de¬nition of σb (see equation (??) in the introduction to this chapter),

b —¦ σ(f ) — 1 ’ 1 — f = 0 for f ∈ EndF (V ).

Since Iσ is generated as a right ideal by the elements f — 1 ’ 1 — σ(f ), and since

g —¦ (f1 — f2 ) = (f2 — f1 ) for f1 , f2 ∈ EndF (V ), it follows that

Iσ ‚ { h ∈ EndF (V — V ) | b —¦ g —¦ h = 0 } = HomF V — V, ker(b —¦ g) .

Dimension count shows that the inclusion is an equality.

As observed in the proof of Proposition (??), J 0 = HomF V — V, ker(Id ’ g) ,

hence the inclusion J 0 ‚ HomF V — V, ker(b —¦ g) holds if and only if ker(Id ’ g) ‚

ker(b —¦ g). Since ker(Id ’ g) is generated by elements of the form v — v, for v ∈ V ,

this condition holds if and only if b is alternating or, equivalently, σ is symplectic.

On the other hand, Jr ‚ HomF V — V, ker(b —¦ g) if and only if

b —¦ g —¦ (Id ’ g)(v1 — v2 ) = 0 for v1 , v2 ∈ V .

Since the left side is equal to b(v2 , v1 ) ’ b(v1 , v2 ), this relation holds if and only if

b is symmetric. Therefore, σ is orthogonal if and only if the corresponding ideal

contains Jr but not J 0 .

(3.9) Remark. If char F = 2, then J 0 = (1 + g) · (A —F A). Indeed, 1 + g ∈ J 0

since (1 ’ g)(1 + g) = 1 ’ g 2 = 0; on the other hand, if x ∈ J 0 then (1 ’ g)x = 0,

hence x = gx = (1 + g)x/2. Therefore, an involution σ is orthogonal if and only if

the corresponding ideal Iσ contains 1 ’ g; it is symplectic if and only Iσ contains

1 + g.

Let deg A = n. The right ideals I ‚ A —F A such that I • (1 — A) = A —F A

then have reduced dimension n2 ’ 1 and form an a¬ne open subvariety

U ‚ SBn2 ’1 (A —F A).

(It is the a¬ne open set denoted by U1—A in the proof of Theorem (??).)

On the other hand, since rdim Jr = n(n ’ 1)/2 by (??) and s2 A = EndA—A (Jr )

0

by de¬nition, Proposition (??) shows that the right ideals of reduced dimension

n2 ’ 1 in A —F A which contain Jr form a closed subvariety So ‚ SBn2 ’1 (A —F A)

isomorphic to SBm (s2 A) where

m = (n2 ’ 1) ’ 2 n(n ’ 1) = 1 n(n + 1) ’ 1 = deg s2 A ’ 1.

1

2

36 I. INVOLUTIONS AND HERMITIAN FORMS

By (??), this variety is also isomorphic to SB (s2 A)op .

Similarly, for n > 1 we write Ss ‚ SBn2 ’1 (A —F A) for the closed subvariety of

right ideals of reduced dimension n2 ’1 which contain J 0 . This variety is isomorphic

to SB 1 n(n’1)’1 (»2 A) and to SB (»2 A)op .

2

With this notation, Theorem (??) can be rephrased as follows:

(3.10) Corollary. There are natural one-to-one correspondences between involu-

tions of orthogonal type on A and the rational points on the variety U © S o , and, if

deg A > 1, between involutions of symplectic type on A and the rational points on

the variety U © Ss .

Inspection of the split case shows that the open subvariety U © So ‚ So is

nonempty, and that U © Ss is nonempty if and only if deg A is even.

We may now complete the proof of part (??) of Theorem (??). We ¬rst observe

that if F is ¬nite, then A is split since the Brauer group of a ¬nite ¬eld is trivial

(see for instance Scharlau [?, Corollary 8.6.3]), hence A has involutions of the ¬rst

kind. We may thus assume henceforth that the base ¬eld F is in¬nite.

Suppose that A —F A is split. Then so are s2 A and »2 A and the varieties

SB (s2 A)op and SB (»2 A)op (when deg A > 1) are projective spaces. It follows

that the rational points are dense in So and Ss . Therefore, U © So has rational

points, so A has involutions of orthogonal type. If deg A is even, then U © Ss also

has rational points8 , so A also has involutions of symplectic type.

(3.11) Remark. Severi-Brauer varieties and density arguments can be avoided in

the proof above by reducing to the case of division algebras: if A —F A is split,

then D —F D is also split, if D is the division algebra Brauer-equivalent to A. Let

I ‚ D —F D be a maximal right ideal containing 1 ’ g. Then dim I = d2 ’ d, where

d = dimF D, and I intersects 1 — D trivially, since it does not contain any invertible

element. Therefore, dimension count shows that D —F D = I • (1 — D). It then

follows from (??) that D has an (orthogonal) involution of the ¬rst kind which we

denote by . An involution — of the ¬rst kind is then de¬ned on Mr (D) by letting

act entrywise on Mr (D) and setting

a— = a t for a ∈ Mr (D).

This involution is transported to A by the isomorphism A Mr (D).

3.B. Existence of involutions of the second kind. Before discussing in-

volutions of the second kind, we recall the construction of the norm of a central

simple algebra in the particular case of interest in this section.

The norm (or corestriction) of central simple algebras. Let K/F be a

¬nite separable ¬eld extension. For every central simple K-algebra A, there is a

central simple F -algebra NK/F (A) of degree (deg A)[K:F ] , called the norm of A,

de¬ned so as to induce a homomorphism of Brauer groups

NK/F : Br(K) ’ Br(F )

which corresponds to the corestriction map in Galois cohomology.

In view of Theorem (??), we shall only discuss here the case where K/F is a

quadratic extension, referring to Draxl [?, §8] or Rowen [?, §7.2] for a more general

treatment along similar lines.

8 If

deg A = 2, the variety Ss has only one point, namely J 0 ; this is a re¬‚ection of the fact

that quaternion algebras have a unique symplectic involution, see (??).

§3. EXISTENCE OF INVOLUTIONS 37

The case of quadratic extensions is particularly simple in view of the fact that

separable quadratic extensions are Galois. Let K/F be such an extension, and let

Gal(K/F ) = {IdK , ι}

be its Galois group. For any K-algebra A, we de¬ne the conjugate algebra

ι

A = { ιa | a ∈ A }

with the following operations:

ι

a + ι b = ι (a + b) ιι

a b = ι (ab) ι

(±a) = ι(±)ιa

for a, b ∈ A and ± ∈ K. The switch map

s : ιA —K A ’ ιA —K A

de¬ned by

s(ιa — b) = ι b — a

is ι-semilinear over K and is an F -algebra automorphism.

(3.12) De¬nition. The norm NK/F (A) of the K-algebra A is the F -subalgebra

of ιA —K A elementwise invariant under the switch map:

NK/F (A) = { u ∈ ιA —K A | s(u) = u }.

Of course, the same construction can be used to de¬ne the norm NK/F (V ) of

any K-vector space V .

(3.13) Proposition. (1) For any K-algebra A,

NK/F (A)K = ιA —K A NK/F (ιA) = NK/F (A).

and

(2) For any K-algebras A, B,

NK/F (A —K B) = NK/F (A) —F NK/F (B).

(3) For any ¬nite dimensional K-vector space V ,

NK/F EndK (V ) = EndF NK/F (V ) .

(4) If A is a central simple K-algebra, the norm NK/F (A) is a central simple F -

algebra of degree deg NK/F (A) = (deg A)2 . Moreover, the norm induces a group

homomorphism

NK/F : Br(K) ’ Br(F ).

(5) For any central simple F -algebra A,

NK/F (AK ) A —F A.

Proof : (??) Since NK/F (A) is an F -subalgebra of ιA —K A, there is a natural map

NK/F (A) —F K ’ ιA —K A induced by multiplication in ιA —K A. This map is

a homomorphism of K-algebras. It is bijective since if ± ∈ K F every element

a ∈ ιA —K A can be written in a unique way as a = a1 + a2 ± with a1 , a2 invariant

under the switch map s by setting

s(a)± ’ aι(±) a ’ s(a)

a1 = and a2 = .

± ’ ι(±) ± ’ ι(±)

In order to prove the second equality, consider the canonical isomorphism of K-

algebras ι (ιA) = A which maps ι (ιa) to a for a ∈ A. In view of this isomorphism,

NK/F (ιA) may be regarded as the set of switch-invariant elements in A —K ιA. The

38 I. INVOLUTIONS AND HERMITIAN FORMS

∼

isomorphism ιA —K A ’ A —K ιA which maps ιa — b to b — ιa commutes with the

’

switch map and therefore induces a canonical isomorphism NK/F (A) = NK/F (ιA).

(??) This is straightforward (Draxl [?, p. 55] or Scharlau [?, Lemma 8.9.7]).

The canonical map NK/F (A) —F NK/F (B) ’ NK/F (A —K B) corresponds, after

scalar extension to K, to the map

(ιA —K A) —K (ι B —K B) ’ ι (A —K B) —K (A —K B)

which carries ιa1 — a2 — ι b1 — b2 to ι (a1 — b1 ) — (a2 — b2 ).

(??) There is a natural isomorphism

ι

EndK (V ) = EndK (ι V )

which identi¬es ιf for f ∈ EndK (V ) with the endomorphism of ι V mapping ι v to

ι

f (v) . We may therefore identify

ι

EndK (V ) —K EndK (V ) = EndK (ι V —K V ),

and check that the switch map s is then identi¬ed with conjugation by sV where

sV : ι V —K V ’ ι V —K V is the ι-linear map de¬ned through

sV (ι v — w) = ι w — v for v, w ∈ V .

The F -algebra NK/F EndK (V ) of ¬xed elements under s is then identi¬ed with

the F -algebra of endomorphisms of the F -subspace elementwise invariant under sV ,

i.e., to EndF NK/F (V ) .

(??) If A is a central simple K-algebra, then ιA —K A also is central simple

over K, hence NK/F (A) is central simple over F , by part (??) and Wedderburn™s

Theorem (??). If A is Brauer-equivalent to A, then we may ¬nd vector spaces

V , V over K such that

A —K EndK (V ) A —K EndK (V ).

It then follows from parts (??) and (??) above that

NK/F (A) —F EndF NK/F (V ) NK/F (A ) —F EndF NK/F (V ) ,

hence NK/F (A) and NK/F (A ) are Brauer-equivalent. Thus NK/F induces a map

on Brauer groups and part (??) above shows that it is a homomorphism.

To prove (??), we ¬rst note that if A is an F -algebra, then ι (AK ) = AK under

the identi¬cation ι (a — ±) = a — ι(±). Therefore,

ι

(AK ) —K AK A — F A —F K

and NK/F (A) can be identi¬ed with the F -algebra elementwise invariant under the

F -algebra automorphism s of A —F A —F K de¬ned through

s (a1 — a2 — ±) = a2 — a1 — ι(±).

On the other hand, A —F A can be identi¬ed with the algebra of ¬xed points under

the automorphism s de¬ned through

s (a1 — a2 — ±) = a1 — a2 — ι(±).

We aim to show that these F -algebras are isomorphic when A is central simple.

Let g ∈ A —F A be the Goldman element (see (??)). By (??), we have

g 2 = 1 and g · (a1 — a2 ) = (a2 — a1 ) · g for all a1 , a2 ∈ A,

hence for all x ∈ A —F A, s (x — 1) = gxg ’1 — 1. In particular

s (g — 1) = g — 1,

§3. EXISTENCE OF INVOLUTIONS 39

and moreover

s (y) = (g — 1) · s (y) · (g — 1)’1 for y ∈ A —F A —F K.

Let ± ∈ K be such that ι(±) = ±± and let

u = ± + (g — 1)ι(±) ∈ A —F A —F K.

This element is invertible, since u · ± ’ (g — 1)ι(±) = ±2 ’ ι(±)2 ∈ K — ; moreover,

s (u) = ι(±) + (g — 1)± = u · (g — 1).

Therefore, for all x ∈ A —F A —F K,

s (uxu’1 ) = u · (g — 1) · s (x) · (g — 1)’1 · u’1 = u · s (x) · u’1 .

This equation shows that conjugation by u induces an isomorphism from the F -

algebra of invariant elements under s onto the F -algebra of invariant elements

under s , hence

A —F A NK/F (AK ).

(3.14) Remark. Property (??) in the proposition above does not hold for ar-

bitrary F -algebras. For instance, one may check as an exercise that NC/R (CC )

R — R — C whereas C —R C C — C. (This simple example is due to M. Ojanguren).

The proof of (??.??) in [?, p. 55] is ¬‚awed; see the correction in Tignol [?] or

Rowen [?, Theorem 7.2.26].

Involutions of the second kind and one-sided ideals. We now come back

to the proof of Theorem (??). As above, let K/F be a separable quadratic extension

of ¬elds with nontrivial automorphism ι. Let B be a central simple K-algebra. As

in the case of involutions of the ¬rst kind, the necessary condition for the existence

of an involution of the second kind on B is easy to prove:

(3.15) Proposition. Suppose that B admits an involution „ of the second kind

whose restriction to K is ι. This involution endows B with a right ι B —K B-module

structure de¬ned by

x —„ (ιa — b) = „ (a)xb for a, b, x ∈ B.

The multiplication —„ induces a right NK/F (B)-module structure on Sym(B, „ ) for

which rdim Sym(B, „ ) = 1. Therefore, NK/F (B) is split.

Proof : It is straightforward to check that —„ de¬nes on B a right ι B —K B-module

structure. For a, b, x ∈ B we have

„ x —„ (ιa — b) = „ (x) —„ (ι b — a).

Therefore, if u ∈ ι B —K B is invariant under the switch map, then multipli-

cation by u preserves Sym(B, „ ). It follows that —„ induces a right NK/F (B)-

module structure on Sym(B, „ ). Since dimF Sym(B, „ ) = deg NK/F (B), we have

rdim Sym(B, „ ) = 1, hence NK/F (B) is split.

(3.16) Remark. Alternately, the involution „ yields a K-algebra isomorphism

„— : B —K ι B ’ EndK (B) de¬ned by „— (a — ι b)(x) = ax„ (b). This isomorphism

restricts to an F -algebra isomorphism NK/F (B) ’ EndF Sym(B, „ ) which shows

that NK/F (B) is split. However, the space Sym(B, „ ) is then considered as a left

NK/F (B)-module; this is less convenient for the discussion below.

40 I. INVOLUTIONS AND HERMITIAN FORMS

Let „ : NK/F (B) ’ Sym(B, „ ) be de¬ned by

„ (u) = 1 —„ u for u ∈ NK/F (A).

Since rdim Sym(B, „ ) = 1, it is clear that the map „ is surjective, hence ker „ is a

right ideal of dimension n4 ’ n2 where n = deg B. We denote this ideal by I„ :

I„ = ker „ .

Extending scalars to K, we have NK/F (B)K = ι B —K B and the map „K : ι B —K

B ’ B induced by „ is

„K (ιa — b) = „ (a)b.

Therefore, the ideal (I„ )K = I„ —F K = ker „K satis¬es (I„ )K © (1 — B) = {0},

hence also

ι

B —K B = (I„ )K • (1 — B).

(3.17) Theorem. The map „ ’ I„ de¬nes a one-to-one correspondence between

involutions of the second kind on B leaving F elementwise invariant and right ideals

I ‚ NK/F (B) such that

ι

B —K B = IK • (1 — B)

where IK = I —F K is the ideal of ι B —K B obtained from I by scalar extension.

Proof : We have already checked that for each involution „ the ideal I„ satis¬es

the condition above. Conversely, suppose I is a right ideal such that ι B —K B =

IK • (1 — B). For each b ∈ B, there is a unique element „I (b) ∈ B such that

ι

(3.18) b — 1 ’ 1 — „I (b) ∈ IK .

The map „I : B ’ B is ι-semilinear and the same arguments as in the proof of

Theorem (??) show that it is an anti-automorphism on B.

2

In order to check that „I (b) = b for all b ∈ B, we use the fact that the ideal IK

is preserved under the switch map s : ι B —K B ’ ι B —K B since it is extended

from an ideal I in NK/F (B). Therefore, applying s to (??) we get

1 — b ’ ι „I (b) — 1 ∈ IK ,

2

hence „I (b) = b.

Arguing as in the proof of Theorem (??), we see that the ideal I„I associated

to the involution „I satis¬es (I„I )K = IK , and conclude that I„I = I, since I (resp.

I„I ) is the subset of invariant elements in IK (resp. (I„I )K ) under the switch map.

On the other hand, for any given involution „ on B we have

ι

b — 1 ’ 1 — „ (b) ∈ (I„ )K for b ∈ B,

hence „I„ = „ .

Let deg B = n. The right ideals I ‚ NK/F (B) such that ι B —K B = IK •

(1 — B) then have reduced dimension n2 ’ 1 and form a dense open subvariety V

in the Severi-Brauer variety SBn2 ’1 NK/F (B) . The theorem above may thus be

reformulated as follows:

(3.19) Corollary. There is a natural one-to-one correspondence between involu-

tions of the second kind on B which leave F elementwise invariant and rational

points on the variety V ‚ SBn2 ’1 NK/F (B) .

§4. HERMITIAN FORMS 41

We may now complete the proof of Theorem (??). If F is ¬nite, the algebras

B and NK/F (B) are split, and (??) shows that B carries unitary involutions. We

may thus assume henceforth that F is in¬nite. If NK/F (B) splits, then the variety

SBn2 ’1 NK/F (B) is a projective space. The set of rational points is therefore

dense in SBn2 ’1 NK/F (B) and so it intersects the nonempty open subvariety V

nontrivially. Corollary (??) then shows B has unitary involutions whose restriction

to K is ι.

(3.20) Remark. As in the case of involutions of the ¬rst kind, density arguments

can be avoided by reducing to division algebras. Suppose that B Mr (D) for some

central division algebra D over K and some integer r. Since the norm map NK/F

is de¬ned on the Brauer group of K, the condition that NK/F (B) splits implies

that NK/F (D) also splits. Let I be a maximal right ideal in NK/F (D). We have

dimF I = dimF NK/F (D) ’ deg NK/F (D) = (dimK D)2 ’ dimK D. Moreover, since

D is a division algebra, it is clear that IK © (1 — D) = {0}, hence

ι

D —K D = IK • (1 — D),

by dimension count. Theorem (??) then shows that D has an involution of the

second kind leaving F elementwise invariant. An involution „ of the same kind

can then be de¬ned on Mr (D) by letting act entrywise and setting

„ (a) = at .

This involution is transported to A by the isomorphism A Mr (D).

Part (??) of Theorem (??) can easily be extended to cover the case of semi-

simple F -algebras E1 — E2 with E1 , E2 central simple over F . The norm N(F —F )/F

is de¬ned by

N(F —F )/F (E1 — E2 ) = E1 —F E2 .

This de¬nition is consistent with (??), and it is easy to check that (??) extends to

the case where K = F — F .

If E1 — E2 has an involution whose restriction to the center F — F interchanges

op

the factors, then E2 E1 , by (??). Therefore, N(F —F )/F (E1 — E2 ) splits. Con-

op

versely, if N(F —F )/F (E1 — E2 ) splits, then E2 E1 and the exchange involution

op

on E1 — E1 can be transported to an involution of the second kind on E1 — E2 .

§4. Hermitian Forms

In this section, we set up a one-to-one correspondence between involutions on

central simple algebras and hermitian forms on vector spaces over division algebras,

generalizing the theorem in the introduction to this chapter.

According to Theorem (??), every central simple algebra A may be viewed as

the algebra of endomorphisms of some ¬nite dimensional vector space V over a

central division algebra D:

A = EndD (V ).

Explicitly, we may take for V any simple left A-module and set D = EndA (V ).

The module V may then be endowed with a right D-vector space structure.

Since D is Brauer-equivalent to A, Theorem (??) shows that A has an invo-

lution if and only if D has an involution. Therefore, in this section we shall work

from the perspective that central simple algebras with involution are algebras of

42 I. INVOLUTIONS AND HERMITIAN FORMS

endomorphisms of vector spaces over division algebras with involution. More gen-

erally, we shall substitute an arbitrary central simple algebra E for D and consider

endomorphism algebras of right modules over E. In the second part of this section,

we discuss extending of involutions from a simple subalgebra B ‚ A in relation to

an analogue of the Scharlau transfer for hermitian forms.

4.A. Adjoint involutions. Let E be a central simple algebra over a ¬eld F

and let M be a ¬nitely generated right E-module. Suppose that θ : E ’ E is

an involution (of any kind) on E. A hermitian form on M (with respect to the

involution θ on E) is a bi-additive map

h: M — M ’ E

subject to the following conditions:

(1) h(x±, yβ) = θ(±)h(x, y)β for all x, y ∈ M and ±, β ∈ E,

(2) h(y, x) = θ h(x, y) for all x, y ∈ M .

It clearly follows from (??) that h(x, x) ∈ Sym(E, θ) for all x ∈ M . If (??) is

replaced by

(?? ) h(y, x) = ’θ h(x, y) for all x, y ∈ M ,

the form h is called skew-hermitian. In that case h(x, x) ∈ Skew(E, θ) for all x ∈ M .

If a skew-hermitian form h satis¬es h(x, x) ∈ Alt(E, θ) for all x ∈ M , it is called

alternating (or even). If char F = 2, every skew-hermitian is alternating since

Skew(E, θ) = Alt(E, θ). If E = F and θ = Id, hermitian (resp. skew-hermitian,

resp. alternating) forms are the symmetric (resp. skew-symmetric, resp. alternating)

bilinear forms.

Similar de¬nitions can be set for left modules. It is then convenient to re-

place (??) by

(?? ) h(±x, βy) = ±h(x, y)θ(β) for all x, y ∈ M and ±, β ∈ E.

The results concerning hermitian forms on left modules are of course essentially the

same as for right modules. We therefore restrict our discussion in this section to

right modules.

The hermitian or skew-hermitian form h on the right E-module M is called

nonsingular if the only element x ∈ M such that h(x, y) = 0 for all y ∈ M is x = 0.

(4.1) Proposition. For every nonsingular hermitian or skew-hermitian form h

on M , there exists a unique involution σh on EndE (M ) such that σh (±) = θ(±) for

all ± ∈ F and

h x, f (y) = h σh (f )(x), y for x, y ∈ M .

The involution σh is called the adjoint involution with respect to h.

Proof : Consider the dual M — = HomE (M, E). It has a natural structure of left

E-module. We de¬ne a right module θ M — by

θ

M — = { θ• | • ∈ M — }

with the operations

θ

• + θ ψ = θ (• + ψ) and (θ •)± = θ θ(±)• for •, ψ ∈ M — and ± ∈ E.

The hermitian or skew-hermitian form h induces a homomorphism of right

E-modules

ˆ

h : M ’ θM —

§4. HERMITIAN FORMS 43

de¬ned by

ˆ

h(x) = θ • where •(y) = h(x, y).

ˆ

If h is nonsingular, the map h is injective, hence bijective since M and θ M — have

the same dimension over F . The unique involution σh for which the condition of

the proposition holds is then given by

ˆ ˆ

σh (f ) = h’1 —¦ θ f t —¦ h

where θ f t : θ M — ’ θ M — is the transpose of f , so that

θ tθ

f ( •) = θ f t (•) = θ (• —¦ f ) for • ∈ M — .

The following theorem is the expected generalization of the result proved in the

introduction.

(4.2) Theorem. Let A = EndE (M ).

(1) If θ is of the ¬rst kind on E, the map h ’ σh de¬nes a one-to-one correspon-

dence between nonsingular hermitian and skew-hermitian forms on M (with respect

to θ) up to a factor in F — and involutions of the ¬rst kind on A.

If char F = 2, the involutions σh on A and θ on E have the same type if h is

hermitian and have opposite types if h is skew-hermitian.

If char F = 2, the involution σh is symplectic if and only if h is alternating.

(2) If θ is of the second kind on E, the map h ’ σh de¬nes a one-to-one correspon-

dence between nonsingular hermitian forms on M up to a factor in F — invariant

under θ and involutions σ of the second kind on A such that σ(±) = θ(±) for all

± ∈ F.

Proof : We ¬rst make some observations which do not depend on the kind of θ. If

h and h are nonsingular hermitian or skew-hermitian forms on M , then the map

v = ˆ ’1 —¦ h ∈ A— is such that

ˆ

h