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(7.11) TrdA1 —A2 x + σ1 — σ2 (x) = TrdA1 —A2 (x) for x ∈ A1 — A2

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

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