∼

(2) The bijection AutX (X) ’ AutY S(X) induced by an equivalence of cat-

’

egories is a group isomorphism if the same convention is used in X and Y for

mapping composition. It is an anti-isomorphism if opposite conventions are used

in X and Y . By contrast, the bijection induced by an anti-equivalence of categories

is an anti-isomorphism if the same convention is used in X and Y , and it is an

isomorphism if opposite conventions are used.

For the rest of this section F is a ¬eld of characteristic di¬erent from 2. For

any integer n ≥ 1, let Bn denote the groupoid of central simple F -algebras of

17 For all the categories we consider in the sequel, this class is a set.

§12. GENERAL PROPERTIES 171

degree 2n + 1 with involution of the ¬rst kind,18 where the morphisms are the F -

algebra isomorphisms which preserve the involutions. Note that these algebras are

necessarily split, and the involution is necessarily of orthogonal type, by (??).

For any integer n ≥ 1, let Qn be the groupoid of all nonsingular quadratic

spaces of dimension n over the ¬eld F , where the morphisms are the isometries,

1

and let Qn be the full subcategory of quadratic spaces with trivial discriminant. For

(V, q) ∈ Qn , let σq denote the adjoint involution on EndF (V ) with respect to (the

polar of) q. If (V, q) ∈ Q2n+1 , then EndF (V ), σq ∈ Bn , and we have a functor

End : Q2n+1 ’ Bn

given by mapping (V, q) to EndF (V ), σq , as observed in (??).

(12.39) Proposition. The functor End de¬nes a bijection between the sets of

isomorphism classes:

∼

1

End : Isom(Q2n+1 ) ’ Isom(Bn ).

’

Proof : By (??), every algebra with involution in Bn is isomorphic to an algebra

with involution of the form EndF (V ), σq for some quadratic space (V, q) of dimen-

sion 2n+1. Since the adjoint involution does not change when the quadratic form is

multiplied by a scalar, we may substitute (disc q)q for q and thus assume disc q = 1.

Therefore, the map induced by End on isomorphism classes is surjective.

On the other hand, suppose

¦ : EndF (V ), σq ’ EndF (V ), σq

1

is an isomorphism, for some quadratic spaces (V, q), (V , q ) ∈ Q2n+1 . By (??), we

may ¬nd a similitude g : (V, q) ’ (V , q ) such that ¦ = g— . This similitude may be

∼

regarded as an isometry (V, ±q) ’ (V , q ) , where ± is the multiplier of g. Since

’

disc q = disc q and dim V = dim V is odd, we must have ± = 1, hence g is an

∼

isometry (V, q) ’ (V , q ).

’

Even though it de¬nes a bijection between the sets of isomorphism classes, the

1

functor End is not an equivalence between Q2n+1 and Bn : this is because the group

of automorphisms of the algebra with involution EndF (V ), σq is

AutF EndF (V ), σq = PGO(V, q) = O+ (V, q)

(the second equality follows from (??)), whereas the group of automorphisms of

(V, q) is O(V, q). However, we may de¬ne some additional structure on quadratic

spaces to restrict the automorphism group and thereby obtain an equivalence of

categories.

(12.40) De¬nition. Let (V, q) be a quadratic space of odd dimension and trivial

discriminant over a ¬eld F of characteristic di¬erent from 2. The center Z of the

Cli¬ord algebra C(V, q) is then an ´tale quadratic extension of F isomorphic to

e

F — F . An orientation of (V, q) is an element ζ ∈ Z F such that ζ 2 = 1. Thus,

each quadratic space (V, q) as above has two possible orientations which di¬er by

a sign. Triples (V, q, ζ) are called oriented quadratic spaces.

18 This notation is motivated by the fact that the automorphism group of each object in this

groupoid is a classical group of type Bn : see Chapter ??. However this groupoid is only de¬ned

for ¬elds of characteristic di¬erent from 2.

172 III. SIMILITUDES

∼

Every isometry g : (V, q) ’ (V , q ) induces an isomorphism g— : C(V, q) ’ ’

C(V , q ) which carries an orientation of (V, q) to an orientation of (V , q ). The

isometries g : (V, q) ’ (V, q) which preserve a given orientation form the group

O+ (V, q).

Let Bn be the groupoid of oriented quadratic spaces of dimension 2n+1 over F .

The objects of Bn are triples (V, q, ζ) where (V, q) is a quadratic space of dimen-

sion 2n + 1 over F with trivial discriminant and ζ is an orientation of (V, q), and

the morphisms are the orientation-preserving isometries. For each (V, q, ζ) ∈ Bn ,

the map ’IdV : V ’ V de¬nes an isomorphism (V, q, ζ) ’ (V, q, ’ζ), hence two

oriented quadratic spaces are isomorphic if and only if the quadratic spaces are iso-

metric. In other words, the functor which forgets the orientation de¬nes a bijection

∼ 1

Isom(Bn ) ’ Isom(Q2n+1 ).

’

(12.41) Theorem. The functor End which maps every oriented quadratic space

(V, q, ζ) in Bn to the algebra with involution EndF (V ), σq ∈ Bn de¬nes an equiv-

alence of categories:

Bn ≡ B n .

1

Proof : Since the isomorphism classes of Bn and Q2n+1 coincide, (??) shows that the

∼

functor End de¬nes a bijection Isom(Bn ) ’ Isom(Bn ). Moreover, as we observed

’

above, for every oriented quadratic space (V, q, ζ) of dimension 2n + 1 we have

Aut EndF (V ), σq = O+ (V, q) = Aut(V, q, ζ).

(12.42)

Therefore, (??) shows that End is an equivalence of categories.

§13. Quadratic Pairs

In this section, (σ, f ) is a quadratic pair on a central simple algebra A over

an arbitrary ¬eld F . If the degree of A is odd, then A is split, char F = 2, and

the group of similitudes of (A, σ, f ) reduces to the orthogonal group of an F -vector

space (see (??)). We therefore assume throughout this section that the degree is

even, and we set

deg A = n = 2m.

Our goal is to obtain additional information on the group GO(A, σ, f ) by relating

similitudes of (A, σ, f ) to the Cli¬ord algebra C(A, σ, f ) and the Cli¬ord bimodule

B(A, σ, f ). We use this to de¬ne a Cli¬ord group “(A, σ, f ), which is a twisted ana-

logue of the special Cli¬ord group of a quadratic space, and also de¬ne an extended

Cli¬ord group „¦(A, σ, f ). These constructions are used to prove an analogue of a

classical theorem of Dieudonn´ on the multipliers of similitudes.

e

13.A. Relation with the Cli¬ord structures. Since the Cli¬ord alge-

bra C(A, σ, f ) and the Cli¬ord bimodule B(A, σ, f ) are canonically associated to

(A, σ, f ), every automorphism in AutF (A, σ, f ) induces automorphisms of C(A, σ, f )

and B(A, σ, f ). Our purpose in this section is to investigate these automorphisms.

The Cli¬ord algebra. Every automorphism θ ∈ Aut F (A, σ, f ) induces an

automorphism

C(θ) ∈ AutF C(A, σ), σ .

Explicitly, C(θ) can be de¬ned as the unique automorphism of C(A, σ, f ) such that

C(θ) c(a) = c θ(a) for a ∈ A,

§13. QUADRATIC PAIRS 173

where c : A ’ C(A, σ, f ) is the canonical map (??). We thereby obtain a canonical

group homomorphism

C : AutF (A, σ, f ) ’ AutF C(A, σ, f ), σ .

Slightly abusing notation, we also call C the homomorphism

C : GO(A, σ, f ) ’ AutF C(A, σ, f ), σ

obtained by composing the preceding map with the epimorphism

Int : GO(A, σ, f ) ’ AutF (A, σ, f )

of (??). Thus, for g ∈ GO(A, σ, f ) and a ∈ A,

C(g) c(a) = c(gag ’1 ).

(13.1) Proposition. Suppose A is split; let (A, σ, f ) = EndF (V ), σq , fq for

some nonsingular quadratic space (V, q). Then, under the standard identi¬cations

GO(A, σ, f ) = GO(V, q) (see (??)) and C(A, σ, f ) = C0 (V, q) (see (??)), the canon-

ical map C : GO(V, q) ’ AutF C0 (V, q) is de¬ned by

C(g)(v1 · · · v2r ) = µ(g)’r g(v1 ) · · · g(v2r )

for g ∈ GO(V, q) and v1 , . . . , v2r ∈ V .

Proof : It su¬ces to check the formula above on generators v · w of C0 (V, q). For

v, w ∈ V , the product v · w in C(V, q) is the image of v — w under the canonical

map c: we thus have v · w = c(v — w), hence

C(g)(v · w) = c g —¦ (v — w) —¦ g ’1 .

Let ± = µ(g) be the multiplier of g; then σ(g)’1 = ±’1 g, hence, for x ∈ V ,

g —¦ (v — w) —¦ g ’1 (x) = g(v)bq w, g ’1 (x) = g(v)bq ±’1 g(w), x .

Therefore, g —¦ (v — w) —¦ g ’1 (x) = ±’1 g(v) — g(w) (x), which shows

g —¦ (v — w) —¦ g ’1 = ±’1 g(v) — g(w),

hence c g —¦ (v — w) —¦ g ’1 = ±’1 g(v) · g(w).

Note that, for g ∈ GO(A, σ, f ), the automorphism C(g) of C(A, σ, f ) is F -linear

but is not necessarily the identity on the center of C(A, σ, f ). The behavior of C(g)

on the center in fact determines whether g is proper, as the next proposition shows.

(13.2) Proposition. A similitude g ∈ GO(A, σ, f ) is proper if and only if C(g)

restricts to the identity map on the center Z of C(A, σ, f ).

Proof : Suppose ¬rst that char F = 2. Choose ∈ A satisfying f (s) = TrdA ( s) for

all s ∈ Sym(A, σ) (see (??)). By (??), we have Z = F c( ) , hence it su¬ces to

show

C(g) c( ) = c( ) + ∆(g) for g ∈ GO(A, σ, f ).

For g ∈ GO(A, σ, f ) we have

∆(g) = f (g ’1 g ’ );

since g is a similitude, the right side also equals

f g(g ’1 g ’ )g ’1 = f (g g ’1 ’ ).

174 III. SIMILITUDES

On the other hand, since g g ’1 ’ ∈ Sym(A, σ), we have

f (g g ’1 ’ ) = c(g g ’1 ’ ),

hence

∆(g) = c(g g ’1 ) ’ c( ) = C(g) c( ) ’ c( ),

proving the proposition when char F = 2.

Suppose now that char F = 2. It su¬ces to check the split case; we may

thus assume (A, σ, f ) = EndF (V ), σq , fq for some nonsingular quadratic space

(V, q), and use the standard identi¬cations and the preceding proposition. Let

(e1 , . . . , e2m ) be an orthogonal basis of (V, q). Recall that e1 · · · e2m ∈ Z F . For

g ∈ GO(A, σ, f ) = GO(V, q), we have

C(g)(e1 · · · e2m ) = µ(g)’m g(e1 ) · · · g(e2m ).

On the other hand, a calculation in the Cli¬ord algebra shows that

g(e1 ) · · · g(e2m ) = det(g)e1 · · · e2m ;

hence e1 · · · e2m is ¬xed by C(g) if and only if det(g) = µ(g)m . This proves the

proposition in the case where char F = 2.

In view of this proposition, the Dickson invariant ∆ : GO(A, σ, f ) ’ Z/2Z

de¬ned in (??) may alternately be de¬ned by

0 if C(g)|Z = IdZ ,

(13.3) ∆(g) =

1 if C(g)|Z = IdZ .

The image of the canonical map C has been determined by Wonenburger in

characteristic di¬erent from 2:

(13.4) Proposition. If deg A > 2, the canonical homomorphism

C : PGO(A, σ, f ) = AutF (A, σ, f ) ’ AutF C(A, σ, f ), σ

is injective. If char F = 2, the image of C is the group of those automorphisms

which preserve the image c(A) of A under the canonical map c : A ’ C(A, σ, f ).

Proof : If θ ∈ AutF (A, σ) lies in the kernel of C, then

c θ(a) = c(a) for a ∈ A,

since the left side is the image of c(a) under C(θ). By applying the map δ of (??),

we obtain

θ a ’ σ(a) = a ’ σ(a) for a ∈ A,

hence θ is the identity on Alt(A, σ). Since deg A > 2, (??) shows that Alt(A, σ)

generates A, hence θ = IdA , proving the injectivity of C.

It follows from the de¬nition that every automorphism of the form C(θ) maps

c(A) to itself. Conversely, suppose ψ is an automorphism of C(A, σ, f ) which pre-

serves c(A), and suppose char F = 2. The map f is then uniquely determined by

σ, so we may denote C(A, σ, f ) simply by C(A, σ). The restriction of ψ to

c(A)0 = c(A) © Skew C(A, σ), σ = { x ∈ c(A) | Trd(x) = 0 }

§13. QUADRATIC PAIRS 175

is a Lie algebra automorphism. By (??), the Lie algebra c(A)0 is isomorphic to

Alt(A, σ) via δ, with inverse isomorphism 1 c; therefore, there is a corresponding

2

Lie automorphism ψ of Alt(A, σ) such that

c ψ (a) = ψ c(a) for a ∈ Alt(A, σ).

Let L be a splitting ¬eld of A. A theorem19 of Wonenburger [?, Theorem 4] shows

that the automorphism ψL = ψ — IdL of C(A, σ)L = C(AL , σL ) is induced by a

similitude g of (AL , σL ), hence

ψL (a) = gag ’1 for a ∈ Alt(AL , σL ).

Therefore, the automorphism ψL of Alt(AL , σL ) extends to an automorphism of

(AL , σL ). By (??), ψ extends to an automorphism θ of (A, σ), and this au-

tomorphism satis¬es C(θ) = ψ since c θ(a) = c ψ (a) = ψ c(a) for all a ∈

Alt(A, σ).

If deg A = 2, then C(A, σ) = Z, so AutF C(A, σ), σ = {Id, ι} and the canon-

ical homomorphism C maps PGO+ (A, σ) to Id, so C is not injective.

The Cli¬ord bimodule. The bimodule B(A, σ, f ) is canonically associated to

(A, σ, f ), just as the Cli¬ord algebra C(A, σ, f ) is. Therefore, every automorphism

θ ∈ AutF (A, σ, f ) induces a bijective linear map

B(θ) : B(A, σ, f ) ’ B(A, σ, f ).

This map satis¬es

B(θ)(ab ) = θ(a)b for a ∈ A

(where b : A ’ B(A, σ, f ) is the canonical map of (??)) and

(13.5) B(θ) a · (c1 — x · c2 ) = θ(a) · C(θ)(c1 ) — B(θ)(x) · C(θ)(c2 )

for a ∈ A, c1 , c2 ∈ C(A, σ, f ) and x ∈ B(A, σ, f ). Explicitly, B(θ) is induced by

the map θ : T + (A) ’ T + (A) such that

θ(a1 — · · · — ar ) = θ(a1 ) — · · · — θ(ar ).

As in the previous case, we modify the domain of de¬nition of B to be the group

GO(A, σ, f ), by letting B(g) = B Int(g) for g ∈ GO(A, σ, f ). We thus obtain a

canonical homomorphism

B : GO(A, σ, f ) ’ GLF B(A, σ, f ).

For g ∈ GO(A, σ, f ), we also de¬ne a map

(13.6) β(g) : B(A, σ, f ) ’ B(A, σ, f )

by

xβ(g) = g · B(g ’1 )(x) for x ∈ B(A, σ, f ).

The map β(g) is a homomorphism of left A-modules, since for a ∈ A and x ∈

B(A, σ, f ),

(a · x)β(g) = g · (g ’1 ag) · B(g ’1 )(x) = a · xβ(g) .

Moreover, the following equation is a straightforward consequence of the de¬nitions:

(1b )β(g) = g b .

(13.7)

19 This theorem is proved under the assumption that char F = 2. See Exercise ?? for a sketch

of proof.

176 III. SIMILITUDES

Since b is injective, it follows that the map

β : GO(A, σ, f ) ’ AutA B(A, σ, f )

is injective. This map also is a homomorphism of groups; to check this, we compute,

for g, h ∈ GO(A, σ, f ) and x ∈ B(A, σ, f ):

β(h)

xβ(g)—¦β(h) = g · B(g ’1 )(x) .

Since β(h) is a homomorphism of left A-modules, the right-hand expression equals

β(h)

g · B(g ’1 )(x) = g · h · B(h’1 ) —¦ B(g ’1 )(x)

= gh · B(h’1 g ’1 )(x)

= xβ(gh) ,

proving the claim.

Let Z be the center of C(A, σ, f ). Recall the right ι C(A, σ, f )op —Z C(A, σ, f )-

module structure on B(A, σ, f ), which yields the canonical map

ν : ι C(A, σ, f ) —Z C(A, σ, f ) ’ EndA—Z B(A, σ, f )

of (??). It follows from (??) that the following equation holds in EndA B(A, σ, f ):

β(g) —¦ ν(ιcop — c2 ) = ν ι C(g)(c1 )op — C(g)(c2 ) —¦ β(g)

(13.8) 1

for all g ∈ GO(A, σ, f ), c1 , c2 ∈ C(A, σ, f ). In particular, it follows by (??) that

β(g) is Z-linear if and only if g is proper.

The following result describes the maps B(g) and β(g) in the split case; it

follows by the same arguments as in (??).

(13.9) Proposition. Suppose A is split; let

(A, σ, f ) = EndF (V ), σq , fq

for some nonsingular quadratic space (V, q). Under the standard identi¬cations

GO(A, σ, f ) = GO(V, q) and B(A, σ, f ) = V — C1 (V, q) (see (??)), the maps B

and β are given by

B(g)(v — w1 · · · w2r’1 ) = µ(g)’r g(v) — g(w1 ) · · · g(w2r’1 )

and

(v — w1 · · · w2r’1 )β(g) = µ(g)r v — g ’1 (w1 ) · · · g ’1 (w2r’1 )

for g ∈ GO(A, σ, f ) and v, w1 , . . . , w2r’1 ∈ V .

13.B. Cli¬ord groups. For a nonsingular quadratic space (V, q) over an ar-

bitrary ¬eld F , the special Cli¬ord group “+ (V, q) is de¬ned by

“+ (V, q) = { c ∈ C0 (V, q)— | c · V · c’1 ‚ V }

where the product c · V · c’1 is computed in the Cli¬ord algebra C(V, q) (see for

instance Knus [?, Ch. 4, §6], or Scharlau [?, §9.3] for the case where char F = 2).

Although there is no analogue of the (full) Cli¬ord algebra for an algebra with

quadratic pair, we show in this section that the Cli¬ord bimodule may be used to

de¬ne an analogue of the special Cli¬ord group. We also show that an extended

Cli¬ord group can be de¬ned by substituting the Cli¬ord algebra for the Cli¬ord

bimodule. These constructions are used to de¬ne spinor norm homomorphisms on

the groups O+ (A, σ, f ) and PGO+ (A, σ, f ).

§13. QUADRATIC PAIRS 177

The special Cli¬ord group in the split case. Let (V, q) be a nonsingular

quadratic space of even20 dimension over an arbitrary ¬eld F , and let “+ (V, q) be

the special Cli¬ord group de¬ned above. Conjugation by c ∈ “+ (V, q) in C(V, q)

induces an isometry of (V, q), since

q(c · v · c’1 ) = (c · v · c’1 )2 = v 2 = q(v) for v ∈ V .

We set χ(c) for this isometry:

χ(c)(v) = c · v · c’1 for v ∈ V .

The map χ : “+ (V, q) ’ O(V, q) is known as the vector representation of the special

Cli¬ord group. The next proposition shows that its image is in O+ (V, q).

(13.10) Proposition. The elements in “+ (V, q) are similitudes of the even Clif-

ford algebra C0 = C0 (V, q) for the canonical involution „0 (see (??)). More pre-

cisely, „0 (c) · c ∈ F — for all c ∈ “+ (V, q). The vector representation χ and the

canonical homomorphism C of (??) ¬t into the following commutative diagram

with exact rows:

χ

O+ (V, q)

1 ’ ’ ’ F— ’ ’ ’ “+ (V, q)

’’ ’’ ’’’

’’ ’’’ 1

’’

¦ ¦ ¦

¦ ¦ ¦

C

Int

1 ’ ’ ’ Z — ’ ’ ’ Sim(C0 , „0 ) ’ ’ ’ AutZ (C0 , „0 ) ’ ’ ’ 1

’’ ’’ ’’ ’’

where Z denotes the center of C0 .

Proof : Let „ be the canonical involution on C(V, q), whose restriction to (the image

of) V is the identity. For c ∈ “+ (V, q) and v ∈ V , we have c · v · c’1 ∈ V , hence

c · v · c’1 = „ (c · v · c’1 ) = „0 (c)’1 · v · „0 (c).

This shows that the element „0 (c) · c centralizes V ; since V generates C(V, q),

it follows that „0 (c) · c is central in C(V, q), hence „0 (c) · c ∈ F — . This proves

“+ (V, q) ‚ Sim(C0 , „0 ).

The elements in ker χ centralize V , hence the same argument as above shows

ker χ = F — .

Let c ∈ “+ (V, q). By (??), the automorphism C χ(c) of C0 maps v1 · · · v2r to

χ(c)(v1 ) · · · χ(c)(v2r ) = c · (v1 · · · v2r ) · c’1

for v1 , . . . , v2r ∈ V , hence

C χ(c) = Int(c).

This automorphism is the identity on Z, hence (??) shows that χ(c) ∈ O+ (V, q).

Moreover, the last equation proves that the diagram commutes. Therefore, it re-

mains only to prove surjectivity of χ onto O+ (V, q).

To prove that every proper isometry is in the image of χ, observe that for every

v, x ∈ V with q(v) = 0,

v · x · v ’1 = v ’1 bq (v, x) ’ x = vq(v)’1 bq (v, x) ’ x,

hence the hyperplane re¬‚ection ρv : V ’ V satis¬es

ρv (x) = ’v · x · v ’1 for all x ∈ V .

20 Cli¬ord groups are also de¬ned in the odd-dimensional case, where results similar to those

of this section can be established. Since we are interested in the generalization to the nonsplit

case, given below, we consider only the even-dimensional case.

178 III. SIMILITUDES

Therefore, for anisotropic v1 , v2 ∈ V , we have v1 · v2 ∈ “+ (V, q) and

χ(v1 · v2 ) = ρv1 —¦ ρv2 .

The Cartan-Dieudonn´ theorem (see Dieudonn´ [?, pp. 20, 42], or Scharlau [?,

e

e

p. 15] for the case where char F = 2) shows that the group O(V, q) is generated by

hyperplane re¬‚ections, except in the case where F is the ¬eld with two elements,

dim V = 4 and q is hyperbolic. Since hyperplane re¬‚ections are improper isometries

(see (??)), it follows that every proper isometry has the form

ρv1 —¦ · · · —¦ ρv2r = χ(v1 · · · v2r )

for some anisotropic vectors v1 , . . . , v2r ∈ V , in the nonexceptional case.

Direct computations, which we omit, prove that χ is surjective in the excep-

tional case as well.

The proof shows that every element in “+ (V, q) is a product of an even number

of anisotropic vectors in V , except when (V, q) is the 4-dimensional hyperbolic space

over the ¬eld with two elements.

The Cli¬ord group of an algebra with quadratic pair. Let (σ, f ) be a

quadratic pair on a central simple algebra A of even degree over an arbitrary ¬eld F .

The Cli¬ord group consists of elements in C(A, σ, f ) which preserve the image Ab

of A in the bimodule B(A, σ, f ) under the canonical map b : A ’ B(A, σ, f ) of (??):

(13.11) De¬nition. The Cli¬ord group “(A, σ, f ) is de¬ned by

“(A, σ, f ) = { c ∈ C(A, σ, f )— | c’1 — Ab · c ‚ Ab }.

Since the C(A, σ, f )-bimodule actions on B(A, σ, f ) commute with the left A-

module action and since the canonical map b is a homomorphism of left A-modules,

the condition de¬ning the Cli¬ord group is equivalent to

c’1 — 1b · c ∈ Ab .

For c ∈ “(A, σ, f ), de¬ne χ(c) ∈ A by the equation

c’1 — 1b · c = χ(c)b .

(The element χ(c) is uniquely determined by this equation, since the canonical

map b is injective: see (??)).

(13.12) Proposition. In the split case (A, σ, f ) = EndF (V ), σq , fq , the standard

identi¬cations C(A, σ, f ) = C0 (V, q), B(A, σ, f ) = V — C1 (V, q) of (??), (??),

induce an identi¬cation “(A, σ, f ) = “+ (V, q), and the map χ de¬ned above is the

vector representation.

Proof : Under the standard identi¬cations, we have A = V — V and Ab = V — V ‚

V — C1 (V, q). Moreover, for c ∈ C(A, σ, f ) = C0 (V, q) and v, w ∈ V ,

c’1 — (v — w)b · c = v — (c’1 · w · c).

(13.13)

Therefore, the condition c’1 — Ab · c ‚ Ab amounts to:

v — (c’1 · w · c) ∈ V — V for v, w ∈ V ,

or c’1 · V · c ‚ V . This proves the ¬rst assertion.

Suppose now that c’1 —1b ·c = g b . Since b is a homomorphism of left A-modules,

we then get for all v, w ∈ V :

b

c’1 — (v — w)b · c = (v — w) · (c’1 — 1b · c) = (v — w) —¦ g .

(13.14)

§13. QUADRATIC PAIRS 179

By evaluating (v — w) —¦ g at an arbitrary x ∈ V , we obtain

vbq w, g(x) = vbq σ(g)(w), x = v — σ(g)(w) (x),

hence (v — w) —¦ g = v — σ(g)(w). Therefore, (??) yields

b

c’1 — (v — w)b · c = v — σ(g)(w) .