sion if char F = 2) and let (σq , fq ) be the associated quadratic pair. The standard

∼

identi¬cation •q : V — V ’ EndF (V ) of (??) induces an identi¬cation

’

∼

·q : C0 (V, q) ’ C EndF (V ), σq , fq .

’

Proof : It su¬ces to show that the isomorphism of tensor algebras

∼

T (•q ) : T (V — V ) ’ T EndF (V )

’

maps I1 (q) to J1 (σq , fq ) and I2 (q) to J2 (σq , fq ).

The ideal T (•q ) I1 (q) is generated by all the elements of the form

•q (v — v) ’ q(v) · 1, for v ∈ V .

Since σq corresponds to the switch map on V — V , the elements s ∈ EndF (V ) such

that σq (s) = s are spanned by elements of the form •q (v — v). Since moreover

q(v) = fq —¦ •q (v — v) by (??), it follows that J1 (σq , fq ) = T (•q ) I1 (q) .

Similarly, (??) shows that the elements u ∈ EndF (V ) — EndF (V ) such that

σ2 (u) = u are spanned by elements of the form •q (x — y) — •q (y — z) for x, y,

z ∈ V . Therefore, in order to show that J2 (σq , fq ) = T (•q ) I2 (q) , it su¬ces to

prove

(8.9) Sand •q (x — y) — •q (y — z) ( ) = q(y)•q (x — z) for x, y, z ∈ V .

1

If char F = 2 we may choose = 2 , hence

Sand •q (x — y) — •q (y — z) ( ) = 1 •q (x — y) —¦ •q (y — z).

2

The right side can be evaluated by (??):

1

— y) —¦ •q (y — z) = 1 bq (y, y)•q (x — z) = q(y)•q (x — z),

2 •q (x 2

proving (??) when char F = 2.

Suppose next char F = 2, hence dim V is even. We may of course assume y = 0

in (??). Let dim V = n = 2m and let (e1 , . . . , en ) be a symplectic basis of V such

that e1 = y. We thus assume

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

§8. THE CLIFFORD ALGEBRA 93

As observed in (??), we may then choose

m

= •q ( i=1 e2i’1 — e2i’1 q(e2i ) + e2i — e2i q(e2i’1 ) + e2i’1 — e2i ).

By (??) we have

Sand •q (x — e1 ) — •q (e1 — z) •q (e2i’1 — e2i’1 ) =

•q (x — e1 ) —¦ •q (e2i’1 — e2i’1 ) —¦ •q (e1 — z) = 0