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for i = 1, . . . , m, and similarly
Sand •q (x — e1 ) — •q (e1 — z) •q (e2i’1 — e2i ) = 0
for i = 1, . . . , m. Moreover,
•q (x — z) for i = 1,
Sand •q (x — e1 ) — •q (e1 — z) •q (e2i — e2i ) =
0 for i > 1,
hence
Sand •q (x — e1 ) — •q (e1 — z) ( ) = q(e1 )•q (x — z).


If char F = 2, the quadratic pair (σ, f ) is entirely determined by the involution
σ, since f (s) = 1 TrdA (s) for all s ∈ Sym(A, σ). We then simply write C(A, σ, f )
2
for C(A, σ) Since we may choose = 1/2, we have Sand(u)( ) = 1 µ(u) where 2
µ : A — A ’ A is the multiplication map: µ(x — y) = xy for x, y ∈ A.
Examples. Cli¬ord algebras of quadratic pairs on nonsplit central simple al-
gebras are not easy to describe explicitly in general. We have the following results
however:
(a) For algebras of degree 2, it readily follows from (??) below that C(A, σ, f )
is the ´tale quadratic F -algebra determined by the discriminant disc(σ, f ).
e
(b) For the tensor product of two quaternion algebras Q1 , Q2 with canonical
involutions γ1 , γ2 it is shown in (??) below that
C(Q1 — Q2 , γ1 — γ2 ) Q1 — Q2 .
More generally, Tao [?] has determined (in characteristic di¬erent from 2)
up to Brauer-equivalence the components of the Cli¬ord algebra of a tensor
product of two central simple algebras with involution: see the notes at the
end of this chapter.
(c) Combining (??) and (??), one sees that the Cli¬ord algebra of a hyper-
bolic quadratic pair on a central simple algebra A of degree divisible by 4
decomposes into a direct product of two central simple algebras, of which
one is split and the other is Brauer-equivalent to A.
Besides the structure theorem in (??) below, additional general information on
Cli¬ord algebras of quadratic pairs is given in (??).
Structure of Cli¬ord algebras. Although the degree of A is arbitrary in
the discussion above (when char F = 2), the case where deg A is odd does not
yield anything beyond the even Cli¬ord algebras of quadratic spaces, since central
simple algebras of odd degree with involutions of the ¬rst kind are split (see (??)).
Therefore, we shall discuss the structure of Cli¬ord algebras only in the case where
deg A = n = 2m.
94 II. INVARIANTS OF INVOLUTIONS


(8.10) Theorem. Let (σ, f ) be a quadratic pair on a central simple algebra A
of even degree n = 2m over a ¬eld F of arbitrary characteristic. The center of
C(A, σ, f ) is an ´tale quadratic F -algebra Z. If Z is a ¬eld, then C(A, σ, f ) is a
e
central simple Z-algebra of degree 2m’1 ; if Z F — F , then C(A, σ, f ) is a direct
product of two central simple F -algebras of degree 2m’1 . Moreover, the center Z is
as follows:
F [X]/(X 2 ’ δ) where δ ∈ F — is a representative of the
(1) If char F = 2, Z
discriminant: disc(σ, f ) = disc σ = δ · F —2 ∈ F — /F —2 .
(2) If char F = 2, Z F [X]/(X 2 + X + δ) where δ ∈ F is a representative of the
discriminant: disc(σ, f ) = δ + „˜(F ) ∈ F/„˜(F ).
Proof : Let L be a splitting ¬eld of A in which F is algebraically closed (for in-
stance the function ¬eld of the Severi-Brauer variety of A). There is a nonsingular
quadratic space (V, q) over L such that
(AL , σL , fL ) EndL (V ), σq , fq ,
by (??). Moreover, (??) shows that disc(σL , fL ) = disc q (in L— /L—2 if char F = 2,
in L/„˜(L) if char F = 2). If δ ∈ F is a representative of disc(σ, f ), we then have
δ · L—2 if char F = 2,
disc q =
δ + „˜(L) if char F = 2.
It is clear from the de¬nition that the construction of the Cli¬ord algebra commutes
with scalar extension, hence by (??)
C(A, σ, f ) —F L = C(AL , σL , fL ) C0 (V, q).
In particular, it follows that the center Z of C(A, σ, f ) is a quadratic ´tale F -
e
algebra which under scalar extension to L becomes isomorphic to L[X]/(X 2 ’ δ) if
char F = 2 and to L[X]/(X 2 + X + δ) if char F = 2. Since F is algebraically closed
in L, it follows that Z F [X]/(X 2 ’ δ) if char F = 2 and Z F [X]/(X 2 + X + δ)
if char F = 2. The other statements also follow from the structure theorem for even
Cli¬ord algebras of quadratic spaces: see (??).
Alternate methods of obtaining the description of Z proven above are given
in (??) and (??).
The canonical involution. Let σ : T (A) ’ T (A) be the involution induced
by σ on the tensor algebra T (A); thus, for a1 , . . . , ar ∈ A,
σ(a1 — · · · — ar ) = σ(ar ) — · · · — σ(a1 ).
Direct computations show that the ideals J1 (σ, f ) and J2 (σ, f ) are preserved un-
der σ. Therefore, σ induces an involution on the factor algebra C(A, σ, f ) which
we also denote by σ and call the canonical involution of C(A, σ, f ).
The following result justi¬es this de¬nition:
(8.11) 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 on EndF (V ).
Under the standard identi¬cation

·q : C0 (V, q) ’ C EndF (V ), σq , fq

of (??), the canonical involution „0 of C0 (V, q) corresponds to the involution σq of
C EndF (V ), σq , fq .
§8. THE CLIFFORD ALGEBRA 95


Proof : The canonical involution „ of C(V, q) is induced by the involution of T (V )
which is the identity on V . Therefore, „0 is induced by the involution of T (V — V )
which switches the factors in V — V . Under the standard identi¬cation of (??), this
involution corresponds to σq .
By extending scalars to a splitting ¬eld of A, we may apply the preceding
proposition and (??) to determine the type of the involution σ on C(A, σ, f ). As
in (??), we only consider the case of even degree.
(8.12) Proposition. Let (σ, f ) be a quadratic pair on a central simple algebra A
of even degree n = 2m over a ¬eld F . The canonical involution σ of C(A, σ, f ) is
unitary if m is odd, orthogonal if m ≡ 0 mod 4 and char F = 2, and symplectic
if m ≡ 2 mod 4 or char F = 2. (In the case where the center of C(A, σ, f ) is
isomorphic to F — F , this means that σ is of orthogonal or symplectic type on each
factor of C(A, σ, f ).)
8.C. Lie algebra structures. We continue with the same notation as in the
preceding section; in particular, (σ, f ) is a quadratic pair on a central simple alge-
bra A over a ¬eld F of arbitrary characteristic and C(A, σ, f ) is the corresponding
Cli¬ord algebra.
Since C(A, σ, f ) is de¬ned as a quotient of the tensor algebra T (A), the canon-
ical map A ’ A ’ T (A) yields a canonical map
(8.13) c : A ’ C(A, σ, f )
which is F -linear but not injective (nor an algebra homomorphism), since c(s) =
1
f (s) for all s ∈ Sym(A, σ). In particular, (??) shows that c(1) = 2 deg A. We will
show that the subspace c(A) ‚ C(A, σ, f ) is a Lie subalgebra of L C(A, σ, f ) , and
relate it to the Lie subalgebra Alt(A, σ) ‚ L(A).
(8.14) Lemma. The kernel of c is ker c = ker f ‚ Sym(A, σ), and dim c(A) =
n(n’1)
+ 1 if deg A = n. Moreover, for x1 , x2 ∈ A we have
2
c(x1 ), c(x2 ) = c x1 ’ σ(x1 ), x2
where [ , ] are the Lie brackets.
Proof : Since c and f have the same restriction to Sym(A, σ), it is clear that
ker f ‚ ker c. Dimension count shows that this inclusion is an equality if we show
dim c(A) = n(n’1) + 1.
2
In order to compute the dimension of c(A), we may extend scalars to a splitting
¬eld of A. Therefore, it su¬ces to consider the case where A is split: let A =
EndF (V ) and (σ, f ) = (σq , fq ) for some nonsingular quadratic space (V, q) (of even

dimension if char F = 2). Under the standard identi¬cations •q : V — V ’ A ’

of (??) and ·q : C0 (V, q) ’ C(A, σ, f ) of (??), the map c : V — V ’ C0 (V, q) is

given by the multiplication in C(V, q):
c(v — w) = v · w ∈ C0 (V, q) for v, w ∈ V .
Let (e1 , . . . , en ) be an arbitrary basis of V . The Poincar´-Birkho¬-Witt theorem
e
(Knus [?, Ch. IV, (1.5.1)]) shows that the elements 1 and ei · ej for i < j are
linearly independent in C0 (V, q). Since these elements span c(V — V ), it follows
that dim c(V — V ) = n(n’1) + 1, completing the proof of the ¬rst part.
2
In order to prove the last relation, we may also assume that A is split. As
above, we identify A with V — V by means of •q . Since both sides of the relation
96 II. INVARIANTS OF INVOLUTIONS


are bilinear in x1 , x2 , it su¬ces to prove it for x1 = v1 — w1 and x2 = v2 — w2 with
v1 , v2 , w1 , w2 ∈ V . Then
x1 ’ σ(x1 ), x2 = (v1 — w1 ’ w1 — v1 ) —¦ (v2 — w2 )
’ (v2 — w2 ) —¦ (v1 — w1 ’ w1 — v1 )
= v1 — w2 bq (w1 , v2 ) ’ w1 — w2 bq (v1 , v2 )
’ v2 — w1 bq (w2 , v1 ) + v2 — v1 bq (w2 , w1 ),
hence
c [x1 ’ σ(x1 ), x2 ] = v1 · w2 bq (w1 , v2 ) ’ w1 · w2 bq (v1 , v2 )
+ v2 · v1 bq (w2 , w1 ) ’ v2 · w1 bq (w2 , v1 ).
For u, v ∈ V , we have u · v + v · u = bq (u, v); therefore, the four terms on the right
side of the last equation can be evaluated as follows:
v1 · w2 bq (w1 , v2 ) = v1 · w1 · v2 · w2 + v1 · v2 · w1 · w2
w1 · w2 bq (v1 , v2 ) = v1 · v2 · w1 · w2 + v2 · v1 · w1 · w2
v2 · v1 bq (w2 , w1 ) = v2 · v1 · w1 · w2 + v2 · v1 · w2 · w1
v2 · w1 bq (w2 , v1 ) = v2 · v1 · w2 · w1 + v2 · w2 · v1 · w1 .
The alternating sum of the right sides is
v1 · w1 · v2 · w2 ’ v2 · w2 · v1 · w1 = c(x1 ), c(x2 ) .


The lemma shows that c(A) is stable under the Lie brackets, and is therefore a
Lie subalgebra of L C(A, σ, f ) . Moreover, it shows that if x, y ∈ A are such that
c(x) = c(y), then x ’ y ∈ Sym(A, σ), hence x ’ σ(x) = y ’ σ(y). We may therefore
de¬ne a map
δ : c(A) ’ Alt(A, σ)
by
δ c(x) = x ’ σ(x) for x ∈ A.
(8.15) Proposition. The map δ is a Lie-algebra homomorphism which ¬ts into
an exact sequence
δ
0 ’ F ’ c(A) ’ Alt(A, σ) ’ 0.

Proof : For x, y ∈ A we have c(x), c(y) = c x ’ σ(x), y by (??), hence
δ c(x), c(y) = x ’ σ(x), y ’ σ x ’ σ(x), y = x ’ σ(x), y ’ σ(y) ,
proving that δ is a Lie-algebra homomorphism. This map is surjective by de¬nition.
In order to show F ‚ ker δ, pick an element a ∈ A such that TrdA (a) = 1; we then
have
c a + σ(a) = f a + σ(a) = TrdA (a) = 1,
hence
δ(1) = a + σ(a) ’ σ a + σ(a) = 0.
Therefore, F ‚ ker δ, and dimension count shows that this inclusion is an equality.
§8. THE CLIFFORD ALGEBRA 97


We proceed to de¬ne on c(A) another Lie-algebra homomorphism, using the
canonical involution σ on C(A, σ, f ).
(8.16) Lemma. For all x ∈ A,
σ c(x) = c σ(x) c(x) + σ c(x) = TrdA (x).
and
In particular, Id + σ maps c(A) onto F . Therefore, c(A) ‚ g C(A, σ, f ), σ .
Proof : The ¬rst equation is clear from the de¬nition of σ. The second equation
follows, since c x + σ(x) = f x + σ(x) .
Let c(A)0 = c(A) © Skew C(A, σ, f ), σ . As an intersection of Lie subalgebras,
c(A)0 is a subalgebra of L C(A, σ, f ) .
(8.17) Proposition. The map Id + σ : c(A) ’ F is a Lie-algebra homomorphism
which ¬ts into an exact sequence
Id+σ
0 ’ c(A)0 ’ c(A) ’ ’ F ’ 0.
’’
n(n’1)
In particular, it follows that dim c(A)0 = if deg A = n.
2

Proof : The de¬nition of c(A)0 shows that this set is the kernel of Id + σ. For x,
y ∈ A, we have c(x), c(y) = c x ’ σ(x), y by (??). The preceding lemma shows
that the image of this under Id + σ is equal to
TrdA x ’ σ(x), y = 0,
hence Id + σ is a Lie-algebra homomorphism.
Special features of the case where char F = 2 are collected in the following
proposition:
(8.18) Proposition. If char F = 2, there is a direct sum decomposition
c(A) = F • c(A)0 .
The restriction of δ to c(A)0 is an isomorphism of Lie algebras

δ : c(A)0 ’ Alt(A, σ) = Skew(A, σ).

1 1
2 c, mapping x ∈ Skew(A, σ) to 2 c(x).
The inverse isomorphism is
Proof : The hypothesis that char F = 2 ensures that F © c(A)0 = {0}, hence c(A) =
F • c(A)0 . For a ∈ Alt(A, σ), we have
σ c(a) = c σ(a) = ’c(a)
by (??), hence c(a) ∈ c(A)0 . On the other hand, the de¬nition of δ yields
δ c(a) = a ’ σ(a) = 2a.
Since c(A)0 and Alt(A, σ) have the same dimension, it follows that δ is bijective
1
and that its inverse is 2 c.
(8.19) Example. Suppose A = Q1 — Q2 is a tensor product of two quaternion
algebras over a ¬eld F of arbitrary characteristic, and let σ = γ1 — γ2 be the tensor
product of the canonical involutions on Q1 and Q2 . Since γ1 and γ2 are symplec-
tic, there is a canonical quadratic pair (σ, f— ) on Q1 — Q2 : see (??). By (??) (if
char F = 2) or (??) (if char F = 2), the discriminant of (σ, f— ) is trivial, hence (??)
shows that C(A, σ, f— ) = C + — C ’ for some quaternion algebras C + , C ’ . More-
over, the canonical involution σ is symplectic (see (??)), hence it is the quaternion
98 II. INVARIANTS OF INVOLUTIONS


conjugation on C + and C ’ . We claim that C + and C ’ are isomorphic to Q1
and Q2 .
Let
(C + — C ’ ) = { (x+ , x’ ) ∈ C + — C ’ | TrdC + (x+ ) = TrdC ’ (x’ ) }
= { ξ ∈ C + — C ’ | TrdC + —C ’ (ξ) ∈ F }

and

(Q1 — Q2 ) = { (x1 , x2 ) ∈ Q1 — Q2 | TrdQ1 (x1 ) = TrdQ2 (x2 ) }.
In view of (??), we have c(A) ‚ (C + —C ’ ) , hence c(A) = (C + —C ’ ) by dimension
count. On the other hand, we may de¬ne a linear map ˜ : A ’ Q1 — Q2 by
˜(x1 — x2 ) = TrdQ2 (x2 )x1 , TrdQ1 (x1 )x2 for x1 ∈ Q1 , x2 ∈ Q2 .
Clearly, im ˜ ‚ (Q1 — Q2 ) ; the converse inclusion follows from the following ob-
servation: if (x1 , x2 ) ∈ Q1 — Q2 and TrdQ1 (x1 ) = TrdQ2 (x2 ) = ±, we have

˜(±’1 x1 — x2 ) if ± = 0,
(x1 , x2 ) =
˜(x1 — 2 + 1 — x2 ) if ± = 0,
where i ∈ Qi is an element of reduced trace 1 for i = 1, 2. A computation shows
that ˜ vanishes on the kernel of the canonical map c : A ’ C(A, σ, f— ) (see (??)),
hence it induces a surjective linear map c(A) ’ (Q1 — Q2 ) which we call again ˜.
Since c(A) and (Q1 — Q2 ) have the same dimension, this map is bijective:

˜ : (C + — C ’ ) = c(A) ’ (Q1 — Q2 ) .

Using (??), one can check that this bijection is an isomorphism of Lie algebras. To
complete the proof, we show that this isomorphism extends to an isomorphism of

(associative) F -algebras C + — C ’ = C(A, σ, f— ) ’ Q1 — Q2 . Since C + — C ’ is

generated by the subspace (C + — C ’ ) , the same argument as in the proof of (??)
shows that it su¬ces to ¬nd an isomorphism extending ˜ over an extension of F .
We may thus assume that Q1 and Q2 are split and identify Q1 = Q2 = EndF (V ) for
some 2-dimensional F -vector space V . Let b be a nonsingular alternating form on V
(such a form is uniquely determined up to a scalar factor) and let q be the quadratic
form on V — V whose polar bilinear form is b — b and such that q(v — w) = 0 for all
v, w ∈ V (see Exercise ?? of Chapter ??). The canonical quadratic pair (γ — γ, f— )
on A = EndF (V ) — EndF (V ) = EndF (V — V ) is then associated with the quadratic
form q, hence the standard identi¬cation •q induces an F -algebra isomorphism

·q : C0 (V — V, q) ’ C(A, σ, f— )

(see (??)). By de¬nition of the canonical map c, we have
c(A) = ·q (V — V ) · (V — V ) .
On the other hand, the map i : V — V ’ M2 EndF (V ) de¬ned by
0 •b (v — w)
i(v — w) = for v, w ∈ V
’•b (w — v) 0
induces an F -algebra homomorphism i— : C(V — V, q) ’ M2 EndF (V ) by the
universal property of Cli¬ord algebras. This homomorphism is injective because
§8. THE CLIFFORD ALGEBRA 99


C(V — V, q) is simple, hence also surjective by dimension count. Under the isomor-
phism i— , the natural gradation of the Cli¬ord algebra corresponds to the checker-
board grading of M2 EndF (V ) , hence i— induces an F -algebra isomorphism
EndF (V ) 0

i— : C0 (V — V, q) ’
’ EndF (V ) — EndF (V ).
0 EndF (V )
For v1 , v2 , w1 , w2 ∈ V , we have •q (v1 —w1 )—(v2 —w2 ) = •b (v1 —v2 )—•b (w1 —w2 ),
hence
˜ ·q (v1 — w1 · v2 — w2 ) =
= tr •b (w1 — w2 ) •b (v1 — v2 ), tr •b (v1 — v2 ) •b (w1 — w2 )
= b(w2 , w1 )•b (v1 — v2 ), b(v2 , v1 )•b (w1 — w2 ) .
On the other hand,
i— (v1 — w1 · v2 — w2 ) =
’•b (v1 — w1 ) —¦ •b (w2 — v2 ) 0
=
0 ’•b (w1 — v1 ) —¦ •b (v2 — w2 )
’b(w1 , w2 )•b (v1 — v2 ) 0
= .
0 ’b(v1 , v2 )•b (w1 — w2 )
Therefore, i— and ˜—¦·q have the same restriction to (V —V )·(V —V ), and it follows

’1
that the F -algebra isomorphism i— —¦ ·q : C(A, σ, f— ) ’ EndF (V ) — EndF (V ) =

Q1 — Q2 extends ˜. This completes the proof of the claim.
In conclusion, we have shown:
C(Q1 — Q2 , γ1 — γ2 , f— ) Q1 — Q2 .
A more general statement is proved in (??) below.
8.D. The center of the Cli¬ord algebra. The center of the Cli¬ord algebra
C(A, σ, f ) of a central simple algebra A with a quadratic pair (σ, f ) is described
in (??) as an ´tale quadratic F -algebra. In this section, we show how elements of
e
the center can be produced explicitly, thus providing another proof of the second
part of (??).
We set Z(A, σ, f ) for the center of C(A, σ, f ). If char F = 2, the map f is
uniquely determined by σ and we use the shorter notation C(A, σ) for the Cli¬ord
algebra and Z(A, σ) for its center.
As may expected from (??), our methods in characteristic 2 and characteristic
not 2 are completely di¬erent. In characteristic di¬erent from 2 they rely on an
analogue of the pfa¬an, viewed as a map from Skew(A, σ) to Z(A, σ). The case
of characteristic 2 is simpler; it turns out then that Z(A, σ, f ) is in the image c(A)
of A in C(A, σ, f ) under the canonical map of §??.
Characteristic not 2. Our ¬rst result yields a standard form for certain skew-
symmetric elements in split algebras with orthogonal involution.
(8.20) Lemma. Let (V, q) be a nonsingular quadratic space of dimension n = 2m
over a ¬eld F of characteristic di¬erent from 2 and let a ∈ EndF (V ) satisfy σq (a) =
’a. Assume moreover that the characteristic polynomial of a splits into pairwise
distinct linear factors:
Pca (X) = (X ’ »1 )(X + »1 ) · · · (X ’ »m )(X + »m )
100 II. INVARIANTS OF INVOLUTIONS


for some »1 , . . . , »m ∈ F — . There exists an orthogonal basis (e1 , . . . , en ) of V such
that the matrix representing a with respect to this basis is
« 
Λ1 0
0 »i
¬ ·
..  where Λi = »
 . 0
i
0 Λm
Letting ±i = q(ei ) for i = 1, . . . , n, we have ±2i = ’±2i’1 for i = 1, . . . , m.

Moreover, with •q : V — V ’ EndF (V ) the standard identi¬cation (??), we have

m
»i
a= •q (e2i’1 — e2i ’ e2i — e2i’1 ).
2±2i
i=1

Proof : For i = 1, . . . , m, let Vi ‚ V be the sum of the eigenspaces of a for
the eigenvalues »i and ’»i . The subspace Vi is thus the eigenspace of a2 for the
eigenvalue »2 . We have
i

V = V1 • · · · • V m
and the subspaces V1 , . . . , Vm are pairwise orthogonal since, for x ∈ Vi and y ∈ Vj ,
»2 bq (x, y) = bq a2 (x), y = bq x, a2 (y) = »2 bq (x, y),
i j

and »2 = »2 for i = j. It follows that the subspaces V1 , . . . , Vm are nonsingular.
i j
For i = 1, . . . , m, pick an anisotropic vector e2i’1 ∈ Vi and let e2i = »’1 a(e2i’1 ).
i
Since σq (a) = ’a, we have
bq (e2i’1 , e2i ) = »’1 bq e2i’1 , a(e2i’1 )
i

= ’»’1 bq a(e2i’1 ), e2i’1 = ’bq (e2i , e2i’1 ),
i

hence (e2i’1 , e2i ) is an orthogonal basis of Vi . It follows that (e1 , . . . , en ) is an
orthogonal basis of V , and the matrix of a with respect to this basis is as stated
above.
The equation a(e2i’1 ) = »i e2i yields
bq a(e2i’1 ), a(e2i’1 ) = »2 bq (e2i , e2i ) = 2»2 q(e2i ).
i i

On the other hand, since σq (a) = ’a and a2 (e2i’1 ) = »2 e2i’1 , the left side is also
i
equal to
bq e2i’1 , ’a2 (e2i’1 ) = ’»2 bq (e2i’1 , e2i’1 ) = ’2»2 q(e2i’1 ),
i i

hence q(e2i ) = ’q(e2i’1 ). Finally, for i, j = 1, . . . , m we have
’2q(e2i’1 )e2i if i = j,
•q (e2i’1 — e2i ’ e2i — e2i’1 )(e2j’1 ) =

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