there exists u ∈ C such that s0 (x) = TrdC (ux) for all x ∈ C. If u is not

invertible, then the annihilator of the left ideal generated by u is a nontrivial right

ideal in the kernel of s0 , contrary to the hypothesis that s is an involution trace.

Finally, observe that for c ∈ C,

s0 θ(c) = θ TrdC cθ(u) = TrdC θ(u)c ,

hence the condition s0 θ(c) = s0 (c) for all c ∈ C implies that θ(u) = u.

(4.5) Corollary. For every involution trace s : E ’ T , there exists an involution

θs on C such that

s θ(c)x = s xθs (c) for c ∈ C, x ∈ E.

The involutions θs and θ have the same restriction to Z.

Proof : Fix a nonzero linear map : Z ’ F which commutes with θ. According

to (??), we have s = IdT — s0 where s0 : C ’ F is de¬ned by s0 (c) = TrdC (uc)

for some symmetric unit u ∈ C — . Let θs = Int(u) —¦ θ. For c, c ∈ C,

TrdC uθ(c)c = TrdC θs (c)uc = TrdC uc θs (c) ,

hence for all t ∈ T ,

s θ(c) · (t — c ) = t · TrdC uθ(c)c = s (t — c ) · θs (c) .

Therefore, the involution θs satis¬es

s θ(c)x = s xθs (c) for c ∈ C, x ∈ E.

The involution θs is uniquely determined by this condition, because if s xθs (c) =

s xθs (c) for all c ∈ C and x ∈ E, then property (??) of involution traces in (??)

implies that θs (c) = θs (c) for all c ∈ C.

Since θs = Int(u) —¦ θ, it is clear that θs (z) = θ(z) for all z ∈ Z.

Using an involution trace s : E ’ T , we may de¬ne a structure of hermitian

module over T on every hermitian module over E, as we proceed to show.

Suppose that M is a ¬nitely generated right module over E. Since T ‚ E, we

may also consider M as a right T -module, and

EndE (M ) ‚ EndT (M ).

The centralizer of EndE (M ) in EndT (M ) is easily determined:

§4. HERMITIAN FORMS 47

(4.6) Lemma. For c in the centralizer C of T in E, let rc ∈ EndT (M ) be the

right multiplication by c. The map cop ’ rc identi¬es C op with the centralizer of

EndE (M ) in EndT (M ).

Proof : Every element f ∈ EndT (M ) in the centralizer of EndE (M ) may be viewed

as an endomorphism of M for its EndE (M )-module structure. By (??), we have

EndEndE (M ) (M ) = E, hence f is right multiplication by some element c ∈ E. Since

f is a T -module endomorphism, c ∈ C.

Suppose now that h : M — M ’ E is a hermitian or skew-hermitian form with

respect to θ. If s : E ’ T is an involution trace, we de¬ne

s— (h) : M — M ’ T

by

s— (h)(x, y) = s h(x, y) for x, y ∈ M .

In view of the properties of s, the form s— (h) is clearly hermitian over T (with

respect to θ) if h is hermitian, and skew-hermitian if h is skew-hermitian. It is

also alternating if h is alternating, since the relation h(x, x) = e ’ θ(e) implies that

s— (h)(x, x) = s(e) ’ θ s(e) .

(4.7) Proposition. If h is nonsingular, then s— (h) is nonsingular and the adjoint

involution σs— (h) on EndT (M ) extends the adjoint involution σh on EndE (M ):

EndE (M ), σh ‚ EndT (M ), σs— (h) .

Moreover, with the notation of (??) and (??),

σs— (h) (rc ) = rθs (c)

for all c ∈ C.

Proof : If x ∈ M is such that s— (h)(x, y) = 0 for all y ∈ M , then h(x, M ) is a right

ideal of E contained in ker s, hence h(x, M ) = {0}. This implies that x = 0 if h is

nonsingular, proving the ¬rst statement.

For f ∈ EndE (M ) and x, y ∈ M we have

h x, f (y) = h σh (f )(x), y .

Hence, applying s to both sides,

s— (h) x, f (y) = s— (h) σh (f )(x), y .

Therefore, σs— (h) (f ) = σh (f ).

On the other hand, for x, y ∈ M and c ∈ C,

s— (h)(xc, y) = s θ(c)h(x, y) .

The de¬ning property of θs shows that the right side is also equal to

s h(x, y)θs (c) = s— (h) x, yθs (c) ,

hence σs— (h) (rc ) = rθs (c) .

(4.8) Example. Suppose that E is central over F , hence the centralizer C of T

in E also is central over F . Let M be a ¬nitely generated right module over E. The

algebra EndE (M ) is a central simple F -subalgebra in EndT (M ), and (??) shows

that its centralizer is isomorphic to C op under the map which carries cop ∈ C op

48 I. INVOLUTIONS AND HERMITIAN FORMS

to the endomorphism rc of right multiplication by c. Hence there is an F -algebra

isomorphism

∼

Ψ : EndE (M ) —F C op ’ EndT (M )

’

which maps f — cop to f —¦ rc = rc —¦ f for f ∈ EndE (M ) and c ∈ C.

Pick an invertible element u ∈ Sym(C, θ) and de¬ne an involution trace s : E ’

T by

s(t — c) = t · TrdC (uc) for t ∈ T , c ∈ C.

The proof of (??) shows that θs = Int(u) —¦ θ. Moreover, (??) shows that for every

nonsingular hermitian or skew-hermitian form h : M —M ’ E, the involution σs— (h)

op

on EndT (M ) corresponds under Ψ to σh —θs where θs (cop ) = θs (c)

op op

for c ∈ C:

∼

Ψ : EndE (M ) —F C op , σh — θs ’ EndT (M ), σs— (h) .

op

’

As a particular case, we may consider T = F , E = C and M = C. Then one

sees that EndE (M ) = C by identifying c ∈ C with left multiplication by c, and the

isomorphism Ψ is the same as in Wedderburn™s theorem (??):

∼

Ψ : C —F C op ’ EndF (C).

’

If h : C — C ’ C is de¬ned by h(x, y) = θ(x)y, then σh = θ and the result above

shows that σTrd— (h) corresponds to θ — θ under Ψ.

(4.9) Example. Suppose that C is the center Z of E, so that

E = T —F Z.

Let N be a ¬nitely generated right module over T and h : N — N ’ T be a

nonsingular hermitian form with respect to θ. Extending scalars to Z, we get a

module NZ = N —F Z over E and a nonsingular hermitian form hZ : NZ —NZ ’ E.

Moreover,

EndE (NZ ) = EndT (N ) —F Z and EndT (NZ ) = EndT (N ) — EndF (Z).

Pick a nonzero linear map : Z ’ F which commutes with θ and let

s = IdT — : E ’ T

be the induced involution trace on E. We claim that under the identi¬cation above,

σs— (hZ ) = σh — σk ,

where k : Z — Z ’ F is the hermitian form de¬ned by

k(z1 , z2 ) = θ(z1 )z2 for z1 , z2 ∈ Z.

Indeed, for x1 , x2 ∈ N and z1 , z2 ∈ Z we have

hZ (x1 — z1 , x2 — z2 ) = h(x1 , x2 ) — θ(z1 )z2 ,

hence

s— (hZ ) = h — k.

We now return to the general case, and show that the involutions on EndT (M )

which are adjoint to transfer forms s— (h) are exactly those which preserve EndE (M )

and induce θs on the centralizer.

§4. HERMITIAN FORMS 49

(4.10) Proposition. Let σ be an involution on EndT (M ) such that

σ EndE (M ) = EndE (M ) σ(re ) = rθs (e)

and

for all e ∈ CE T . There exists a nonsingular hermitian or skew-hermitian form

h : M — M ’ E with respect to θ such that σ = σs— (h) .

Proof : Since θs |Z = θ|Z by (??), it follows that σ(rz ) = rθs (z) = rθ(z) for all

z ∈ Z. Therefore, Theorem (??) shows that the restriction of σ to EndE (M ) is the

adjoint involution with respect to some nonsingular hermitian or skew-hermitian

form h0 : M — M ’ E. Proposition (??) (if θ|F = IdF ) or (??) (if θ|F = IdF )

yields an invertible element u ∈ EndT (M ) such that

σ = Int(u) —¦ σs— (h0 )

and σs— (h0 ) (u) = ±u. By (??), the restriction of σs— (h0 ) to EndE (M ) is σh0 which

is also the restriction of σ to EndE (M ). Therefore, u centralizes EndE (M ). It

follows from (??) that u = re for some e ∈ C — . Proposition (??) shows that

σs— (h0 ) (rc ) = rθs (c) for all c ∈ C, hence

σ(rc ) = u —¦ rθs (c) —¦ u’1 = re’1 θs (c)e .

Since we assume that σ(rc ) = rθs (c) for all c ∈ C, it follows that e ∈ Z — . Moreover,

θ(e) = ±e since σs— (h0 ) (u) = ±u. We may then de¬ne a nonsingular hermitian or

skew-hermitian form h : M — M ’ E by

h(x, y) = e’1 h0 (x, y) for x, y ∈ M .

If δ = θ(e)e’1 (= ±1), we also have

h(x, y) = δh0 (xe’1 , y) = δh0 re’1 (x), y ,

hence

σs— (h) = Int(re ) —¦ σs— (h0 ) = σ.

(4.11) Example. Suppose E is commutative, so that E = Z = C and suppose

that T = F . Assume further that θ = IdE . Let V be a ¬nite dimensional vector

space over F and ¬x some F -algebra embedding

i : Z ’ EndF (V ).

We may then consider V as a vector space over Z by de¬ning

v · z = i(z)(v) for v ∈ V , z ∈ Z.

By de¬nition, the centralizer of i(Z) in EndF (V ) is EndZ (V ).

Suppose that σ is an involution on EndF (V ) which leaves i(Z) elementwise

invariant and that s : Z ’ F is a nonzero linear map. By (??), we have θs = θ =

IdZ . On the other hand, since σ preserves i(Z), it also preserves its centralizer

EndZ (V ). We may therefore apply (??) to conclude that there exists a nonsingular

symmetric or skew-symmetric bilinear form b : V — V ’ Z such that σ = σs— (b) .

By (??), the restriction of σ to EndZ (V ) is σb . If b is symmetric, skew-

symmetric or alternating, then s— (b) has the same property. If char F = 2, the

bilinear form s— (b) cannot be simultaneously symmetric and skew-symmetric, or it

would be singular. Therefore, b and s— (b) are of the same type, and it follows that

σ has the same type as its restriction to EndZ (V ). If char F = 2, it is still true that

50 I. INVOLUTIONS AND HERMITIAN FORMS

σ is symplectic if its restriction to EndZ (V ) is symplectic, since s— (b) is alternating

if b is alternating, but the converse is not true without some further hypotheses.

To construct a speci¬c example, consider a ¬eld Z which is ¬nite dimensional

over its sub¬eld of squares Z 2 , and let F = Z 2 . Pick a nonzero linear map s : Z ’ F

such that s(1) = 0. The nonsingular symmetric bilinear form b on V = Z de¬ned

by b(z1 , z2 ) = z1 z2 is not alternating, but s— (b) is alternating since s vanishes on Z 2 .

Therefore, the involution σs— (b) on EndF (Z) is symplectic, but its restriction σb to

EndZ (Z) is orthogonal. (Indeed, EndZ (Z) = Z and σb = IdZ .)

These observations on the type of an involution compared with the type of its

restriction to a centralizer are generalized in the next proposition.

(4.12) Proposition. Let A be a central simple F -algebra with an involution σ of

the ¬rst kind and let L ‚ A be a sub¬eld containing F . Suppose that σ leaves L

elementwise invariant, so that it restricts to an involution of the ¬rst kind „ on the

centralizer CA L.

(1) If char F = 2, the involutions σ and „ have the same type.

(2) Suppose that char F = 2. If „ is symplectic, then σ is symplectic. If L/F is

separable, then σ and „ have the same type.

Proof : We ¬rst consider the simpler case where char F = 2. If „ is symplectic,

then (??) shows that the centralizer CA L contains an element c such that c+„ (c) =

1. Since „ (c) = σ(c), it also follows from (??) that σ is symplectic.

If „ is orthogonal, then TrdCA L Sym(CA L, „ ) = L by (??). If L/F is separa-

ble, the trace form TL/F is nonzero, hence

TL/F —¦ TrdCA L Sym(CA L, „ ) = F.

Since TL/F —¦ TrdCA L (c) = TrdA (c) for all c ∈ CA L (see Draxl [?, p. 150]), we have

TL/F —¦ TrdCA L Sym(CA L, „ ) ‚ TrdA Sym(A, σ) ,

hence TrdA Sym(A, σ) = {0}, and σ is orthogonal. This completes the proof in

the case where char F = 2.

In arbitrary characteristic, let F be a splitting ¬eld of A in which F is alge-

braically closed and such that the ¬eld extension F /F is separable (for instance, the

function ¬eld of the Severi-Brauer variety SB(A)). The composite L·F (= L—F F )

is then a ¬eld, and it su¬ces to prove the proposition after extending scalars to F .

We may thus assume that A = EndF (V ) for some F -vector space V . If char F = 2

the result then follows from the observations in (??).

(4.13) Corollary. Let M be a maximal sub¬eld of degree n in a central simple

F -algebra A of degree n. Suppose that char F = 2 or that M/F is separable. Every

involution which leaves M elementwise invariant is orthogonal.

Proof : We have CA M = M by (??). Therefore, if σ is an involution on A which

leaves M elementwise invariant, then σ|CA M = IdM and (??) shows that σ is

orthogonal if char F = 2 or M/F is separable.

The result does not hold in characteristic 2 when M/F is not separable, as

example (??) shows.

§4. HERMITIAN FORMS 51

Extension of involutions. The following theorem is a kind of “Skolem-

Noether theorem” for involutions. The ¬rst part is due to M. Kneser [?, p. 37].

(For a di¬erent proof, see Scharlau [?, §8.10].)

(4.14) Theorem. Let B be a simple subalgebra of a central simple algebra A over

a ¬eld F . Suppose that A and B have involutions σ and „ respectively which have

the same restriction to F . Then A has an involution σ whose restriction to B is „ .

If σ is of the ¬rst kind, the types of σ and „ are related as follows:

(1) If char F = 2, then σ can be arbitrarily chosen of orthogonal or symplectic

type, except when the following two conditions hold : „ is of the ¬rst kind and the

degree of the centralizer CA B of B in A is odd. In that case, every extension σ

of „ has the same type as „ .

(2) Suppose that char F = 2. If „ is of symplectic or unitary type, then σ is

symplectic. If „ is of orthogonal type and the center of B is a separable extension

of F , then σ can be arbitrarily chosen of orthogonal or symplectic type, except when

the degree of the centralizer CA B is odd. In that case σ is orthogonal.

Proof : In order to show the existence of σ , we ¬rst reduce to the case where the

centralizer CA B is a division algebra. Let Z = B © CA B be the center of B,

hence also of CA B by the double centralizer theorem (see (??)). Wedderburn™s

theorem (??) yields a decomposition of CA B:

CA B = M · D M —Z D

where M is a matrix algebra: M Mr (Z) for some integer r, and D is a division

algebra with center Z. Let B = B · M B —Z M be the subalgebra of A generated

by B and M . An involution — on Mr (Z) of the same kind as „ can be de¬ned by

letting „ |Z act entrywise and setting

a— = „ (a)t for a ∈ Mr (Z).

The involution „ — — on B — Mr (Z) extends „ and is carried to an involution „

on B through an isomorphism B —Z Mr (Z) B · M = B . It now remains to

extend „ to A. Note that the centralizer of B is a division algebra, i.e., CA B = D.

Since σ and „ have the same restriction to the center F of A, the Skolem-

Noether theorem shows that σ —¦ „ is an inner automorphism. Let σ —¦ „ = Int(u)

for some u ∈ A— , so that

(4.15) σ —¦ „ (x)u = ux for x ∈ B .

Substituting „ (x) for x, we get

σ(x)u = u„ (x) for x ∈ B

and, applying σ to both sides,

(4.16) σ —¦ „ (x)σ(u) = σ(u)x for x ∈ B .

By comparing (??) and (??), we obtain u’1 σ(u) ∈ CA B . At least one of the

elements a+1 = 1 + u’1 σ(u), a’1 = 1 ’ u’1 σ(u) is nonzero, hence invertible since

CA B is a division algebra. If aµ is invertible (where µ = ±1), we have

σ —¦ „ = Int(u) —¦ Int(aµ ) = Int(uaµ )

since aµ ∈ CA B , and uaµ = u + µσ(u) ∈ Sym(A, σ) ∪ Skew(A, σ). Therefore,

σ —¦ Int(uaµ ) (= Int (uaµ )’1 —¦ σ) is an involution on A whose restriction to B is

„ . This completes the proof of the existence of an extension σ of „ to A.

52 I. INVOLUTIONS AND HERMITIAN FORMS

We now discuss the type of σ (assuming that it is of the ¬rst kind, i.e., „ |F =

IdF ). Suppose ¬rst that char F = 2. Since σ extends „ , it preserves B, hence also

its centralizer CA B. It therefore restricts to an involution on CA B. If „ is of the

second kind, we may ¬nd some v ∈ Z — such that σ (v) = ’v. Similarly, if „ is

of the ¬rst kind, then (??) shows that we may ¬nd some v ∈ (CA B)— such that

σ (v) = ’v, except when the degree of CA B is odd. Assuming we have such a v,

the involution σ = Int(v) —¦ σ also extends „ since v ∈ CA B, and it is of the type

opposite to σ since v ∈ Skew(A, σ ) (see (??)). Therefore, „ has extensions of both

types to A.

If the degree of CA B is odd and σ leaves Z elementwise invariant, consider the

restriction of σ to the centralizer CA Z. Since B has center Z we have, by (??),

CA Z = B —Z CA B.

The restriction σ |CA Z preserves both factors, hence it decomposes as

σ |C A Z = „ — σ |C A B .

Since the degree of CA B is odd, the involution σ |CA B is orthogonal by (??). There-

fore, it follows from (??) that the involution σ |CA Z has the same type as „ . Propo-

sition (??) shows that σ and σ |CA Z have the same type and completes the proof

in the case where char F = 2.

Suppose next that char F = 2. If „ is unitary, then it induces a nontrivial

automorphism of order 2 on Z, hence there exists z ∈ Z such that z + „ (z) = 1.

For every extension σ of „ to A we have that z + σ (z) = 1, hence 1 ∈ Alt(A, σ )

and σ is symplectic by (??). Similarly, if „ is symplectic then 1 ∈ Alt(B, „ ) hence

also 1 ∈ Alt(A, σ ) for every extension σ of „ . Therefore, every extension of „ is

symplectic.

Suppose ¬nally that „ is orthogonal and that Z/F is separable. As above, we

have CA B = B —Z CA B. If deg CA B is even, then (??) shows that we may ¬nd an

involution θ1 on CA B of orthogonal type and an involution θ2 of symplectic type.

By (??), the involution „ — θ1 (resp. „ — θ2 ) is orthogonal (resp. symplectic). It

follows from (??) that every extension of this involution to A has the same type.

If deg CA B is odd, the same arguments as in the case where char F = 2 show

that every extension of „ is orthogonal.

If the subalgebra B ‚ A is not simple, much less is known on the possibility

of extending an involution from B to A. We have however the following general

result:

(4.17) Proposition. Let A be a central simple algebra with involution of the ¬rst

kind. Every element of A is invariant under some involution.

Proof : Let a ∈ A and let σ be an arbitrary orthogonal involution on A. Consider

the vector space

V = { x ∈ A | σ(x) = x, xσ(a) = ax }.

It su¬ces to show that V contains an invertible element u, for then Int(u) —¦ σ is an

involution on A which leaves a invariant.

Let L be a splitting ¬eld of A. Fix an isomorphism AL Mn (L). The existence

of g ∈ GLn (L) such that g t = g and gat = ag is shown in Kaplansky [?, Theorem 66]

(see also Exercise ??). The involution „ = Int(g) —¦ t leaves a invariant, and it is

orthogonal if char F = 2. By (??), there exists an invertible element u ∈ AL such

§5. QUADRATIC FORMS 53

that „ = Int(u) —¦ σL and σL (u) = u. Since „ leaves a invariant, we have u ∈ V — L.

We have thus shown that V —L contains an element whose reduced norm is nonzero,

hence the Zariski-open subset in V consisting of elements whose reduced norm is

nonzero is nonempty. If F is in¬nite, we may use density of the rational points to

conclude that V contains an invertible element. If F is ¬nite, we may take L = F

in the discussion above.

Note that if char F = 2 the proof yields a more precise result: every element is

invariant under some orthogonal involution. Similar arguments apply to involutions

of the second kind, as the next proposition shows.

(4.18) Proposition. Let (B, „ ) be a central simple F -algebra with involution of

the second kind of degree n and let K be the center of B. For every b ∈ B whose

minimal polynomial over K has degree n and coe¬cients in F , there exists an

involution of the second kind on B which leaves b invariant.

Proof : Consider the F -vector space

W = { x ∈ B | „ (x) = x, x„ (b) = bx }.

As in the proof of (??), it su¬ces to show that W contains an invertible element.

Mn (L) — Mn (L)op .

By (??), we may ¬nd a ¬eld extension L/F such that BL

Fix such an isomorphism. Since the minimal polynomial of b has degree n and

coe¬cients in F , its image in Mn (L) — Mn (L)op has the form (m1 , mop ) where

2

m1 , m2 are matrices which have the same minimal polynomial of degree n. We

may then ¬nd a matrix u ∈ GLn (L) such that um2 u’1 = m1 . If µ is the exchange

involution on Mn (L) — Mn (L)op , the involution Int(u, uop ) —¦ µ leaves (m1 , mop )

2

invariant. This involution has the form Int(v) —¦ „L for some invertible element

v ∈ W — L. If F is in¬nite, we may conclude as in the proof of (??) that W also

contains an invertible element, completing the proof.

If F is ¬nite and K F — F , the arguments above apply with L = F . The

remaining case where F is ¬nite and K is a ¬eld is left to the reader. (See Exer-

cise ??.)

§5. Quadratic Forms

This section introduces the notion of a quadratic pair which is a twisted ana-

logue of quadratic form in the same way that involutions are twisted analogues of

symmetric, skew-symmetric or alternating forms (up to a scalar factor). The full

force of this notion is in characteristic 2, since quadratic forms correspond bijec-

tively to symmetric bilinear forms in characteristic di¬erent from 2. Nevertheless

we place no restrictions on the characteristic of our base ¬eld F .

As a preparation for the proof that quadratic pairs on a split algebra EndF (V )

correspond to quadratic forms on the vector space V (see (??)), we ¬rst show

that every nonsingular bilinear form on V determines a standard identi¬cation

EndF (V ) = V —F V . This identi¬cation is of central importance for the de¬nition

of the Cli¬ord algebra of a quadratic pair in §??.

5.A. Standard identi¬cations. In this subsection, D denotes a central di-

vision algebra over F and θ denotes an involution (of any kind) on D. Let V be

a ¬nite dimensional right vector space over D. We de¬ne9 a left vector space θ V

9 Note that this de¬nition is consistent with those in § ?? and § ??.

54 I. INVOLUTIONS AND HERMITIAN FORMS

over D by

θ

V = { θv | v ∈ V }

with the operations

θ

v + θ w = θ (v + w) and ± · θ v = θ v · θ(±)

for v, w ∈ V and ± ∈ D. We may then consider the tensor product V —D θ V which

is a vector space over F of dimension

dimF V —F V

dimF V —D θ V = = (dimF V )2 dimF D.

dimF D

Now, let h : V — V ’ D be a nonsingular hermitian or skew-hermitian form on V

with respect to θ. There is an F -linear map

•h : V —D θ V ’ EndD (V )

such that

•h (v — θ w)(x) = v · h(w, x) for v, w, x ∈ V .

(5.1) Theorem. The map •h is bijective. Letting σh denote the adjoint involution

on EndD (V ) with respect to h, we have

σh •h (v — θ w) = δ•h (w — θ v) for v, w ∈ V ,

where δ = +1 if h is hermitian and δ = ’1 if h is skew-hermitian. Moreover,

TrdEndD (V ) •h (v — θ w) = TrdD h(w, v) for v, w ∈ V

and, for v1 , v2 , w1 , w2 ∈ V ,

•h (v1 — θ w1 ) —¦ •h (v2 — θ w2 ) = •h v1 h(w1 , v2 ) — θ w2 .

Proof : Let (e1 , . . . , en ) be a basis of V over D. Since h is nonsingular, for i ∈

{1, . . . , n} there exists a unique vector ei ∈ V such that

1 if i = j,

h(ei , ej ) =

0 if i = j,

and (e1 , . . . , en ) is a basis of V over D. Every element x ∈ V —D θ V therefore has

a unique expression of the form

n

ei aij — θ ej

x= for some aij ∈ D.

i,j=1

The map •h takes the element x to the endomorphism of V with the matrix

(aij )1¤i,j¤n (with respect to the basis (e1 , . . . , en )), hence •h is bijective. Moreover,

we have

n n

TrdEndD (V ) •h (x) = TrdD (aii ) = TrdD h(ej , ei aij ) ,

i=1 i,j=1

hence in particular

TrdEndD (V ) •h (v — θ w) = TrdD h(w, v) for v, w ∈ V .

For v, w, x, y ∈ V we have

h x, •h (v — θ w)(y) = h(x, v)h(w, y)

§5. QUADRATIC FORMS 55

and

h •h (w — θ v)(x), y = h wh(v, x), y = θ h(v, x) h(w, y),

hence σh •h (v — θ w) = δ•h (w — θ v).

Finally, for v1 , w1 , v2 , w2 , x ∈ V ,

•h (v1 — θ w1 ) —¦ •h (v2 — θ w2 )(x) = v1 h(w1 , v2 )h(w2 , x)

= •h v1 h(w1 , v2 ) — θ w2 (x),

hence •h (v1 — θ w1 ) —¦ •h (v2 — θ w2 ) = •h v1 h(w1 , v2 ) — θ w2 .

Under •h , the F -algebra with involution EndD (V ), σh is thus identi¬ed with

V —D θ V endowed with the product

(v1 — θ w1 ) —¦ (v2 — θ w2 ) = v1 h(w1 , v2 ) — θ w2 for v1 , v2 , w1 , w2 ∈ V

and the involution σ de¬ned by

σ(v — θ w) = δw — θ v for v, w ∈ V ,

where δ = +1 if h is hermitian and δ = ’1 if h is skew-hermitian. We shall refer

to the map •h in the sequel as the standard identi¬cation of EndD (V ), σh with

(V —D θ V, σ). Note that the map •h depends on the choice of h and not just on the

involution σh . Indeed, for any ± ∈ F — ¬xed by θ we have σ±h = σh but •±h = ±•h .

The standard identi¬cation will be used mostly in the split case where D = F .

If moreover θ = IdF , then θ V = V , hence the standard identi¬cation associated to

a nonsingular symmetric or skew-symmetric bilinear form b on V is

∼

(5.2) •b : (V —F V, σ) ’ EndF (V ), σb ,

’ •b (v — w)(x) = vb(w, x),

where σ(v — w) = w — v if b is symmetric and σ(v — w) = ’w — v if b is skew-

symmetric.

(5.3) Example. As in (??), for all integers r, s let — : Mr,s (D) ’ Ms,r (D) be the

map de¬ned by

t

(aij )— = θ(aij ) .

i,j i,j

Let V = Dr (= Mr,1 (D)) and let h : Dr — Dr ’ D be the hermitian form de¬ned

by

h(x, y) = x— · g · y for x, y ∈ Dr ,

where g ∈ Mr (D) is invertible and satis¬es g — = g.

Identify Mr (D) with EndD (Dr ) by mapping m ∈ Mr (D) to the endomorphism

x ’ m · x. The standard identi¬cation

∼

•h : Dr —D Dr ’ Mr (D)

’

carries v — w ∈ D r —D Dr to the matrix v · w — · g, since for all x ∈ D r

v · w— · g · x = vh(w, x).

56 I. INVOLUTIONS AND HERMITIAN FORMS

5.B. Quadratic pairs. Let A be a central simple algebra of degree n over a

¬eld F of arbitrary characteristic.

(5.4) De¬nition. A quadratic pair on A is a couple (σ, f ), where σ is an involution

of the ¬rst kind on A and f : Sym(A, σ) ’ F is a linear map, subject to the

following conditions:

(1) dimF Sym(A, σ) = n(n + 1)/2 and TrdA Skew(A, σ) = {0}.

(2) f x + σ(x) = TrdA (x) for all x ∈ A.

Note that the equality x + σ(x) = x + σ(x ) holds for x, x ∈ A if and only

if x ’ x ∈ Skew(A, σ). Therefore, condition (??) makes sense only if the reduced

trace of every skew-symmetric element is zero, as required in (??).

If char F = 2, the equality TrdA Skew(A, σ) = {0} holds for every involution

of the ¬rst kind, by (??), hence condition (??) simply means that σ is of orthogonal

type. On the other hand, the map f is uniquely determined by (??) since for

1 1

s ∈ Sym(A, σ) we have s = 2 s + σ(s) , hence f (s) = 2 TrdA (s).

If char F = 2, Proposition (??) shows that condition (??) holds if and only

if σ is symplectic (which implies that n is even). Condition (??) determines the

value of f on the subspace Symd(A, σ) but not on Sym(A, σ). Indeed, in view

of (??), condition (??) simply means that f is an extension of the linear form

Trpσ : Symd(A, σ) ’ F . Therefore, there exist several quadratic pairs with the

same symplectic involution.

(5.5) Example. Let „ be an involution of orthogonal type on A. Every element

a ∈ A such that a+„ (a) is invertible determines a quadratic pair (σa , fa ) as follows:

let g = a + „ (a) ∈ A— and de¬ne

σa = Int(g ’1 ) —¦ „ and fa (s) = TrdA (g ’1 as) for s ∈ Sym(A, σa ).

From (??), it follows that σa is orthogonal if char F = 2 and symplectic if char F =

2. In order to check condition (??) of the de¬nition of a quadratic pair, we compute

fa x + σa (x) = TrdA (g ’1 ax) + TrdA g ’1 ag ’1 „ (x)g for x ∈ A.

Since ag ’1 „ (x) and „ ag ’1 „ (x) = xg ’1 „ (a) have the same trace, the last term on

the right side is also equal to TrdA xg ’1 „ (a) , hence

fa x + σa (x) = TrdA xg ’1 a + „ (a) = TrdA (x).

We will show in (??) that every quadratic pair is of the form (σa , fa ) for some

a ∈ A such that a + „ (a) is invertible.

We start with a couple of general results:

(5.6) Proposition. For every quadratic pair (σ, f ) on A,

1

f (1) = deg A.

2

Proof : If char F = 2, we have f (1) = 1 TrdA (1) = 1

deg A. If char F = 2, the

2 2

proposition follows from (??) since f (1) = Trpσ (1).

(5.7) Proposition. For every quadratic pair (σ, f ) on A, there exists an element

∈ A, uniquely determined up to the addition of an element in Alt(A, σ), such that

f (s) = TrdA ( s) for all s ∈ Sym(A, σ).

satis¬es + σ( ) = 1.

The element

§5. QUADRATIC FORMS 57

Proof : Since the bilinear reduced trace form on A is nonsingular by (??), every

linear form A ’ F is of the form x ’ TrdA (ax) for some a ∈ A. Therefore,

extending f arbitrarily to a linear form on A, we may ¬nd some ∈ A such that

f (s) = TrdA ( s) for all s ∈ Sym(A, σ). If , ∈ A both satisfy this relation, then

TrdA ( ’ )s = 0 for all s ∈ Sym(A, σ), hence (??) shows that ’ ∈ Alt(A, σ).

Condition (??) of the de¬nition of a quadratic pair yields

TrdA x + σ(x) = TrdA (x) for x ∈ A.

Since TrdA σ(x) = TrdA xσ( ) , it follows that

TrdA + σ( ) x = TrdA (x) for x ∈ A,

hence + σ( ) = 1 since the bilinear reduced trace form on A is nonsingular.

(5.8) Proposition. Let „ be an orthogonal involution on A. Every quadratic pair

on A is of the form (σa , fa ) for some a ∈ A such that a + „ (a) ∈ A— . If a, b ∈ A

are such that a + „ (a) ∈ A— and b + „ (b) ∈ A— , then (σa , fa ) = (σb , fb ) if and only

if there exists » ∈ F — and c ∈ Alt(A, „ ) such that a = »b + c.

Proof : Let (σ, f ) be a quadratic pair on A. By (??), there exists an invertible

element g ∈ A— such that σ = Int(g ’1 ) —¦ „ and g ∈ Sym(A, „ ) if char F = 2, and

g ∈ Alt(A, „ ) if char F = 2. Moreover, (??) yields an element ∈ A such that

σ( ) + = 1 and f (s) = TrdA ( s) for all s ∈ Sym(A, σ). Let a = g ∈ A. Since

„ (g) = g and σ( ) + = 1, we have a + „ (a) = g, hence σa = σ. Moreover, for

s ∈ Sym(A, σ),

fa (s) = TrdA (g ’1 as) = TrdA ( s) = f (s),

hence (σ, f ) = (σa , fa ).

Suppose now that a, b ∈ A are such that a + „ (a), b + „ (b) are each invertible

and that (σa , fa ) = (σb , fb ). Writing g = a + „ (a) and h = b + „ (b), we have

σa = Int(g ’1 ) —¦ „ and σb = Int(h’1 ) —¦ „ , hence the equality σa = σb yields g = »h

for some » ∈ F — . On the other hand, since fa = fb we have

TrdA (g ’1 as) = TrdA (h’1 bs) for s ∈ Sym(A, σa ) = Sym(B, σb ).

Since Sym(A, σb ) = Sym(A, „ )h and g = »h, it follows that

»’1 TrdA (ax) = TrdA (bx) for x ∈ Sym(A, „ ),

hence a ’ »b ∈ Alt(A, „ ), by (??).

This proposition shows that quadratic pairs on A are in one-to-one correspon-

dence with equivalence classes of elements a + Alt(A, „ ) ∈ A/ Alt(A, „ ) such that

a + „ (a) is invertible, modulo multiplication by a factor in F — .

In the particular case where A = Mn (F ) and „ = t is the transpose involution,

the elements in A/ Alt(A, „ ) may be regarded as quadratic forms of dimension n,

by identifying a + Alt(A, „ ) with the quadratic form q(X) = X · a · X t where

X = (x1 , . . . , xn ). The matrix a + at is invertible if and only if the corresponding

quadratic form is nonsingular (of even dimension if char F = 2). Therefore, quad-

ratic pairs on Mn (F ) are in one-to-one correspondence with equivalence classes of

nonsingular quadratic forms of dimension n modulo a factor in F — (with n even if

char F = 2).

58 I. INVOLUTIONS AND HERMITIAN FORMS

(5.9) Example. Suppose that n = 2 and char F = 2. Straightforward compu-

tations show that the quadratic pair on M2 (F ) associated to the quadratic form

aX 2 + bXY + cY 2 is (σ, f ) with

a11 a12 a22 a12

σ =

a21 a22 a21 a11

and

a11 a12

= a11 + ab’1 a12 + cb’1 a21 .

f

a21 a11

The observation above explains why quadratic pairs on central simple algebras

may be thought of as twisted forms of nonsingular quadratic forms up to a scalar

factor. We next give another perspective on this result by relating quadratic forms

on a vector space V to quadratic pairs on the endomorphism algebra EndF (V ).

Let V be a vector space of dimension n over F and let q : V ’ F be a quadratic

form on V . We recall that bq is the polar symmetric bilinear form of q,

bq (x, y) = q(x + y) ’ q(x) ’ q(y) for x, y ∈ V ,

which we assume to be nonsingular. This hypothesis implies that n is even if

char F = 2, since the bilinear form bq is alternating in this case. We write simply

σq for the adjoint involution σbq on EndF (V ) and

(5.10) •q : V —F V ’ EndF (V ), •q (v — w)(x) = vbq (w, x)

for the standard identi¬cation •bq of (??). Under this identi¬cation, we have

(V — V, σ) = EndF (V ), σq ,

where σ : V — V ’ V — V is the switch.

(5.11) Proposition. There is a unique linear map fq : Sym EndF (V ), σq ’ F

such that

fq —¦ •q (v — v) = q(v) for x, y ∈ V .

The couple (σq , fq ) is a quadratic pair on EndF (V ). Moreover, assuming that

dimF V is even if char F = 2, every quadratic pair on EndF (V ) is of the form

(σq , fq ) for some nonsingular quadratic form q on V which is uniquely determined

up to a factor in F — .

Proof : Let (e1 , . . . , en ) be a basis of V . The elements •q (ei —ei ) for i = 1, . . . , n and

•q (ei —ej +ej —ei ) for i, j ∈ {1, . . . , n}, j = i, form a basis of Sym EndF (V ), σq =

•q Sym(V — V, σ) . De¬ne

fq •q (ei — ei ) = q(ei ), fq •q (ei — ej + ej — ei ) = bq (ei , ej )

n

and extend by linearity to a map fq : Sym EndF (V ), σq ’ F . For v = e i ±i ∈

i=1

V we have

— e i ±2 +

fq —¦ •q (v — v) = fq —¦ •q 1¤i¤n ei 1¤i<j¤n (ei — ej + ej — ei )±i ±j

i

2

= 1¤i¤n q(ei )±i + 1¤i<j¤n bq (ei , ej )±i ±j = q(v),

hence the map fq thus de¬ned satis¬es the required condition. Uniqueness of fq is

clear, since Sym EndF (V ), σq is spanned by elements of the form •q (v — v).

§5. QUADRATIC FORMS 59

Since the bilinear form q is symmetric, and alternating if char F = 2, the

involution σq is orthogonal if char F = 2 and symplectic if char F = 2. To check

that (σq , fq ) is a quadratic pair, it remains to prove that

fq x + σ(x) = TrdEndF (V ) (x) for x ∈ EndF (V ).

Since both sides are linear, it su¬ces to check this formula for x = •q (v — w) with

v, w ∈ V . The left side is then

fq —¦ •q (v — w + w — v) = fq —¦ •q (v + w) — (v + w) ’ fq —¦ •q (v — v)

’ fq —¦ •q (w — w)

= bq (v, w)

and the claim follows, since (??) shows that bq (v, w) = bq (w, v) = Trd •q (v — w) .

Suppose now that (σ, f ) is an arbitrary quadratic pair on EndF (V ). As shown

in the introduction to this chapter (and in (??)), the involution σ is the adjoint

involution with respect to some nonsingular symmetric bilinear form b : V — V ’ F

which is uniquely determined up to a factor in F — . Use the standard identi¬cation

•b (see (??)) to de¬ne a map q : V ’ F by

q(v) = f —¦ •b (v — v) for v ∈ V .

From the de¬nition, it is clear that q(v±) = q(v)±2 for ± ∈ F . Moreover, for v,

w ∈ V we have

q(v + w) ’ q(v) ’ q(w) = f —¦ •b (v — w) + f —¦ •b (w — v)

= f •b (v — w) + σ •b (v — w) .

Since (σ, f ) is a quadratic pair, the right side is equal to

TrdEndF (V ) •b (v — w) = b(w, v) = b(v, w).

Therefore, q is a quadratic form with associated polar form b, and it is clear that

the corresponding quadratic pair (σq , fq ) is (σ, f ). Since b is uniquely determined

up to a factor in F — , the same property holds for q.

For later use, we give an explicit description of an element ∈ EndF (V ) satis-

fying property (??) for the quadratic pair (σq , fq ). It su¬ces to consider the case

where char F = 2, since otherwise we may take = 1 . 2

(5.12) Proposition. Let (V, q) be a nonsingular quadratic space of even dimension

n = 2m over a ¬eld F of characteristic 2 and let (e1 , . . . , en ) be a symplectic basis

of V for the bilinear form bq , i.e., a basis such that

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

and

Set

m

= •q e2i’1 — e2i’1 q(e2i ) + e2i — e2i q(e2i’1 ) + e2i’1 — e2i ∈ EndF (V ).

i=1

(1) The element satis¬es tr( s) = f (s) for all s ∈ Sym EndF (V ), σq .

(2) The characteristic polynomial of equals

Pc (X) = X 2 + X + q(e1 )q(e2 ) · · · X 2 + X + q(e2m’1 )q(e2m ) ,

hence

m m(m’1)

s2 ( ) = q(e2i’1 )q(e2i ) + .

i=1 2

60 I. INVOLUTIONS AND HERMITIAN FORMS

Proof : It su¬ces to check the equation for s of the form •q (v —v) with v ∈ V , since

n

these elements span Sym EndF (V ), σ . If v = i=1 ei ±i , we have bq (e2i’1 , v) = ±2i

and bq (e2i , v) = ±2i’1 , hence

•q (v — v) =

m

•q i=1 (e2i’1 — v)±2i q(e2i ) + (e2i — v)±2i’1 q(e2i’1 ) + (e2i’1 — v)±2i’1 .

It follows that

m

tr •q (v — v) = bq (v, e2i’1 )±2i q(e2i ) + bq (v, e2i )±2i’1 q(e2i’1 )

i=1

+ bq (v, e2i’1 )±2i’1

n m

±2 q(ei )

= + ±2i’1 ±2i = q(v).

i

i=1 i=1

Since q(v) = f —¦ •q (v — v), the ¬rst assertion is proved. Identifying EndF (V ) with

Mn (F ) by means of the basis (e1 , . . . , en ) maps to the matrix

«

0

1

1 q(e2i )

¬ ·

..

= where i = q(e

.

2i’1 ) 0

0 m

The characteristic polynomial of is the product of the characteristic polynomials

of 1 , . . . , m . This implies the second assertion.

(5.13) Example. Suppose that char F = 2 and let (1, u, v, w) be a quaternion

basis of a quaternion F -algebra Q = [a, b)F . In every quadratic pair (σ, f ) on Q,

the involution σ is symplectic. It is therefore the canonical involution γ. The space

Sym(Q, γ) is the span of 1, v, w, and Alt(Q, γ) = F . Since 1 = u + γ(u) and

TrdQ (u) = 1, the map f may be any linear form on Sym(Q, γ) such that f (1) = 1.

An element corresponding to f as in (??) is

= u + f (w)b’1 v + f (v)b’1 w.

(For a given f , the element is uniquely determined up to the addition of an element

in F .)

Quadratic pairs on tensor products. Let A1 , A2 be central simple F -

algebras. Given a quadratic pair (σ1 , f1 ) on A1 and an involution σ2 on A2 , we

aim to de¬ne a quadratic pair on the tensor product A1 —F A2 . If char F = 2, this

amounts to de¬ning an orthogonal involution on A1 —F A2 , and it su¬ces to take

σ1 — σ2 , assuming that σ2 is orthogonal, see (??). For the rest of this section, we

may thus focus on the case where char F = 2.

(5.14) Lemma. Let (A1 , σ1 ) and (A2 , σ2 ) be central simple algebras with involu-

tion of the ¬rst kind over a ¬eld F of characteristic 2.

(5.15) Symd(A1 , σ1 ) — Symd(A2 , σ2 ) =

Symd(A1 , σ1 ) — Sym(A2 , σ2 ) © Sym(A1 , σ1 ) — Symd(A2 , σ2 );

(5.16) Symd(A1 — A2 , σ1 — σ2 ) © Sym(A1 , σ1 ) — Sym(A2 , σ2 ) =

Symd(A1 , σ1 ) — Sym(A2 , σ2 ) + Sym(A1 , σ1 ) — Symd(A2 , σ2 );

§5. QUADRATIC FORMS 61

(5.17) Sym(A1 — A2 , σ1 — σ2 ) =

Symd(A1 — A2 , σ1 — σ2 ) + Sym(A1 , σ1 ) — Sym(A2 , σ2 ) .

Proof : Equation (??) is clear. For x1 ∈ A1 and s2 ∈ Sym(A2 , σ2 ),

x1 + σ1 (x1 ) — s2 = x1 — s2 + (σ1 — σ2 )(x1 — s2 ),

hence Symd(A1 , σ1 ) — Sym(A2 , σ2 ) ‚ Symd(A1 — A2 , σ1 — σ2 ). Similarly,

Sym(A1 , σ1 ) — Symd(A2 , σ2 ) ‚ Symd(A1 — A2 , σ1 — σ2 ),

hence the left side of (??) contains the right side. To prove the reverse inclusion,

consider x ∈ A1 — A2 . If x + (σ1 — σ2 )(x) ∈ Sym(A1 , σ1 ) — Sym(A2 , σ2 ), then

x + (σ1 — σ2 )(x) is invariant under σ1 — IdA2 , hence

x + (σ1 — IdA2 )(x) + (IdA1 — σ2 )(x) + (σ1 — σ2 )(x) = 0.

Therefore, the element u = x + (σ1 — IdA2 )(x) is invariant under IdA1 — σ2 , hence it

lies in A1 — Sym(A2 , σ2 ). Similarly, the element v = (σ1 — IdA2 )(x) + (σ1 — σ2 )(x)

is in Sym(A1 , σ1 ) — A2 . On the other hand, it is clear by de¬nition that u ∈

Symd(A1 , σ1 ) — A2 and v ∈ A1 — Symd(A2 , σ2 ), hence

u ∈ Symd(A1 , σ1 ) — Sym(A2 , σ2 ) and v ∈ Sym(A1 , σ1 ) — Symd(A2 , σ2 ).

Since x + (σ1 — σ2 )(x) = u + v, the proof of (??) is complete.

Since the left side of equation (??) obviously contains the right side, it suf-

¬ces to prove that both sides have the same dimension. Let ni = deg Ai , so that

dimF Sym(Ai , σi ) = 2 ni (ni + 1) and dimF Symd(Ai , σi ) = 1 ni (ni ’ 1) for i = 1, 2.

1

2

From (??), it follows that

dimF Symd(A1 , σ1 ) — Sym(A2 , σ2 ) + Sym(A1 , σ1 ) — Symd(A2 , σ2 ) =

1

’ 1)(n2 + 1) + 1 n1 n2 (n1 + 1)(n2 ’ 1) ’ 1 n1 n2 (n1 ’ 1)(n2 ’ 1)

4 n1 n2 (n1 4 4

= 1 n1 n2 (n1 n2 + n1 + n2 ’ 3).

4

Therefore, (??) yields

dimF Symd(A1 — A2 , σ1 — σ2 ) + Sym(A1 , σ1 ) — Sym(A2 , σ2 ) =

1

’ 1) + 1 n1 n2 (n1 + 1)(n2 + 1) ’ 1 n1 n2 (n1 n2 + n1 + n2 ’ 3)

2 n1 n2 (n1 n2 4 4

= 1 n1 n2 (n1 n2 + 1) = dimF Sym(A1 — A2 , σ1 — σ2 ).

2

(5.18) Proposition. Suppose that char F = 2. Let (σ1 , f1 ) be a quadratic pair

on a central simple F -algebra A1 and let (A2 , σ2 ) be a central simple F -algebra

with involution of the ¬rst kind. There is a unique quadratic pair (σ1 — σ2 , f1— ) on

A1 —F A2 such that

f1— (s1 — s2 ) = f1 (s1 ) TrdA2 (s2 )

for s1 ∈ Sym(A1 , σ1 ) and s2 ∈ Sym(A2 , σ2 ).

Proof : Since σ1 is symplectic, (??) shows that σ1 — σ2 is symplectic. To prove the

existence of a quadratic pair (σ1 —σ2 , f1— ) as above, we have to show that the values

that f1— is required to take on Symd(A1 — A2 , σ1 — σ2 ) because of the quadratic

pair conditions agree with the prescribed values on Sym(A1 , σ1 ) — Sym(A2 , σ2 ). In

view of the description of Symd(A1 — A2 , σ1 — σ2 ) © Sym(A1 , σ1 ) — Sym(A2 , σ2 )

in the preceding lemma, it su¬ces to consider the values of f1— on elements of the

62 I. INVOLUTIONS AND HERMITIAN FORMS

form x1 + σ1 (x1 ) — s2 = x1 — s2 + (σ1 — σ2 )(x1 — s2 ) or s1 — x2 + σ2 (x2 ) =

s1 — x2 + (σ1 — σ2 )(s1 — x2 ) with xi ∈ Ai and si ∈ Sym(Ai , σi ) for i = 1, 2. Since

(σ1 , f1 ) is a quadratic pair on A1 , we have

f1 x1 + σ1 (x1 ) TrdA2 (s2 ) = TrdA1 (x1 ) TrdA2 (s2 ) = TrdA1 —A2 (x1 — s2 ),

as required. For the second type of element we have

f1 (s1 ) TrdA2 x2 + σ2 (x2 ) = 0.

On the other hand, since σ1 is symplectic we have TrdA1 (s1 ) = 0, hence

TrdA1 —A2 (s1 — x2 ) = TrdA1 (s1 ) TrdA2 (x2 ) = 0.

Therefore,

f1 (s1 ) TrdA2 x2 + σ2 (x2 ) = TrdA1 —A2 (s1 — x2 )

for s1 ∈ Sym(A1 , σ1 ) and x2 ∈ A2 , and the existence of the quadratic pair (σ1 —

σ2 , f1— ) is proved.

Uniqueness is clear, since the values of the linear map f1— are determined on

the set Symd(A1 — A2 , σ1 — σ2 ) and on Sym(A1 , σ1 ) — Sym(A2 , σ2 ), and (??) shows

that these subspaces span Sym(A1 — A2 , σ1 — σ2 ).

(5.19) Example. Let (V1 , q1 ) be a nonsingular quadratic space of even dimen-

sion and let (V2 , b2 ) be a nonsingular symmetric bilinear space over a ¬eld F of

characteristic 2. Let (σ1 , f1 ) be the quadratic pair on A1 = EndF (V1 ) associ-

ated with q1 (see (??)) and let σ2 = σb2 denote the adjoint involution with re-

spect to b2 on A2 = EndF (V2 ). We claim that, under the canonical isomorphism

A1 — A2 = EndF (V1 — V2 ), the quadratic pair (σ1 — σ2 , f1— ) is associated with the

quadratic form q1 — b2 on V1 — V2 whose polar form is bq1 — b2 and such that

(q1 — b2 )(v1 — v2 ) = q1 (v1 )b2 (v2 , v2 ) for v1 ∈ V1 and v2 ∈ V2 .

∼

Indeed, letting •1 , •2 and • denote the standard identi¬cations V1 — V1 ’ ’

∼ ∼

EndF (V1 ), V2 — V2 ’ EndF (V2 ) and (V1 — V2 ) — (V1 — V2 ) ’ EndF (V1 — V2 )

’ ’

associated with the bilinear forms bq1 , b2 and bq1 — b2 , we have

•(v1 — v2 — v1 — v2 ) = •1 (v1 — v1 ) — •2 (v2 — v2 )

and

f1 •1 (v1 — v1 ) TrdA2 •2 (v2 — v2 ) = q1 (v1 )b2 (v2 , v2 ),

hence

f1— •(v1 — v2 — v1 — v2 ) = q1 — b2 (v1 — v2 ).

(5.20) Corollary. Let (A1 , σ1 ), (A2 , σ2 ) be central simple algebras with symplectic

involutions over a ¬eld F of arbitrary characteristic. There is a unique quadratic

pair (σ1 — σ2 , f— ) on A1 — A2 such that f— (s1 — s2 ) = 0 for all s1 ∈ Skew(A1 , σ1 ),

s2 ∈ Skew(A2 , σ2 ).

1

Proof : If char F = 2, the linear form f— which is the restriction of TrdA1 —A2 to

2

Sym(A1 — A2 , σ1 — σ2 ) satis¬es

1

f— (s1 — s2 ) = TrdA1 (s1 ) TrdA2 (s2 ) = 0

2

EXERCISES 63

for all s1 ∈ Skew(A1 , σ1 ), s2 ∈ Skew(A2 , σ2 ). Suppose next that char F = 2.

For any linear form f1 on Sym(A1 , σ1 ) we have f1 (s1 ) TrdA2 (s2 ) = 0 for all s1 ∈

Sym(A1 , σ1 ), s2 ∈ Sym(A2 , σ2 ), since σ2 is symplectic. Therefore, we may set

(σ1 — σ2 , f— ) = (σ1 — σ2 , f1— )

for any quadratic pair (σ1 , f1 ) on A1 . Uniqueness of f— follows from (??).

(5.21) De¬nition. The quadratic pair (σ1 — σ2 , f— ) of (??) is called the canonical

quadratic pair on A1 — A2 .

Exercises

1. Let A be a central simple algebra over a ¬eld F and ¬x a ∈ A. Show that there

is a canonical F -algebra isomorphism EndA (aA) EndA (Aa) which takes f ∈

EndA (aA) to the endomorphism f ∈ EndA (Aa) de¬ned by (xa)f = xf (a) for

x ∈ A, and the inverse takes g ∈ EndA (Aa) to the endomorphism g ∈ EndA (aA)

de¬ned by g(ax) = ag x for x ∈ A.

∼

Show that there is a canonical F -algebra isomorphism EndA (Aa)op ’ ’

EndAop (aop Aop ) which, for f ∈ EndA (Aa), maps f op to the endomorphism

˜ ˜

f de¬ned by f (mop ) = (mf )op . Therefore, there is a canonical isomorphism

op

EndAop (Aop aop ). Use it to identify (»k A)op = »k (Aop ), for

EndA (Aa)

k = 1, . . . , deg A.

2. Let Q be a quaternion algebra over a ¬eld F of arbitrary characteristic. Show

that the conjugation involution is the only linear map σ : Q ’ Q such that

σ(1) = 1 and σ(x)x ∈ F for all x ∈ F .

3. (Rowen-Saltman [?]) Let V be a vector space of dimension n over a ¬eld F and

let „ be an involution of the ¬rst kind on EndF (V ). Prove that „ is orthogonal

if and only if there exist n symmetric orthogonal10 idempotents in EndF (V ).

Find a similar characterization of the symplectic involutions on EndF (V ).

4. Let A be a central simple algebra with involution σ of the ¬rst kind. Show

that σ is orthogonal if and only if it restricts to the identity on a maximal ´tale

e

(commutative) subalgebra of A.

Hint: Extend scalars and use the preceding exercise.

5. Show that in a central simple algebra with involution, every left or right ideal is

generated by a symmetric element, unless the algebra is split and the involution

is symplectic.

6. (Albert) Let b be a symmetric, nonalternating bilinear form on a vector space V

over a ¬eld of characteristic 2. Show that V contains an orthogonal basis for b.

7. Let (ai )i=1,...,n2 be an arbitrary basis of a central simple algebra A, and let

(bi )i=1,...,n2 be the dual basis for the bilinear form TA , which means that

TrdA (ai bj ) = δij for i, j = 1, . . . , n2 . Show that the Goldman element of

n2

A is ai — b i .

i=1

Hint: Reduce by scalar extension to the split case and show that it su¬ces

to prove the assertion for the standard basis of Mn (F ).

10 Two idempotents e, f are called orthogonal if ef = f e = 0.

64 I. INVOLUTIONS AND HERMITIAN FORMS

8. Let (1, i, j, k) be a quaternion basis in a quaternion algebra Q of characteristic

di¬erent from 2. Show that the Goldman element in Q — Q is g = 1 (1 — 1 + i —

2

i’1 + j — j ’1 + k — k ’1 ). Let (1, u, v, w) be a quaternion basis in a quaternion

algebra Q of characteristic 2. Show that the Goldman element in Q — Q is

g = 1 — 1 + u — 1 + 1 — u + w — v ’1 + v ’1 — w.

9. Let K/F be a quadratic extension of ¬elds of characteristic di¬erent from 2,

and let a ∈ F — , b ∈ K — . Prove the “projection formula” for the norm of the

quaternion algebra (a, b)K :

NK/F (a, b)K ∼ a, NK/F (b) F

(where ∼ denotes Brauer-equivalence). Prove corresponding statements in

characteristic 2: if K/F is a separable quadratic extension of ¬elds and a ∈ F ,

b ∈ K — , c ∈ K, d ∈ F — ,

NK/F [a, b)K ∼ a, NK/F (b) and NK/F [c, d)K ∼ TK/F (c), d .

F F

Hint (when char F = 2): Let ι be the non-trivial automorphism of K/F

and let (1, i1 , j1 , k1 ) (resp. (1, i2 , j2 , k2 )) denote the usual quaternion basis of

a, ι(b) K = ι (a, b)K (resp. (a, b)K ). Let s be the switch map on ι (a, b)K —K

1

(a, b)K . Let u = 2 1 + a’1i1 — i2 + b’1 j1 — j2 ’ (ab)’1 k1 — k2 ∈ a, ι(b) K —K

(a, b)K . Show that s(u)u = 1, and that s = Int(u) —¦ s is a semi-linear auto-

morphism of order 2 of ι (a, b)K —K (a, b)K which leaves invariant i1 — 1, 1 — i2

and j1 — j2 . Conclude that the F -subalgebra of elements invariant under s is

Brauer-equivalent to a, NK/F (b) F . If b ∈ F , let v = 1 + u; if b ∈ F , pick

c ∈ K such that c2 ∈ F and let v = c + uι(c). Show that u = s (v)’1 v and

that Int(v) maps the subalgebra of s-invariant elements onto the subalgebra of

s -invariant elements.

10. (Knus-Parimala-Srinivas [?, Theorem 4.1]) Let A be a central simple algebra

over a ¬eld F , let V be an F -vector space and let

ρ : A —F A ’ EndF (V )

be an isomorphism of F -algebras. We consider V as a left A-module via av =

ρ(a — 1)(v) and identify EndA (A) = A, HomA (A, V ) = V by mapping every

homomorphism f to 1f . Moreover, we identify EndA (V ) = Aop by setting

op

va = ρ(1 — a)(v) for v ∈ V , a ∈ A

and HomA (V, A) = V — = HomF (V, F ) by mapping h ∈ HomA (V, A) to the

linear form

v ’ TrdA (v h ) for v ∈ V .

Let

EndA (A) HomA (A, V ) A V

B = EndA (A • V ) = = ;

V— Aop

HomA (V, A) EndA (V )

this is a central simple F -algebra which is Brauer-equivalent to A, by Proposi-

tion (??). Let γ = ρ(g) ∈ EndF (V ), where g ∈ A—F A is the Goldman element.

Show that

av b γ(v)

’

op

γ (f ) aop

t

fb

is an involution of orthogonal type of B.

EXERCISES 65

11. (Knus-Parimala-Srinivas [?, Theorem 4.2]) Let K/F be a separable quadratic

extension with nontrivial automorphism ι, let A be a central simple K-algebra,

let V be an F -vector space and let

ρ : NK/F (A) ’ EndF (V )

be an isomorphism of F -algebras. Set W = V —F K and write

ρ : A —K ιA ’ EndK (W )

for the isomorphism induced from ρ by extension of scalars. We consider W as

a left A-module via av = ρ(a — 1)(v). Let

EndA (A) HomA (A, W ) A W

B = EndA (A • W ) = =

W— ι op

HomA (W, A) EndA (W ) A

with identi¬cations similar to those in Exercise ??. The K-algebra B is Brauer

equivalent to A by (??). Show that

a w b ι(w)

’

ι op ι op

f b ι(f ) a

is an involution of the second kind of B.

12. Let A be a central simple F -algebra with involution σ of the ¬rst kind. Recall

the F -algebra isomorphism

σ— : A —F A ’ EndF A

de¬ned in the proof of Corollary (??) by

σ— (a — b)(x) = axσ(b) for a, b, x ∈ A.

Show that the image of the Goldman element g ∈ A—A under this isomorphism

is δσ where δ = +1 if σ is orthogonal and δ = ’1 if σ is symplectic. Use this

result to de¬ne canonical F -algebra isomorphisms

op

EndF A/ Alt(A, σ) EndF Sym(A, σ) if δ = +1,

2

sA op

EndF A/ Symd(A, σ) EndF Skew(A, σ) if δ = ’1,

EndF Alt(A, σ) if δ = +1,

»2 A

EndF Symd(A, σ) if δ = ’1.

13. (Saltman [?, Proposition 5]) Let A, B be central simple algebras of degrees m, n

over a ¬eld F . For every F -algebra homomorphism f : A ’ B, de¬ne a map

f : Aop —F B ’ B by f (aop — b) = f (a)b. Show that f is a homomorphism of

right Aop —F B-modules if B is endowed with the following Aop —F B-module

structure:

x —f (aop — b) = f (a)xb for a ∈ A and b, x ∈ B.

Show that ker f ‚ Aop —F B is a right ideal of reduced dimension mn ’ (n/m)

generated by the elements aop — 1 ’ 1 — f (a) for a ∈ A and that

Aop —F B = (1 — B) • ker f .

Conversely, show that every right ideal I ‚ Aop —F B of reduced dimension

mn ’ (n/m) such that

Aop —F B = (1 — B) • I

de¬nes an F -algebra homomorphism f : A ’ B such that I = ker f .

66 I. INVOLUTIONS AND HERMITIAN FORMS

Deduce from the results above that there is a natural one-to-one correspon-

dence between the set of F -algebra homomorphisms A ’ B and the rational

points in an open subset of the Severi-Brauer variety SBd (Aop —F B) where

d = mn ’ (n/m).

14. Let (A, σ) be a central simple algebra with involution of the ¬rst kind over

a ¬eld F of characteristic di¬erent from 2. Let a ∈ A be an element whose

minimal polynomial over F is separable. Show that a is symmetric for some

symplectic involution on A if and only if its reduced characteristic polynomial

is a square.

15. Let (A, σ) be a central simple algebra with involution of orthogonal type. Show

that every element in A is the product of two symmetric elements, one of which

is invertible.

Hint: Use (??).

16. Let V be a ¬nite dimensional vector space over a ¬eld F of arbitrary character-

istic and let a ∈ EndF (V ). Extend the notion of involution trace by using the

structure of V as an F [a]-module to de¬ne a nonsingular symmetric bilinear

form b : V — V ’ F such that a is invariant under the adjoint involution σb .

17. Let K/F be a separable quadratic extension of ¬elds with nontrivial automor-

phism ι. Let V be a vector space of dimension n over K and let b ∈ EndK (V )

be an endomorphism whose minimal polynomial has degree n and coe¬cients

in F . Show that V is a free F [b]-module of rank 1 and de¬ne a nonsingular her-

mitian form h : V —V ’ K such that b is invariant under the adjoint involution

σh .

Show that a matrix m ∈ Mn (K) is symmetric under some involution of the

second kind whose restriction to K is ι if and only if all the invariant factors

of m have coe¬cients in F .

18. Let q be a nonsingular quadratic form of dimension n over a ¬eld F , with n even

if char F = 2, and let a ∈ Mn (F ) be a matrix representing q, in the sense that

q(X) = X · a · X t . After identifying Mn (F ) with EndF (F n ) by mapping every

matrix m ∈ Mn (F ) to the endomorphism x ’ m · x, show that the quadratic

pair (σq , fq ) on EndF (F n ) associated to the quadratic map q : F n ’ F is the

same as the quadratic pair (σa , fa ) associated to a.

19. Let Q1 , Q2 be quaternion algebras with canonical involutions γ1 , γ2 over a ¬eld

F of arbitrary characteristic. Show that Alt(Q1 —Q2 , γ1 —γ2 ) = { q1 —1’1—q2 |

TrdQ1 (q1 ) = TrdQ2 (q2 ) }. If char F = 2, show that f (q1 — 1 + 1 — q2 ) =

TrdQ1 (q1 ) = TrdQ2 (q2 ) for all q1 — 1 + 1 — q2 ∈ Symd(Q1 — Q2 , γ1 — γ2 ) and for

all quadratic pairs (γ1 — γ2 , f ) on Q1 — Q2 .

20. The aim of this exercise is to give a description of the variety of quadratic

pairs on a central simple algebra in the spirit of (??). Let σ be a symplectic

involution on a central simple algebra A over a ¬eld F of characteristic 2 and

let σ— : A — A ’ EndF (A) be the isomorphism of Exercise ??. Let Iσ ‚ A — A

denote the right ideal corresponding to σ by (??) and let J ‚ A — A be the left

ideal generated by 1 ’ g, where g is the Goldman element. Denote by A0 ‚ A

the kernel of the reduced trace: A0 = { a ∈ A | TrdA (a) = 0 }. Show that

σ— (Iσ ) = Hom(A, A0 ), σ— (J ) = Hom A/ Sym(A, σ), A

and

σ— (J 0 ) = Hom A, Sym(A, σ) .

NOTES 67

Let now I ‚ A — A be a left ideal containing J , so that

σ— (I) = Hom(A/U, A)

for some vector space U ‚ Sym(A, σ). Show that σ— I·(1+g) = Hom(A/W, A),

where W = { a ∈ A | a + σ(a) ∈ U }, and deduce that σ— [I · (1 + g)]0 = Iσ if

and only if W = A0 , if and only if U © Symd(A, σ) = ker Trpσ .

Observe now that the set of rational points in SB s2 (Aop ) is in canonical

one-to-one correspondence with the set of left ideals I ‚ A — A containing J 0

and such that rdim I ’ rdim J = 1. Consider the subset U of such ideals which

satisfy [I · (1 + g)]0 • (1 — A) = A — A. Show that the map which carries every

’1

quadratic pair (σ, f ) on A to the left ideal σ— Hom(A/ ker f, A) de¬nes a

bijection from the set of quadratic pairs on A onto U.

Hint: For I ∈ U, the right ideal [I · (1 + g)]0 corresponds by (??) to

some symplectic involution σ. If U ‚ A is the subspace such that σ— (I) =

Hom(A/U, A), there is a unique quadratic pair (σ, f ) such that U = ker f .

21. Let (V1 , b1 ) and (V2 , b2 ) be vector spaces with nonsingular alternating forms

over an arbitrary ¬eld F . Show that there is a unique quadratic form q on

V1 — V2 whose polar form is b1 — b2 and such that q(v1 — v2 ) = 0 for all

v1 ∈ V1 , v2 ∈ V2 . Show that the canonical quadratic pair (σb1 — σb2 , f— ) on

EndF (V1 ) — EndF (V2 ) = EndF (V1 — V2 ) is associated with the quadratic form

q.

22. Let (A, σ) be a central simple algebra with involution of the ¬rst kind over an

arbitrary ¬eld F . Assume σ is symplectic if char F = 2. By (??), there exists

an element ∈ A such that +σ( ) = 1. De¬ne a quadratic form q(A,σ) : A ’ F

by

q(A,σ) (x) = TrdA σ(x) x for x ∈ A.

Show that this quadratic form does not depend on the choice of such that

+ σ( ) = 1. Show that the associated quadratic pair on EndF (A) corresponds

to the canonical quadratic pair (σ — σ, f— ) on A — A under the isomorphism

∼

σ— : A — A ’ EndF (A) such that σ— (a — b)(x) = axσ(b) for a, b, x ∈ A.

’

Notes

§??. Additional references for the material in this section include the classical

books of Albert [?], Deuring [?] and Reiner [?]. For Severi-Brauer varieties, see

Artin™s notes [?]. A self-contained exposition of Severi-Brauer varieties can be

found in Jacobson™s book [?, Chapter 3].

§??. The ¬rst systematic investigations of involutions of central simple alge-

bras are due to Albert. His motivation came from the theory of Riemann matrices:

on a Riemann surface of genus g, choose a basis (ω± )1¤±¤g of the space of holo-

morphic di¬erentials and a system of closed curves (γβ )1¤β¤2g which form a basis

of the ¬rst homology group, and consider the matrix of periods P = ( γβ ω± ). This

is a complex g — 2g matrix which satis¬es Riemann™s period relations: there exists

t

a nonsingular alternating matrix C ∈ M2g (Q) such that P CP t = 0 and iP CP is

positive de¬nite hermitian. The study of correspondences on the Riemann surface

leads one to consider the matrices M ∈ M2g (Q) for which there exists a matrix

68 I. INVOLUTIONS AND HERMITIAN FORMS

K ∈ Mg (C) such that KP = P M . Following Weyl™s simpler formulation [?],

P

one considers the matrix W = ∈ M2g (C) and the so-called Riemann matrix

P

’iIg 0

R = W ’1 W ∈ M2g (R). The matrices M such that KP = P M for

0 iIg

some K ∈ Mg (C) are exactly those which commute with R. They form a subal-

gebra of M2g (Q) known as the multiplication algebra. As observed by Rosati [?],

this algebra admits the involution X ’ C ’1 X t C (see Weyl [?]). Albert completely

determined the structure of the multiplication algebra in three papers in the An-

nals of Mathematics in 1934“1935. An improved version, [?], see also [?], laid the

foundations of the theory of simple algebras with involutions.

Corollary (??) was observed independently by several authors: see Tits [?,

Proposition 3], Platonov [?, Proposition 5] and Rowen [?, Proposition 5.3].

The original proof of Albert™s theorem on quaternion algebras with involution

of the second kind (??) is given in [?, Theorem 10.21]. This result will be put in a

broader perspective in §??: the subalgebra Q0 is the discriminant algebra of (Q, σ)

(see (??)).

There is an extensive literature on Lie and Jordan structures in rings with

involution; we refer the reader to Herstein™s monographs [?] and [?]. In particu-

lar, Lemma (??) can be proved by ring-theoretic arguments which do not involve

scalar extension (and therefore hold for more general simple rings): see Herstein [?,

Theorem 2.2, p. 28]. In the same spirit, extension of Lie isomorphisms has been

investigated for more general rings: see11 Martindale [?], Rosen [?] and Beidar-

Martindale-Mikhalev [?].

§??. Part (??) of Theorem (??) is due to Albert [?, Theorem 10.19]. Albert

also proved part (??) for crossed products of a special kind: Albert assumes in [?,

Theorem 10.16] the existence of a splitting ¬eld of the form L —F K where L is

Galois over F . Part (??) was stated in full generality by Riehm [?] and proved by

Scharlau [?] (see also [?, §8.9]). In order to see that every central simple algebra

which is Brauer-equivalent to an algebra with involution also has an involution, it

is not essential to use (??): see Albert [?, Theorem 10.12] or Scharlau [?, Corol-

lary 8.8.3]. By combining this result with Exercises ?? and ??, we obtain an

alternate proof of Theorem (??).

§??. If E and E are Brauer-equivalent central simple F -algebras, then Morita

theory yields an E-E -bimodule P and an E -E-bimodule Q such that P —E Q E

and Q —E P E . If M is a right E-module, then there is a natural isomorphism

EndE (M ) = EndE (M —E P ).

Therefore, if E (hence also E ) has an involution, (??) yields one-to-one correspon-

dences between hermitian or skew-hermitian forms on M (up to a central factor),

involutions on EndE (M ) = EndE (M —E P ) and hermitian or skew-hermitian forms

on M —E P (up to a central factor). The correspondence between hermitian forms

can be made more precise and explicit; it is part of a Morita equivalence between

categories of hermitian modules which is discussed in Knus™ book [?, §1.9].

The notion of involution trace was introduced by Fr¨hlich and McEvett [?,

o

§7]. Special cases of the extension theorem (??) have been proved by Rowen [?,

Corollary 5.5] and by Lam-Leep-Tignol [?, Proposition 5.1]. Kneser™s theorem has

11 We are indebted to W. S. Martindale III for references to the recent literature.

NOTES 69

been generalized by Held and Scharlau [?] to the case where the subalgebra is

semisimple. (A particular case of this situation had also been considered by Kneser

in [?, p. 37].)

The existence of involutions for which a given element is symmetric or skew-

symmetric is discussed in Shapiro [?], which also contains an extensive discussion

of the literature.

§??. The de¬nition of quadratic pair in (??) is new. While involutions on

arbitrary central simple algebras have been related to hermitian forms in §??, the

relation between quadratic pairs and quadratic forms is described only in the split

case. The nonsplit case requires an extension of the notion of quadratic form.

For quaternion algebras such an extension was given by Seip-Hornix [?]. Tits [?]

de¬nes a (generalized) quadratic form as an element in the factor group A/ Alt(A, „ )

(compare with (??)); a more geometric viewpoint which also extends this notion

further, was proposed by Bak [?] (see also for instance Hahn-O™Meara [?, 5.1C],

Knus [?, Ch. 1, §5] or Scharlau [?, Ch. 7, §3]).

70 I. INVOLUTIONS AND HERMITIAN FORMS

CHAPTER II

Invariants of Involutions

In this chapter, we de¬ne various kinds of invariants of central simple algebras

with involution (or with quadratic pair) and we investigate their basic properties.

The invariants considered here are analogues of the classical invariants of quadratic

forms: the Witt index, the discriminant, the Cli¬ord algebra and the signature.

How far the analogy can be pushed depends of course on the nature of the in-

volution: an index is de¬ned for arbitrary central simple algebras with involution

or quadratic pair, but the discriminant is de¬ned only for orthogonal involutions

and quadratic pairs, and the Cli¬ord algebra just for quadratic pairs. The Cli¬ord

algebra construction actually splits into two parts: while it is impossible to de¬ne a

full Cli¬ord algebra for quadratic pairs, the even and the odd parts of the Cli¬ord

algebra can be recovered in the form of an algebra and a bimodule. For unitary

involutions, the notion of discriminant turns out to lead to a rich structure: we as-

sociate in §?? a discriminant algebra (with involution) to every unitary involution

on a central simple algebra of even degree. Finally, signatures can be de¬ned for

arbitrary involutions through the associated trace forms. These trace forms also

have relations with the discriminant or discriminant algebra. They yield higher

invariants for algebras with unitary involution of degree 3 in Chapter ?? and for

Jordan algebras in Chapter ??.

The invariants considered in this chapter are produced by various techniques.

The index is derived from a representation of the algebra with involution as the

endomorphism algebra of some hermitian or skew-hermitian space over a division

algebra, while the de¬nitions of discriminant and Cli¬ord algebra are based on

the fact that scalar extension reduces the algebra with quadratic pair to the endo-

morphism algebra of a quadratic space. We show that the discriminant and even

Cli¬ord algebra of the corresponding quadratic form can be de¬ned in terms of

the adjoint quadratic pair, and that the de¬nitions generalize to yield invariants

of arbitrary quadratic pairs. A similar procedure is used to de¬ne the discrimi-

nant algebra of a central simple algebra of even degree with unitary involution.

Throughout most of this chapter, our method of investigation is thus based on

scalar extension: after specifying the de¬nitions “rationally” (i.e., over an arbitrary

base ¬eld), the main properties are proven by extending scalars to a splitting ¬eld.

This method contrasts with Galois descent, where constructions over a separable

closure are shown to be invariant under the action of the absolute Galois group and

are therefore de¬ned over the base ¬eld.

§6. The Index

According to (??), every central simple F -algebra with involution (A, σ) can be

represented as EndD (V ), σh for some division algebra D, some D-vector space V

and some nonsingular hermitian form h on V . Since this representation is essentially

71

72 II. INVARIANTS OF INVOLUTIONS

unique, it is not di¬cult to check that the Witt index w(V, h) of the hermitian space

(V, h), de¬ned as the maximum of the dimensions of totally isotropic subspaces

of V , is an invariant of (A, σ). In this section, we give an alternate de¬nition of this

invariant which does not depend on a representation of (A, σ) as EndD (V ), σh ,

and we characterize the involutions which can be represented as adjoint involutions

with respect to a hyperbolic form. We de¬ne a slightly more general notion of

index which takes into account the Schur index of the algebra. A (weak) analogue

of Springer™s theorem on odd degree extensions is discussed in the ¬nal subsection.

6.A. Isotropic ideals. Let (A, σ) be a central simple algebra with involution

(of any kind) over a ¬eld F of arbitrary characteristic.

(6.1) De¬nition. For every right ideal I in A, the orthogonal ideal I ⊥ is de¬ned

by

I ⊥ = { x ∈ A | σ(x)y = 0 for y ∈ I }.

It is clearly a right ideal of A, which may alternately be de¬ned as the annihilator

of the left ideal σ(I):

I ⊥ = σ(I)0 .

(6.2) Proposition. Suppose the center of A is a ¬eld. For every right ideal I ‚ A,

rdim I + rdim I ⊥ = deg A and I ⊥⊥ = I. Moreover, if (A, σ) = EndD (V ), σh and

I = HomD (V, W ) for some subspace W ‚ V , then

I ⊥ = HomD (V, W ⊥ ).

Proof : Since rdim σ(I) = rdim I, the ¬rst relation follows from the corresponding

statement for annihilators (??). This ¬rst relation implies that rdim I ⊥⊥ = rdim I.

Since the inclusion I ‚ I ⊥⊥ is obvious, we get I = I ⊥⊥ . Finally, suppose I =

HomD (V, W ) for some subspace W ‚ V . For every f ∈ EndD (V ), g ∈ I we have

g(y) ∈ W and h f (x), g(y) = h x, σ(f ) —¦ g(y) for x, y ∈ V .

Therefore, σ(f ) —¦ g = 0 if and only if f (x) ∈ W ⊥ , hence

I ⊥ = HomD (V, W ⊥ ).

A similar result holds if (A, σ) = (E — E op , µ), where E is a central simple F -

algebra and µ is the exchange involution, although the reduced dimension of a right

op

ideal is not de¬ned in this case. Every right ideal I ‚ A has the form I = I1 — I2

where I1 (resp. I2 ) is a right (resp. left) ideal in E, and

op

(I1 — I2 )⊥ = I2 — (I1 )op .

0 0

Therefore, by (??),

dimF I ⊥ = dimF A ’ dimF I and I ⊥⊥ = I

for every right ideal I ‚ A.

In view of the proposition above, the following de¬nitions are natural:

(6.3) De¬nitions. Let (A, σ) be a central simple algebra with involution over a

¬eld F . A right ideal I ‚ A is called isotropic (with respect to the involution

σ) if I ‚ I ⊥ . This inclusion implies rdim I ¤ rdim I ⊥ , hence (??) shows that

rdim I ¤ 1 deg A for every isotropic right ideal.

2

§6. THE INDEX 73

The algebra with involution (A, σ)”or the involution σ itself”is called isotropic

if A contains a nonzero isotropic ideal.

If the center of A is a ¬eld, the index of the algebra with involution (A, σ) is

de¬ned as the set of reduced dimensions of isotropic right ideals:

ind(A, σ) = { rdim I | I ‚ I ⊥ }.

Since the (Schur) index of A divides the reduced dimension of every right ideal,

the index ind(A, σ) is a set of multiples of ind A. More precisely, if (A, σ)

EndD (V ), σh for some hermitian or skew-hermitian space (V, h) over a division

algebra D and w(V, h) denotes the Witt index of (V, h), then ind A = deg D and

ind(A, σ) = { deg D | 0 ¤ ¤ w(V, h) }

since (??) shows that the isotropic ideals of EndD (V ) are of the form HomD (V, W )

with W a totally isotropic subspace of V , and rdim HomD (V, W ) = dimD W deg D.

Thus, if ind(A, σ) contains at least two elements, the di¬erence between two con-

secutive integers in ind(A, σ) is ind A. If ind(A, σ) has only one element, then

ind(A, σ) = {0}, which means that (A, σ) is anisotropic; this is the case for in-

stance when A is a division algebra.

We extend the de¬nition of ind(A, σ) to the case where the center of A is

(E — E op , µ) for some central simple F -algebra E where µ

F — F ; then (A, σ)

is the exchange involution, and we de¬ne ind(A, σ) as the set of multiples of the

1

Schur index of E in the interval [0, 2 deg E]:

deg E

ind(A, σ) = ind E 0¤ ¤ .

2 ind E

(Note that deg E = deg(A, σ) is not necessarily even).

(6.4) Proposition. For every ¬eld extension L/F ,

ind(A, σ) ‚ ind(AL , σL ).

Proof : This is clear if the center of AL = A —F L is a ¬eld, since scalar extensions

preserve the reduced dimension of ideals and since isotropic ideals remain isotropic

under scalar extension. If the center of A is not a ¬eld, the proposition is also clear.

Suppose the center of A is a ¬eld K properly containing F and contained in L;

then by (??),

(A —K L) — (A —K L)op , µ .

(AL , σL )

Since the reduced dimension of every right ideal in A is a multiple of ind A and

since ind(A —K L) divides ind A, the reduced dimension of every isotropic ideal of

(A, σ) is a multiple of ind(A —K L). Moreover, the reduced dimension of isotropic

ideals is bounded by 1 deg A, hence ind(A, σ) ‚ ind(AL , σL ).

2

For central simple algebras with a quadratic pair, we de¬ne isotropic ideals by

a more restrictive condition.

(6.5) De¬nition. Let (σ, f ) be a quadratic pair on a central simple algebra A over

a ¬eld F . A right ideal I ‚ A is called isotropic with respect to the quadratic pair

(σ, f ) if the following two conditions hold:

(1) σ(x)y = 0 for all x, y ∈ I.

(2) f (x) = 0 for all x ∈ I © Sym(A, σ).

74 II. INVARIANTS OF INVOLUTIONS

The ¬rst condition means that I ‚ I ⊥ , hence isotropic ideals for the quadratic

pair (σ, f ) are also isotropic for the involution σ. Condition (??) implies that every

x ∈ I ©Sym(A, σ) satis¬es x2 = 0, hence also TrdA (x) = 0. If char F = 2, the map f

1

is the restriction of 2 TrdA to Sym(A, σ), hence condition (??) follows from (??).

Therefore, in this case the isotropic ideals for (σ, f ) are exactly the isotropic ideals

for σ.

The algebra with quadratic pair (A, σ, f )”or the quadratic pair (σ, f ) itself”is

called isotropic if A contains a nonzero isotropic ideal for the quadratic pair (σ, f ).

(6.6) Example. Let (V, q) be an even-dimensional quadratic space over a ¬eld F

of characteristic 2, and let (σq , fq ) be the corresponding quadratic pair on EndF (V )

(see (??)). A subspace W ‚ V is totally isotropic for q if and only if the right ideal

HomF (V, W ) ‚ EndF (V ) is isotropic for (σq , fq ).

Indeed, the standard identi¬cation •q (see (??)) identi¬es HomF (V, W ) with

W —F V . The elements in W — V which are invariant under the switch involution

(which corresponds to σq under •q ) are spanned by elements of the form w —w with

w ∈ W , and (??) shows that fq —¦ •q (w — w) = q(w). Therefore, the subspace W

is totally isotropic with respect to q if and only if fq vanishes on HomF (V, W ) ©

Sym EndF (V ), σq . Proposition (??) shows that this condition also implies that

HomF (V, W ) is isotropic with respect to the involution σq .

Mimicking (??), we de¬ne the index of a central simple algebra with quadratic

pair (A, σ, f ) as the set of reduced dimensions of isotropic ideals:

ind(A, σ, f ) = { rdim I | I is isotropic with respect to (σ, f ) }.

6.B. Hyperbolic involutions. Let E be a central simple algebra with invo-

lution θ (of any kind) and let U be a ¬nitely generated right E-module. As in (??),

we use the involution θ to endow the dual of U with a structure of right E-module

θ—

U . For » = ±1, de¬ne

h» : ( θ U — • U ) — ( θ U — • U ) ’ E

by

h» (θ • + x, θ ψ + y) = •(y) + »θ ψ(x)

for •, ψ ∈ U — and x, y ∈ U . Straightforward computations show that h1 (resp.

h’1 ) is a nonsingular hermitian (resp. alternating) form on θ U — • U with respect

to the involution θ on E. The hermitian or alternating module (θ U — • U, h» ) is

denoted H» (U ). A hermitian or alternating module (M, h) over (E, θ) is called

hyperbolic if it is isometric to some H» (U ).

The following proposition characterizes the adjoint involutions with respect to

hyperbolic forms:

(6.7) Proposition. Let (A, σ) be a central simple algebra with involution (of any

kind ) over a ¬eld F of arbitrary characteristic. Suppose the center K of A is a

¬eld. The following conditions are equivalent:

(1) for every central simple K-algebra E Brauer-equivalent to A and every involu-

tion θ on E such that θ|K = σ|K , every hermitian or skew-hermitian module (M, h)

over (E, θ) such that (A, σ) EndE (M ), σh is hyperbolic;

(2) there exists a central simple K-algebra E Brauer-equivalent to A, an involution

θ on E such that θ|K = σ|K and a hyperbolic hermitian or skew-hermitian module

(M, h) over (E, θ) such that (A, σ) EndE (M ), σh ;

§6. THE INDEX 75

(3) 1 deg A ∈ ind(A, σ) and further, if char F = 2, the involution σ is either sym-

2

plectic or unitary;

(4) there is an idempotent e ∈ A such that σ(e) = 1 ’ e.

Proof : (??) ’ (??) This is clear.

(??) ’ (??) In the hyperbolic module M = θ U — • U , the submodule U is

totally isotropic, hence the same argument as in (??) shows that the right ideal

HomE (M, U ) is isotropic in EndE (M ). Moreover, since rdim U = 1 rdim M , we

2

have rdim HomE (M, U ) = 2 deg EndE (M ). Therefore, (??) implies that 1 deg A ∈

1

2

ind(A, σ). If char F = 2, the hermitian form h+1 = h’1 is alternating, hence σ is

symplectic if the involution θ on E is of the ¬rst kind, and is unitary if θ is of the

second kind.

1

(??) ’ (??) Let I ‚ A be an isotropic ideal of reduced dimension 2 deg A.

By (??), there is an idempotent f ∈ A such that I = f A. Since I is isotropic, we

have σ(f )f = 0. We shall modify f into an idempotent e such that I = eA and

σ(e) = 1 ’ e.

The ¬rst step is to ¬nd u ∈ A such that σ(u) = 1 ’ u. If char F = 2, we

may choose u = 1/2; if char F = 2 and σ is symplectic, the existence of u follows

from (??); if char F = 2 and σ is unitary, we may choose u in the center K of A,

since K/F is a separable quadratic extension and the restriction of σ to K is the

nontrivial automorphism of K/F .

We next set e = f ’f uσ(f ) and proceed to show that this element satis¬es (??).

Since f 2 = f and σ(f )f = 0, it is clear that e is an idempotent and σ(e)e = 0.

Moreover, since σ(u) + u = 1,

eσ(e) = f σ(f ) ’ f uσ(f ) ’ f σ(u)σ(f ) = 0.

Therefore, e and σ(e) are orthogonal idempotents; it follows that e + σ(e) also is an

idempotent, and e + σ(e) A = eA • σ(e)A. To complete the proof of (??), observe

that e ∈ f A and f = ef ∈ eA, hence eA = f A = I. Since rdim I = 1 deg A, 2

1

it follows that dimF eA = dimF σ(e)A = 2 dimF A, hence e + σ(e) A = A and

therefore e + σ(e) = 1.

(??) ’ (??) Let E be a central simple K-algebra Brauer-equivalent to A and

θ be an involution on E such that θ|K = σ|K . Let also (M, h) be a hermitian or

skew-hermitian module over (E, θ) such that (A, σ) EndE (M ), σh . Viewing

this isomorphism as an identi¬cation, we may ¬nd for every idempotent e ∈ A a

pair of complementary submodules U = im e, W = ker e in M such that e is the

projection M ’ U parallel to W ; then 1 ’ e is the projection M ’ W parallel

to U and σ(e) is the projection M ’ W ⊥ parallel to U ⊥ . Therefore, if σ(e) = 1 ’ e

∼

we have U = U ⊥ and W = W ⊥ . We then de¬ne an isomorphism W ’ θ U — by ’

mapping w ∈ W to θ •w where •w ∈ U — is de¬ned by •w (u) = h(w, u) for u ∈ U .

∼

This isomorphism extends to an isometry M = W • U ’ H» (U ) where » = +1 if

’

h is hermitian and » = ’1 if h is skew-hermitian.

(6.8) De¬nition. A central simple algebra with involution (A, σ) over a ¬eld F ”

or the involution σ itself”is called hyperbolic if either the center of A is isomorphic

to F — F or the equivalent conditions of (??) hold. If the center is F — F , then

the idempotent e = (1, 0) satis¬es σ(e) = 1 ’ e; therefore, in all cases (A, σ) is

hyperbolic if and only if A contains an idempotent e such that σ(e) = 1 ’ e. This

condition is also equivalent to the existence of an isotropic ideal I of dimension

1

dimF I = 2 dimF A if char F = 2, but if char F = 2 the extra assumption that σ is

76 II. INVARIANTS OF INVOLUTIONS

symplectic or unitary is also needed. For instance, if A = M2 (F ) (with char F = 2)

1

and σ is the transpose involution, then 2 deg A ∈ ind(A, σ) since the right ideal

I = { x x | x, y ∈ F } is isotropic, but (A, σ) is not hyperbolic since σ is of

yy

orthogonal type.

Note that (A, σ) may be hyperbolic without A being split; indeed we may have

1 1

ind(A, σ) = {0, 2 deg A}, in which case the index of A is 2 deg A.

From any of the equivalent characterizations in (??), it is clear that hyper-

bolic involutions remain hyperbolic over arbitrary scalar extensions. Characteriza-

tion (??) readily shows that hyperbolic involutions are also preserved by transfer.

Explicitly, consider the situation of §??: Z/F is a ¬nite extension of ¬elds, E is

a central simple Z-algebra and T is a central simple F -algebra contained in E, so

that E = T —F C where C is the centralizer of T in E. Let θ be an involution on E

which preserves T and let s : E ’ T be an involution trace. Recall from (??) that

for every hermitian or skew-hermitian module (M, h) over (E, θ) there is a transfer

M, s— (h) which is a hermitian or skew-hermitian module over (T, θ).

(6.9) Proposition. If h is hyperbolic, then s— (h) is hyperbolic.

Proof : If h is hyperbolic, (??) yields an idempotent e ∈ EndE (M ) such that

σh (e) = 1 ’ e. By (??), the involution σs— (h) on EndT (M ) extends σh , hence

e also is an idempotent of EndT (M ) such that σs— (h) (e) = 1 ’ e. Therefore, s— (h)

is hyperbolic.

In the same spirit, we have the following transfer-type result:

(6.10) Corollary. Let (A, σ) be a central simple algebra with involution (of any

kind ) over a ¬eld F and let L/F be a ¬nite extension of ¬elds. Embed L ’

EndF (L) by mapping x ∈ L to multiplication by x, and let ν be an involution on

EndF (L) leaving the image of L elementwise invariant. If (AL , σL ) is hyperbolic,

then A —F EndF (L), σ — ν is hyperbolic.

Proof : The embedding L ’ EndF (L) induces an embedding

(AL , σL ) = (A —F L, σ — IdL ) ’ A —F EndF (L), σ — ν .

The same argument as in the proof of (??) applies. (Indeed, (??) may be regarded

as the special case of (??) where C = Z = L: see ??).

(6.11) Example. Interesting examples of hyperbolic involutions can be obtained

as follows: let (A, σ) be a central simple algebra with involution (of any kind) over a

¬eld F of characteristic di¬erent from 2 and let u ∈ Sym(A, σ)©A— be a symmetric

unit in A. De¬ne an involution νu on M2 (A) by

’σ(a21 )u’1

a11 a12 σ(a11 )

νu =

’uσ(a12 ) uσ(a22 )u’1

a21 a22

for a11 , a12 , a21 , a22 ∈ A, i.e.,

10

νu = Int —¦ (σ — t)

0 ’u

where t is the transpose involution on M2 (F ).

Claim. The involution νu is hyperbolic if and only if u = vσ(v) for some v ∈ A.

§6. THE INDEX 77

Proof : Let D be a division algebra Brauer-equivalent to A and let θ be an involution

on D of the same type as σ. We may identify (A, σ) = EndD (V ), σh for some

hermitian space (V, h) over (D, σ), by (??). De¬ne a hermitian form h on V by

h (x, y) = h u’1 (x), y = h x, u’1 (y) for x, y ∈ V .

Under the natural identi¬cation M2 (A) = EndD (V • V ), the involution νu is the

adjoint involution with respect to h⊥(’h ). The form h⊥(’h ) is hyperbolic if and

only if (V, h) is isometric to (V, h ). Therefore, by Proposition (??), νu is hyperbolic

if and only if (V, h) is isometric to (V, h ). This condition is also equivalent to the

existence of a ∈ A— such that h (x, y) = h a(x), a(y) for all x, y ∈ V ; in view of

the de¬nition of h , this relation holds if and only if u = a’1 σ(a’1 ).