and

2q(e2i )e2i’1 if i = j,

•q (e2i’1 — e2i ’ e2i — e2i’1 )(e2j ) =

0 if i = j.

The last equation in the statement of the lemma follows.

Let (A, σ) be a central simple algebra with orthogonal involution over a ¬eld F

of characteristic di¬erent from 2. We assume throughout this subsection that the

degree of A is even and let deg A = n = 2m. Our ¬rst observations also require

the ¬eld F to be in¬nite. Under this hypothesis, we denote by S(A, σ) the set of

§8. THE CLIFFORD ALGEBRA 101

skew-symmetric units in A— whose reduced characteristic polynomials are separable

(i.e., have no repeated root in an algebraic closure). This set is Zariski-open in

Skew(A, σ), since it is de¬ned by the condition that the discriminant of the reduced

characteristic polynomial does not vanish. By scalar extension to a splitting ¬eld L

such that (AL , σL ) Mn (L), t , we can see that this open set is not empty, since

S Mn (L), t = ….

Over an algebraic closure, the reduced characteristic polynomial of every a ∈

S(A, σ) splits into a product of pairwise distinct linear factors of the form

Prda (X) = (X ’ »1 )(X + »1 ) · · · (X ’ »m )(X + »m )

since σ(a) = ’a. Therefore, the subalgebra F [a] ‚ A generated by a has dimen-

sion n and F [a2 ] = F [a]©Sym(A, σ) has dimension m. Clearly, F [a]©Skew(A, σ) =

a · F [a2 ].

(8.21) Lemma. Let a ∈ S(A, σ). Denote H = F [a] © Skew(A, σ) = a · F [a2 ] and

E = F [a2 ]. The bilinear form T : H — H ’ F de¬ned by

T (x, y) = TE/F (xy) for x, y ∈ H

is nonsingular. Moreover, the elements in the image c(H) of H in C(A, σ) under

the canonical map c : A ’ C(A, σ) commute.

Proof : It su¬ces to check the lemma over a scalar extension. We may therefore

assume that A and the reduced characteristic polynomial of a are split12 . We

identify (A, σ) = EndF (V ), σq for some nonsingular quadratic space (V, q) of

dimension n. By (??), there is an orthogonal basis (e1 , . . . , en ) of V such that,

letting q(ei ) = ±i ∈ F — ,

m

»i

a= •q (e2i’1 — e2i ’ e2i — e2i’1 )

2±2i

i=1

(and ±2i’1 = ’±2i for i = 1, . . . , m). For i = 1, . . . , m, let

1

hi = •q (e2i’1 — e2i ’ e2i — e2i’1 ) ∈ A.

2±2i

By using the matrix representation with respect to the basis (e1 , . . . , en ), it is easily

seen that every skew-symmetric element in A which commutes with a is a linear

combination of h1 , . . . , hm . In particular, H is contained in the span of h1 , . . . , hm .

Since dim H = dim F [a2 ] = m, it follows that (h1 , . . . , hm ) is a basis of H.

With the same matrix representation, it is easy to check that h2 , . . . , h2 are

1 m

primitive orthogonal idempotents in E which form a basis of E and that hi hj = 0

for i = j. Therefore, the bilinear form T satis¬es

m m m

T( xi h i , y j hj ) = xi y i

i=1 j=1 i=1

for x1 , . . . , xm , y1 , . . . , ym ∈ F . It is therefore nonsingular.

Consider now the subspace c(H) of C(A, σ) spanned by c(h1 ), . . . , c(hm ). For

i = 1, . . . , m we have

1 1

c(hi ) = (e2i’1 · e2i ’ e2i · e2i’1 ) = e2i’1 · e2i .

2±2i ±2i

These elements commute, since (e1 , . . . , en ) is an orthogonal basis of V .

12 In fact, the algebra A splits as soon as the reduced characteristic polynomial of a splits.

102 II. INVARIANTS OF INVOLUTIONS

Since the bilinear form T on H is nonsingular, every linear form on H is of the

type x ’ TE/F (hx) for some h ∈ H. Therefore, every homogeneous polynomial

map P : H ’ F of degree d has the form

P (x) = TE/F (hi1 x) · · · TE/F (hid x)

i

for some hi1 , . . . , hid ∈ H, and in the d-th symmetric power S d H the element

i hi1 · · · hid is uniquely determined by P . In particular, there is a uniquely deter-

mined element ν = i hi1 · · · him ∈ S m H such that

NE/F (ax) = TE/F (hi1 x) · · · TE/F (him x) for x ∈ H,

i

since the map x ’ NE/F (ax) is a homogeneous polynomial map of degree m on H.

Since the elements in c(H) commute, the canonical map c induces a well-de¬ned

linear map S m H ’ C(A, σ). We set π(a) for the image under this induced map of

the element ν ∈ S m H de¬ned above.

In summary, the element π(a) ∈ C(A, σ) is de¬ned as follows:

(8.22) De¬nition. For a ∈ S(A, σ), we let

π(a) = c(hi1 ) · · · c(him )

i

where hi1 , . . . , him ∈ H = a · F [a2 ] satisfy

NF [a2 ]/F (ax) = TF [a2 ]/F (hi1 x) · · · TF [a2 ]/F (him x)

i

for all x ∈ a · F [a2 ].

(8.23) Lemma. Let ι be the nontrivial automorphism of the center Z(A, σ) of

C(A, σ). For a ∈ S(A, σ) we have π(a) ∈ Z(A, σ), ι π(a) = ’π(a) and

π(a)2 = (’1)m NrdA (a).

Proof : It su¬ces to verify the assertions over a scalar extension. We may thus

assume that A and the reduced characteristic polynomial of a are split, and use the

same notation as in (??). In particular, we let (A, σ) = EndF (V ), σq and choose

an orthogonal basis (e1 , . . . , em ) of V such that

1

m

a= »i hi where hi = •q (e2i’1 — e2i ’ e2i — e2i’1 ).

i=1

2±2i

m

As observed in (??), the elements h1 , . . . , hm form a basis of H. For x = xi h i ,

i=1

we have

m

» i xi h 2 ) = » 1 · · · » m x1 · · · x m .

NF [a2 ]/F (ax) = NF [a2 ]/F ( i

i=1

On the other hand, TF [a2 ]/F (hi x) = xi , hence ν = h1 . . . hm and

π(a) = »1 · · · »m c(h1 ) · · · c(hm ).

It was also seen in (??) that c(hi ) = ±1 e2i’1 · e2i where ±2i = q(e2i ) = ’q(e2i’1 ),

2i

hence

»1 · · · » m

π(a) = m e1 · · · en .

i=1 ±2i

It is then clear that π(a) ∈ Z(A, σ) and ι π(a) = ’π(a).

m

Since (e1 · · · en )2 = (’1)m e2 · · · e2 = i=1 ±2 and NrdA (a) = (’1)m »2 · · · »2 ,

n m

1 1

2i

the last equation in the statement of the proposition follows.

§8. THE CLIFFORD ALGEBRA 103

To extend the de¬nition of π to the whole of Skew(A, σ), including also the

case where the base ¬eld F is ¬nite (of characteristic di¬erent from 2), we adjoin

indeterminates to F and apply π to a generic skew-symmetric element.

Pick a basis (a1 , . . . , ad ) of Skew(A, σ) where d = m(n ’ 1) = dim Skew(A, σ),

d

and let ξ = i=1 ai xi where x1 , . . . , xd are indeterminates over F . We have

ξ ∈ S AF (x1 ,...,xd ) , σF (x1 ,...,xd ) and

π(ξ) ∈ Z(AF (x1 ,...,xd ) , σF (x1 ,...,xd ) ) = Z(A, σ) —F F (x1 , . . . , xd ).

Since π(ξ)2 = Nrd(ξ) is a polynomial in x1 , . . . , xd , we have in fact

π(ξ) ∈ Z(A, σ) — F [x1 , . . . , xd ].

We may then de¬ne π(a) for all a ∈ Skew(A, σ) by specializing the indeterminates.

We call the map π : Skew(A, σ) ’ Z(A, σ) thus de¬ned the generalized pfa¬an of

(A, σ) in view of Example (??) below.

(8.24) Proposition. The map π : Skew(A, σ) ’ Z(A, σ) is a homogeneous poly-

nomial map of degree m. Denoting by ι the nontrivial automorphism of Z(A, σ)

over F , we have

π(a)2 = (’1)m NrdA (a)

ι π(a) = ’π(a) and

for all a ∈ Skew(A, σ). Moreover, for all x ∈ A, a ∈ Skew(A, σ),

π xaσ(x) = NrdA (x)π(a).

Proof : For the generic element ξ we have by (??)

and π(ξ)2 = (’1)m Nrd(ξ).

ι π(ξ) = ’π(ξ)

The same formulas follow for all a ∈ Skew(A, σ) by specialization. Since the reduced

norm is a homogeneous polynomial map of degree n, the second formula shows that

π is a homogeneous polynomial map of degree m. It also shows that an element

a ∈ Skew(A, σ) is invertible if and only if π(a) = 0.

In order to prove the last property, ¬x some element a ∈ Skew(A, σ). If a is

not invertible, then π(a) = π xaσ(x) = 0 for all x ∈ A and the property is clear.

Suppose a ∈ A— . Since the F -vector space of elements z ∈ Z(A, σ) such that

ι(z) = ’z has dimension 1, we have for all x ∈ A

π xaσ(x) = P (x)π(a) for some P (x) ∈ F .

The map P : A ’ F is polynomial and satis¬es

2

π xaσ(x) Nrd xaσ(x)

2

= Nrd(x)2 .

P (x) = =

π(a)2 Nrd(a)

By adjoining indeterminates to F if necessary, we may assume F is in¬nite. The

algebra of polynomial maps on A is then a domain, hence the preceding equation

yields the alternative: P (x) = Nrd(x) for all x or P (x) = ’ Nrd(x) for all x. Since

P (1) = 1, we have P (x) = Nrd(x) for all x ∈ A.

Using the map π, we may give an alternate proof of (??) and of part of (??):

(8.25) Corollary. For all a, b ∈ Skew(A, σ) © A— ,

NrdA (a) ≡ NrdA (b) mod F —2

F [X]/ X 2 ’ (’1)m NrdA (a) .

and Z(A, σ)

104 II. INVARIANTS OF INVOLUTIONS

Proof : Since the F -vector space of elements z ∈ Z(A, σ) such that ι(z) = ’z is 1-

dimensional, we have π(a) ≡ π(b) mod F — . By squaring both sides we obtain the

¬rst equation. The second equation follows from the fact that Z(A, σ) = F π(a) .

(8.26) Example. In the case where A is split, the map π can be described ex-

plicitly in terms of the pfa¬an. Let (V, q) be a nonsingular quadratic space of

dimension n = 2m over F . For (A, σ) = EndF (V ), σq , we identify A = V — V as

in (??) and C(A, σ) = C0 (V, q) as in (??). Let (e1 , . . . , en ) be an orthogonal basis

1

of V . The elements 2 (ei — ej ’ ej — ei ) for 1 ¤ i < j ¤ n form a basis of Skew(A, σ).

aij

For a = i<j 2 (ei — ej ’ ej — ei ) ∈ Skew(A, σ), de¬ne a skew-symmetric matrix

a = (aij ) ∈ Mn (F ) by

±

aij if i < j,

aij = 0 if i = j

’aji if i > j.

Claim. The element π(a) ∈ Z(A, σ) is related to the pfa¬an pf(a ) as follows:

π(a) = pf(a )e1 · · · en .

Proof : A computation shows that the matrix representing a with respect to the

basis (e1 , . . . , en ) is a · d where

«

q(e1 ) 0

¬ ·

..

d=

.

0 q(en )

Since det a = pf(a )2 , it follows that

2

det a = pf(a )2 q(e1 ) · · · q(en ) = (’1)m pf(a )e1 · · · en .

Therefore, π(a) = ± pf(a )e1 · · · en , since both sides are polynomial maps of de-

gree m whose squares are equal. To prove that the equality holds with the + sign,

it su¬ces to evaluate both sides on a particular unit in Skew(A, σ). Adjoin inde-

terminates z1 , . . . , zm to F and consider

m

zi

ζ= (e2i’1 — e2i ’ e2i — e2i’1 ) ∈ Skew AF (z1 ,...,zm ) , σF (z1 ,...,zm ) .

2

i=1

The same computation as in (??) shows that

π(ζ) = z1 · · · zm e1 · · · en .

m1

By setting z1 = · · · = zm = 1, we get for a = i=1 2 (e2i’1 — e2i ’ e2i — e2i’1 ):

π(a) = e1 · · · en .

On the other hand, the corresponding matrix a is

«

J 0

01

¬ ·

..

a = where J = ,

. ’1 0

0 J

hence pf(a ) = 1.

§8. THE CLIFFORD ALGEBRA 105

Characteristic 2. Let (σ, f ) be a quadratic pair on a central simple algebra A

over a ¬eld F of characteristic 2. We write Z(A, σ, f ) for the center of the Cli¬ord

algebra C(A, σ, f ) and by ι the nontrivial automorphism of Z(A, σ, f ) over F .

Consider the set

Λ = { ∈ A | f (s) = TrdA ( s) for s ∈ Sym(A, σ) }.

By (??), this set is nonempty; it is a coset of Alt(A, σ).

The following proposition shows that the canonical map c : A ’ C(A, σ, f ),

restricted to Λ, plays a rˆle analogous to the map π in characteristic di¬erent

o

from 2.

(8.27) Proposition. Let deg A = n = 2m. For all ∈ Λ, we have c( ) ∈

Z(A, σ, f ), ι c( ) = c( ) + 1 and

m(m’1)

c( )2 + c( ) = SrdA ( ) + .

2

Proof : It su¬ces to check the equations above after a scalar extension. We may

therefore assume that A is split. Moreover, it su¬ces to consider a particular choice

of ; indeed, if 0 , ∈ Λ, then = 0 + x + σ(x) for some x ∈ A, hence

c( ) = c( 0 ) + c x + σ(x) .

Since c and f have the same restriction to Sym(A, σ), the last term on the right

side is f x + σ(x) = TrdA (x), hence

c( ) = c( 0 ) + TrdA (x).

Therefore, we have c( ) ∈ Z(A, σ, f ) and ι c( ) = c( ) + 1 if and only if c( 0 ) satis-

¬es the same conditions. Moreover, (??) yields SrdA ( ) = SrdA ( 0 ) + „˜ TrdA (x) ,

hence „˜ c( ) = SrdA ( ) + m(m’1) if and only if the same equation holds for 0 .

2

We may thus assume A = EndF (V ) and (σ, f ) = (σq , fq ) for some nonsingular

quadratic space (V, q), and consider only the case of

m

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

0 i=1

where (e1 , . . . , en ) is a symplectic basis of V for the polar form bq :

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

Using the standard identi¬cation C(A, σ, f ) = C0 (V, q) of (??), we then have

m m

e2 q(e2i ) + e2 q(e2i’1 ) + e2i’1 · e2i =

c( 0 ) = e2i’1 · e2i

2i’1 2i

i=1 i=1

and the required equations follow by computation (compare with (??)).

(8.28) Corollary. For any ∈ Λ,

m(m’1)

F [X]/ X 2 + X + SrdA ( ) +

Z(A, σ, f ) .

2

Proof : The proposition above shows that Z(A, σ, f ) = F c( ) and c( )2 + c( ) +

SrdA ( ) + m(m’1) = 0.

2

106 II. INVARIANTS OF INVOLUTIONS

8.E. The Cli¬ord algebra of a hyperbolic quadratic pair. Let (σ, f ) be

a hyperbolic quadratic pair on a central simple algebra A over an arbitrary ¬eld F

and let deg A = n = 2m. By (??), the discriminant of (σ, f ) is trivial, hence (??)

shows that the Cli¬ord algebra C(A, σ, f ) decomposes as a direct product of two

central simple algebras of degree 2m’1 :

C(A, σ, f ) = C + (A, σ, f ) — C ’ (A, σ, f ).

Our aim is to show that one of the factors C± (A, σ, f ) is split if m is even.

We start with some observations on isotropic ideals in a central simple algebra

with an arbitrary quadratic pair: suppose I ‚ A is a right ideal of even reduced

dimension rdim I = r = 2s with respect to a quadratic pair (σ, f ). Consider the

image c Iσ(I) ‚ C(A, σ, f ) of Iσ(I) under the canonical map c : A ’ C(A, σ, f )

and let

s

ρ(I) = c Iσ(I) = x1 · · · x s x1 , . . . , xs ∈ c Iσ(I) .

(8.29) Lemma. The elements in c Iσ(I) commute. The F -vector space ρ(I) ‚

C(A, σ, f ) is 1-dimensional; it satis¬es σ ρ(I) · ρ(I) = {0} and

dimF ρ(I) · C(A, σ, f ) = dimF C(A, σ, f ) · ρ(I) = 2n’r’1 .

Proof : It su¬ces to check the lemma over a scalar extension. We may therefore

assume that A is split and identify A = EndF (V ), C(A, σ, f ) = C0 (V, q) for some

nonsingular quadratic space (V, q) of dimension n, by (??). The ideal I then has the

form I = HomF (V, U ) for some m-dimensional totally isotropic subspace U ‚ V .

Let (u1 , . . . , ur ) be a basis of U . Under the identi¬cation A = V — V described

in (??), the vector space Iσ(I) is spanned by the elements ui — uj for i, j = 1,

. . . , r, hence c Iσ(I) is spanned by the elements ui · uj in C0 (V, q). Since U is

totally isotropic, we have u2 = 0 and ui · uj + uj · ui = 0 for all i, j = 1, . . . , r, hence

i

the elements ui · uj commute. Moreover, the space ρ(I) is spanned by u1 · · · ur .

The dimensions of ρ(I) · C0 (V, q) and C0 (V, q) · ρ(I) are then easily computed, and

since σ(u1 · · · ur ) = ur · · · u1 we have σ(x)x = 0 for all x ∈ ρ(I).

(8.30) Corollary. Let deg A = n = 2m and suppose the center Z = Z(A, σ, f ) of

C(A, σ, f ) is a ¬eld. For any even integer r, the relation r ∈ ind(A, σ, f ) implies

2m’r’1 ∈ ind C(A, σ, f ), σ .

Proof : If r is an even integer in ind(A, σ, f ), then A contains an isotropic right

ideal I of even reduced dimension r. The lemma shows that ρ(I) · C(A, σ, f ) is an

isotropic ideal for the involution σ. Its reduced dimension is

1

dimF ρ(I) · C(A, σ, f )

dimZ ρ(I) · C(A, σ, f )

= 2m’r’1 .

2

=

2m’1

deg C(A, σ, f )

We next turn to the case of hyperbolic quadratic pairs:

(8.31) Proposition. Let (σ, f ) be a hyperbolic quadratic pair on a central simple

algebra A of degree 2m over an arbitrary ¬eld F . If m is even, then one of the

factors C± (A, σ, f ) of the Cli¬ord algebra C(A, σ, f ) is split.

§9. THE CLIFFORD BIMODULE 107

Proof : Let I ‚ A be a right ideal of reduced dimension m which is isotropic with

respect to (σ, f ), and consider the 1-dimensional vector space ρ(I) ‚ C(A, σ, f )

de¬ned above. Multiplication on the left de¬nes an F -algebra homomorphism

» : C(A, σ, f ) ’ EndF C(A, σ, f ) · ρ(I) .

Dimension count shows that this homomorphism is not injective, hence the kernel

is one of the nontrivial ideals C + (A, σ, f ) — {0} or {0} — C ’ (A, σ, f ). Assuming

for instance ker » = C + (A, σ, f ) — {0}, the homomorphism » factors through an

injective F -algebra homomorphism C ’ (A, σ, f ) ’ EndF C(A, σ, f ) · ρ(I) . This

homomorphism is surjective by dimension count.

§9. The Cli¬ord Bimodule

Although the odd part C1 (V, q) of the Cli¬ord algebra of a quadratic space (V, q)

is not invariant under similarities, it turns out that the tensor product V — C1 (V, q)

is invariant, and therefore an analogue can be de¬ned for a central simple algebra

with quadratic pair (A, σ, f ). The aim of this section is to de¬ne such an analogue.

This construction will be used at the end of this section to obtain fundamental

relations between the Cli¬ord algebra C(A, σ, f ) and the algebra A (see (??)); it

will also be an indispensable tool in the de¬nition of spin groups in the next chapter.

We ¬rst review the basic properties of the vector space V — C1 (V, q) that we

want to generalize.

9.A. The split case. Let (V, q) be a quadratic space over a ¬eld F (of arbi-

trary characteristic). Let C1 (V, q) be the odd part of the Cli¬ord algebra C(V, q).

Multiplication in C(V, q) endows C1 (V, q) with a C0 (V, q)-bimodule structure. Since

V is in a natural way a left End(V )-module, the tensor product V — C1 (V, q) is

at the same time a left End(V )-module and a C0 (V, q)-bimodule: for f ∈ End(V ),

v ∈ V , c0 ∈ C0 (V, q) and c1 ∈ C1 (V, q) we set

f · (v — c1 ) = f (v) — c1 , c0 — (v — c1 ) = v — c0 c1 , (v — c1 ) · c0 = v — c1 c0 .

These various actions clearly commute.

We summarize the basic properties of V — C1 (V, q) in the following proposition:

(9.1) Proposition. Let dim V = n.

(1) The vector space V — C1 (V, q) carries natural structures of left End(V )-module

and C0 (V, q)-bimodule, and the various actions commute.

(2) The standard identi¬cation End(V ) = V — V induced by the quadratic form q

(see (??)) and the embedding V ’ C1 (V, q) de¬ne a canonical map

b : End(V ) ’ V — C1 (V, q)

which is an injective homomorphism of left End(V )-modules.

(3) dimF V — C1 (V, q) = 2n’1 n.

The proof follows by straightforward veri¬cation.