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 ) =