volution of the ¬rst kind.

§2. INVOLUTIONS 17

(1) Suppose that char F = 2, hence Symd(A, σ) = Sym(A, σ) and Alt(A, σ) =

Skew(A, σ). If σ is of orthogonal type, then

n(n+1) n(n’1)

dimF Sym(A, σ) = dimF Skew(A, σ) = .

and

2 2

If σ is of symplectic type, then

n(n’1) n(n+1)

dimF Sym(A, σ) = dimF Skew(A, σ) = .

and

2 2

Moreover, in this case n is necessarily even.

(2) Suppose that char F = 2, hence Sym(A, σ) = Skew(A, σ) and Symd(A, σ) =

Alt(A, σ); then

n(n+1) n(n’1)

dimF Sym(A, σ) = dimF Alt(A, σ) = .

and

2 2

The involution σ is of symplectic type if and only if TrdA Sym(A, σ) = {0}, which

holds if and only if 1 ∈ Alt(A, σ). In this case n is necessarily even.

Proof : The only statement which has not been observed before is that, if char F =

2, the reduced trace of every symmetric element is 0 if and only if 1 ∈ Alt(A, σ).

This follows from the characterization of alternating elements in (??).

Given an involution of the ¬rst kind on a central simple algebra A, all the other

involutions of the ¬rst kind on A can be obtained by the following proposition:

(2.7) Proposition. Let A be a central simple algebra over a ¬eld F and let σ be

an involution of the ¬rst kind on A.

(1) For each unit u ∈ A— such that σ(u) = ±u, the map Int(u) —¦ σ is an involution

of the ¬rst kind on A.

(2) Conversely, for every involution σ of the ¬rst kind on A, there exists some

u ∈ A— , uniquely determined up to a factor in F — , such that

σ = Int(u) —¦ σ σ(u) = ±u.

and

We then have

u · Sym(A, σ) = Sym(A, σ) · u’1 if σ(u) = u

Sym(A, σ ) =

u · Skew(A, σ) = Skew(A, σ) · u’1 if σ(u) = ’u

and

u · Skew(A, σ) = Skew(A, σ) · u’1 if σ(u) = u

Skew(A, σ ) =

u · Sym(A, σ) = Sym(A, σ) · u’1 if σ(u) = ’u.

If σ(u) = u, then Alt(A, σ ) = u · Alt(A, σ) = Alt(A, σ) · u’1 .

(3) Suppose that σ = Int(u) —¦ σ where u ∈ A— is such that u = ±u. If char F = 2,

then σ and σ are of the same type if and only if σ(u) = u. If char F = 2, the

involution σ is symplectic if and only if u ∈ Alt(A, σ).

2

Proof : A computation shows that Int(u) —¦ σ = Int uσ(u)’1 , proving (??).

If σ is an involution of the ¬rst kind on A, then σ —¦ σ is an automorphism

of A which leaves F elementwise invariant. The Skolem-Noether theorem then

yields an element u ∈ A— , uniquely determined up to a factor in F — , such that

2

σ —¦ σ = Int(u), hence σ = Int(u) —¦ σ. It follows that σ = Int uσ(u)’1 , hence

2

the relation σ = IdA yields

σ(u) = »u for some » ∈ F — .

18 I. INVOLUTIONS AND HERMITIAN FORMS

Applying σ to both sides of this relation and substituting »u for σ(u) in the resulting

equation, we get u = »2 u, hence » = ±1. If σ(u) = u, then for all x ∈ A,

x ’ σ (x) = u · u’1 x ’ σ(u’1 x) = xu ’ σ(xu) · u’1 .

This proves Alt(A, σ ) = u · Alt(A, σ) = Alt(A, σ) · u’1 . The relations between

Sym(A, σ ), Skew(A, σ ) and Sym(A, σ), Skew(A, σ) follow by straightforward com-

putations, completing the proof of (??).

If char F = 2, the involutions σ and σ have the same type if and only if

Sym(A, σ) and Sym(A, σ ) have the same dimension. Part (??) shows that this

condition holds if and only if σ(u) = u. If char F = 2, the involution σ is symplectic

if and only if TrdA (s ) = 0 for all s ∈ Sym(A, σ ). In view of (??), this condition

may be rephrased as

TrdA (us) = 0 for s ∈ Sym(A, σ).

Lemma (??) shows that this condition holds if and only if u ∈ Alt(A, σ).

(2.8) Corollary. Let A be a central simple F -algebra with an involution σ of the

¬rst kind.

(1) If deg A is odd, then A is split and σ is necessarily of orthogonal type. Moreover,

the space Alt(A, σ) contains no invertible elements.

(2) If deg A is even, then the index of A is a power of 2 and A has involutions of

both types. Whatever the type of σ, the space Alt(A, σ) contains invertible elements

and the space Sym(A, σ) contains invertible elements which are not in Alt(A, σ).

Proof : De¬ne a homomorphism of F -algebras

σ— : A —F A ’ EndF (A)

by σ— (a — b)(x) = axσ(b) for a, b, x ∈ A. This homomorphism is injective since

A —F A is simple and surjective by dimension count, hence it is an isomorphism.

Therefore, A —F A splits2 , and the exponent of A is 1 or 2. Since the index and

the exponent of a central simple algebra have the same prime factors (see Draxl [?,

Theorem 11, p. 66]), it follows that the index of A, ind A, is a power of 2. In

particular, if deg A is odd, then A is split. In this case, Proposition (??) shows

that every involution of the ¬rst kind has orthogonal type. If Alt(A, σ) contains an

invertible element u, then Int(u)—¦σ has symplectic type, by (??); this is impossible.

Suppose henceforth that the degree of A is even. If A is split, then it has

involutions of both types, since a vector space of even dimension carries nonsingular

alternating bilinear forms as well as nonsingular symmetric, nonalternating bilinear

forms. Let σ be an involution of the ¬rst kind on A. In order to show that Alt(A, σ)

contains invertible elements, we consider separately the case where char F = 2. If

char F = 2, consider an involution σ whose type is di¬erent from the type of σ.

Proposition (??) yields an invertible element u ∈ A such that σ = Int(u) —¦ σ

and σ(u) = ’u. Note also that 1 is an invertible element which is symmetric

but not alternating. If char F = 2, consider a symplectic involution σ and an

orthogonal involution σ . Again, (??) yields invertible elements u, v ∈ A— such

that σ = Int(u) —¦ σ and σ = Int(v) —¦ σ, and shows that u ∈ Alt(A, σ) and

v ∈ Sym(A, σ) Alt(A, σ).

2 Alternately, Aop by mapping a ∈ A to

the involution σ yields an isomorphism A

op op, hence the Brauer class [A] of A satis¬es [A] = [A]’1 .

∈A

σ(a)

§2. INVOLUTIONS 19

Assume next that A is not split. The base ¬eld F is then in¬nite, since the

Brauer group of a ¬nite ¬eld is trivial (see for instance Scharlau [?, Corollary 8.6.3]).

Since invertible elements s are characterized by NrdA (s) = 0 where NrdA is the

reduced norm in A, the set of invertible alternating elements is a Zariski-open subset

of Alt(A, σ). Our discussion above of the split case shows that this open subset is

nonempty over an algebraic closure. Since F is in¬nite, rational points are dense,

hence this open set has a rational point. Similarly, the set of invertible symmetric

elements is a dense Zariski-open subset in Sym(A, σ), hence it is not contained in

the closed subset Sym(A, σ)©Alt(A, σ). Therefore, there exist invertible symmetric

elements which are not alternating.

If u ∈ Alt(A, σ) is invertible, then Int(u)—¦σ is an involution of the type opposite

to σ if char F = 2, and is a symplectic involution if char F = 2. If char F = 2 and

v ∈ Sym(A, σ) is invertible but not alternating, then Int(v) —¦ σ is an orthogonal

involution.

The existence of involutions of both types on central simple algebras of even

degree with involution can also be derived from the proof of (??) below.

The following proposition highlights a special feature of symplectic involutions:

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

plectic type. The reduced characteristic polynomial of every element in Symd(A, σ)

is a square. In particular, NrdA (s) is a square in F for all s ∈ Symd(A, σ).

Proof : Let K be a Galois extension of F which splits A and let s ∈ Symd(A, σ).

It su¬ces to show that the reduced characteristic polynomial PrdA,s (X) ∈ F [X]

is a square in K[X], for then its monic square root is invariant under the action

of the Galois group Gal(K/F ), hence it is in F [X]. Extending scalars from F to

K, we reduce to the case where A is split. We may thus assume that A = Mn (F ).

Proposition (??) then yields σ = Int(u) —¦ t for some invertible alternating matrix

u ∈ A— , hence Symd(A, σ) = u · Alt Mn (F ), t . Therefore, there exists a matrix

a ∈ Alt Mn (F ), t such that s = ua. The (reduced) characteristic polynomial of s

is then

PrdA,s (X) = det(X · 1 ’ s) = (det u) det(X · u’1 ’ a) .

Since u and X · u’1 ’ a are alternating, their determinants are the squares of their

pfa¬an pf u, pf(X · u’1 ’ a) (see for instance E. Artin [?, Theorem 3.27]), hence

PrdA,s (X) = [(pf u) pf(X · u’1 ’ a)]2 .

Let deg A = n = 2m. In view of the preceding proposition, for every s ∈

Symd(A, σ) there is a unique monic polynomial, the pfa¬an characteristic polyno-

mial, Prpσ,s (X) ∈ F [X] of degree m such that

PrdA,s (X) = Prpσ,s (X)2 .

For s ∈ Symd(A, σ), we de¬ne the pfa¬an trace Trpσ (s) and the pfa¬an norm

Nrpσ (s) ∈ F as coe¬cients of Prpσ,s (X):

Prpσ,s (X) = X m ’ Trpσ (s)X m’1 + · · · + (’1)m Nrpσ (s).

(2.10)

20 I. INVOLUTIONS AND HERMITIAN FORMS

Thus, Trpσ and Nrpσ are polynomial maps of degree 1 and m respectively on

Symd(A, σ), and we have

NrdA (s) = Nrpσ (s)2

(2.11) TrdA (s) = 2 Trpσ (s) and

for all s ∈ Symd(A, σ). Moreover, we have PrdA,1 (X) = (X ’ 1)2m , hence

Prpσ,1 (X) = (X ’ 1)m and therefore

(2.12) Trpσ (1) = m and Nrpσ (1) = 1.

Since polynomial maps on Symd(A, σ) form a domain, the map Nrpσ is uniquely

determined by (??) and (??). (Of course, if char F = 2, the map Trpσ is also

uniquely determined by (??); in all characteristics it is uniquely determined by the

property in (??) below.) Note that the pfa¬an norm Nrpσ (or simply pfa¬an) is

an analogue of the classical pfa¬an. However it is de¬ned on the space Symd(A, σ)

whereas pf is de¬ned on alternating matrices (under the transpose involution).

Nevertheless, it shares with the pfa¬an the fundamental property demonstrated in

the following proposition:

(2.13) Proposition. For all s ∈ Symd(A, σ) and all a ∈ A,

Trpσ σ(a) + a = TrdA (a) Nrpσ σ(a)sa = NrdA (a) Nrpσ (s).

and

Proof : We ¬rst prove the second equation. If s is not invertible, then NrdA (s) =

NrdA σ(a)sa = 0, hence Nrpσ (s) = Nrpσ σ(a)sa = 0, proving the equation in

this particular case. For s ∈ Symd(A, σ) © A— ¬xed, consider both sides of the

equality to be proved as polynomial maps on A:

f1 : a ’ Nrpσ σ(a)sa and f2 : a ’ NrdA (a) Nrpσ (s).

Since NrdA σ(a)sa = NrdA (a)2 NrdA (s), we have f1 = f2 , hence (f1 + f2 )(f1 ’

2 2

f2 ) = 0. Since polynomial maps on A form a domain, it follows that f1 = ±f2 .

Taking into account the fact that f1 (1) = Nrpσ (s) = f2 (1), we get f1 = f2 .

The ¬rst equation follows from the second. For, let t be an indeterminate over

F and consider the element 1 + ta ∈ AF (t) . By the equation just proven, we have

1 ’ tσ(a) (1 ’ ta) = NrdA (1 ’ ta) = 1 ’ TrdA (a)t + SrdA (a)t2 ’ . . .

Nrpσ

On the other hand, for all s ∈ Symd(A, σ) we have

Nrpσ (1 ’ s) = Prpσ,s (1) = 1 ’ Trpσ (s) + · · · + (’1)m Nrpσ (s),

hence the coe¬cient of ’t in Nrpσ 1 ’ tσ(a) (1 ’ ta) is Trpσ σ(a) + a ; the ¬rst

equality is thus proved.

2.B. Involutions of the second kind. In the case of involutions of the

second kind on a simple algebra B, the base ¬eld F is usually not the center of the

algebra, but the sub¬eld of central invariant elements which is of codimension 2

in the center. Under scalar extension to an algebraic closure of F , the algebra B

decomposes into a direct product of two simple factors. It is therefore convenient

to extend our discussion of involutions of the second kind to semisimple F -algebras

of the form B1 — B2 , where B1 , B2 are central simple F -algebras.

Throughout this section, we thus denote by B a ¬nite dimensional F -algebra

whose center K is a quadratic ´tale3 extension of F , and assume that B is either

e

simple (if K is a ¬eld) or a direct product of two simple algebras (if K F — F ).

3 Thissimply means that K is either a ¬eld which is a separable quadratic extension of F , or

F — F . See § ?? for more on ´tale algebras.

e

K

§2. INVOLUTIONS 21

We denote by ι the nontrivial automorphism of K/F and by „ an involution of the

second kind on B, whose restriction to K is ι. For convenience, we refer to (B, „ )

as a central simple F -algebra with involution of the second kind, even though

its center is not F and the algebra B may not be simple.4 A homomorphism

f : (B, „ ) ’ (B , „ ) is then an F -algebra homomorphism f : B ’ B such that

„ —¦ f = f —¦ „.

If L is any ¬eld containing F , the L-algebra BL = B —F L has center KL =

K —F L, a quadratic ´tale extension of L, and carries an involution of the second

e

kind „L = „ — IdL . Moreover, (BL , „L ) is a central simple L-algebra with involution

of the second kind.

As a parallel to the terminology of types used for involutions of the ¬rst kind,

and because of their relation with unitary groups (see Chapter ??), involutions of

the second kind are also called of unitary type (or simply unitary).

We ¬rst examine the case where the center K is not a ¬eld.

(2.14) Proposition. If K F — F , there is a central simple F -algebra E such

that

(E — E op , µ),

(B, „ )

where the involution µ is de¬ned by µ(x, y op ) = (y, xop ). This involution is called

the exchange involution.

Proof : Let B = B1 — B2 where B1 , B2 are central simple F -algebras. Since the

restriction of „ to the center K = F — F interchanges the two factors, it maps (1, 0)

to (0, 1), hence

„ B1 — {0} = „ (B1 — B2 ) · (1, 0) = (0, 1) · (B1 — B2 ) = {0} — B2 .

It follows that B1 and B2 are anti-isomorphic. We may then de¬ne an F -algebra

op ∼

isomorphism f : B1 ’ B2 by the relation

’

„ (x, 0) = 0, f (xop ) ,

op

and identify B1 — B2 with B1 — B1 by mapping (x1 , x2 ) to x1 , f ’1 (x2 ) . Under

this map, „ is identi¬ed with the involution µ.

In view of this proposition, we may de¬ne the degree of the central simple

F -algebra (B, „ ) with involution of the second kind by

deg B if K is a ¬eld,

deg(B, „ ) =

(E — E op , µ).

deg E if K F — F and (B, „ )

2

Equivalently, deg(B, „ ) is de¬ned by the relation dimF B = 2 deg(B, „ ) .

If the center K of B is a ¬eld, (??) applies to BK = B —F K, since its center

is KK = K —F K K — K. In this case we get a canonical isomorphism:

(2.15) Proposition. Suppose that the center K of B is a ¬eld. There is a canon-

ical isomorphism of K-algebras with involution

∼

• : (BK , „K ) ’ (B — B op , µ)

’

op

which maps b — ± to b±, „ (b)± for b ∈ B and ± ∈ K.

4 We thus follow Jacobson™s convention in [?, p. 208]; it can be justi¬ed by showing that (B, „ )

is indeed central simple as an algebra-with-involution.

22 I. INVOLUTIONS AND HERMITIAN FORMS

Proof : It is straightforward to check that • is a homomorphism of central simple

K-algebras with involution of the second kind. It thus su¬ces to prove that • has

an inverse. Let ± ∈ K F . Then the map Ψ : B — B op ’ BK de¬ned by

„ (y)± ’ xι(±) x ’ „ (y)

Ψ(x, y op ) = —1+ —±

± ’ ι(±) ± ’ ι(±)

is the inverse of •.

In a semisimple algebra of the form B1 — B2 , where B1 , B2 are central sim-

ple F -algebras of the same degree, the reduced characteristic polynomial of an

element (b1 , b2 ) may be de¬ned as PrdB1 ,b1 (X), PrdB2 ,b2 (X) ∈ (F — F )[X],

where PrdB1 ,b1 (X) and PrdB2 ,b2 (X) are the reduced characteristic polynomials of

b1 and b2 respectively (see Reiner [?, p. 121]). Since the reduced characteristic

polynomial of an element does not change under scalar extension (see Reiner [?,

Theorem (9.27)]), the preceding proposition yields:

(2.16) Corollary. For every b ∈ B, the reduced characteristic polynomials of b

and „ (b) are related by

PrdB,„ (b) = ι(PrdB,b ) in K[X].

In particular, TrdB „ (b) = ι TrdB (b) and NrdB „ (b) = ι NrdB (b) .

Proof : If K F — F , the result follows from (??); if K is a ¬eld, it follows

from (??).

As for involutions of the ¬rst kind, we may de¬ne the sets of symmetric, skew-

symmetric, symmetrized and alternating elements in (B, „ ) by

Sym(B, „ ) = { b ∈ B | „ (b) = b },

Skew(B, „ ) = { b ∈ B | „ (b) = ’b },

Symd(B, „ ) = { b + „ (b) | b ∈ B },

Alt(B, „ ) = { b ’ „ (b) | b ∈ B }.

These sets are vector spaces over F . In contrast with the case of involutions of the

¬rst kind, there is a straightforward relation between symmetric, skew-symmetric

and alternating elements, as the following proposition shows:

(2.17) Proposition. Symd(B, „ ) = Sym(B, „ ) and Alt(B, „ ) = Skew(B, „ ). For

any ± ∈ K — such that „ (±) = ’±,

Skew(B, „ ) = ± · Sym(B, „ ).

If deg(B, „ ) = n, then

dimF Sym(B, „ ) = dimF Skew(B, „ ) = dimF Symd(B, „ ) = dimF Alt(B, „ ) = n2 .

Proof : The relations Skew(B, „ ) = ± · Sym(B, „ ) and Symd(B, „ ) ‚ Sym(B, „ ),

Alt(B, „ ) ‚ Skew(B, „ ) are clear. If β ∈ K is such that β + ι(β) = 1, then every

element s ∈ Symd(B, „ ) may be written as s = βs + „ (βs), hence Sym(B, „ ) =

Symd(B, „ ). Similarly, every element s ∈ Skew(B, „ ) may be written as s = βs ’

„ (βs), hence Skew(B, „ ) = Alt(B, „ ). Therefore, the vector spaces Sym(B, „ ),

Skew(B, „ ), Symd(B, „ ) and Alt(B, „ ) have the same dimension. This dimension

1

is 2 dimF B, since Alt(B, „ ) is the image of the F -linear endomorphism Id ’ „

of B, whose kernel is Sym(B, „ ). Since dimF B = 2 dimK B = 2n2 , the proof is

complete.

§2. INVOLUTIONS 23

As for involutions of the ¬rst kind, all the involutions of the second kind on B

which have the same restriction to K as „ are obtained by composing „ with an

inner automorphism, as we now show.

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

the second kind, and let K denote the center of B.

(1) For every unit u ∈ B — such that „ (u) = »u with » ∈ K — , the map Int(u) —¦ „ is

an involution of the second kind on B.

(2) Conversely, for every involution „ on B whose restriction to K is ι, there

exists some u ∈ B — , uniquely determined up to a factor in F — , such that

„ = Int(u) —¦ „ „ (u) = u.

and

In this case,

Sym(B, „ ) = u · Sym(B, „ ) = Sym(B, „ ) · u’1 .

2

Proof : Computation shows that Int(u) —¦ „ = Int u„ (u)’1 , and (??) follows.

If „ is an involution on B which has the same restriction to K as „ , the

composition „ —¦ „ is an automorphism which leaves K elementwise invariant. The

Skolem-Noether theorem shows that „ —¦ „ = Int(u0 ) for some u0 ∈ B — , hence

2

„ = Int(u0 ) —¦ „ . Since „ = Id, we have u0 „ (u0 )’1 ∈ K — . Let » ∈ K — be such

that „ (u0 ) = »u0 . Applying „ to both sides of this relation, we get NK/F (») = 1.

Hilbert™s theorem 90 (see (??)) yields an element µ ∈ K — such that » = µι(µ)’1 .

Explicitly one can take µ = ± + »ι(±) for ± ∈ K such that ± + »ι(±) is invertible.

The element u = µu0 then satis¬es the required conditions.

2.C. Examples.

Endomorphism algebras. Let V be a ¬nite dimensional vector space over

a ¬eld F . The involutions of the ¬rst kind on EndF (V ) have been determined

in the introduction to this chapter: every such involution is the adjoint involution

with respect to some nonsingular symmetric or skew-symmetric bilinear form on V ,

uniquely determined up to a scalar factor. Moreover, it is clear from De¬nition (??)

that the involution is orthogonal (resp. symplectic) if the corresponding bilinear

form is symmetric and nonalternating (resp. alternating).

Involutions of the second kind can be described similarly. Suppose that V

is a ¬nite dimensional vector space over a ¬eld K which is a separable quadratic

extension of some sub¬eld F with nontrivial automorphism ι. A hermitian form

on V (with respect to ι) is a bi-additive map

h: V — V ’ K

such that

h(v±, wβ) = ι(±)h(v, w)β for v, w ∈ V and ±, β ∈ K

and

h(w, v) = ι h(v, w) for v, w ∈ V .

The form h is called nonsingular if the only element x ∈ V such that h(x, y) = 0

for all y ∈ V is x = 0. If this condition holds, an involution σh on EndK (V ) may

be de¬ned by the following condition:

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

24 I. INVOLUTIONS AND HERMITIAN FORMS

The involution σh on EndK (V ) is the adjoint involution with respect to the hermit-

ian form h. From the de¬nition of σh , it follows that σh (±) = ι(±) for all ± ∈ K,

hence σh is of the second kind. As for involutions of the ¬rst kind, one can prove

that every involution „ of the second kind on EndK (V ) whose restriction to K is

ι is the adjoint involution with respect to some nonsingular hermitian form on V ,

uniquely determined up to a factor in F — .

We omit the details of the proof, since a more general statement will be proved

in §?? below (see (??)).

Matrix algebras. The preceding discussion can of course be translated to

matrix algebras, since the choice of a basis in an n-dimensional vector space V

over F yields an isomorphism EndF (V ) Mn (F ). However, matrix algebras are

endowed with a canonical involution of the ¬rst kind, namely the transpose involu-

tion t. Therefore, a complete description of involutions of the ¬rst kind on Mn (F )

can also be easily derived from (??).

(2.19) Proposition. Every involution of the ¬rst kind σ on Mn (F ) is of the form

σ = Int(u) —¦ t

for some u ∈ GLn (F ), uniquely determined up to a factor in F — , such that ut = ±u.

Moreover, the involution σ is orthogonal if ut = u and u ∈ Alt Mn (F ), t , and it

is symplectic if u ∈ Alt Mn (F ), t .

If Mn (F ) is identi¬ed with EndF (F n ), the involution σ = Int(u) —¦ t is the

adjoint involution with respect to the nonsingular form b on F n de¬ned by

b(x, y) = xt · u’1 · y for x, y ∈ F n .

Suppose now that A is an arbitrary central simple algebra over a ¬eld F and

that is an involution (of any kind) on A. We de¬ne an involution — on Mn (A) by

(aij )— t

1¤i,j¤n = (aij )1¤i,j¤n .

(2.20) Proposition. The involution — is of the same type as . Moreover, the

involutions σ on Mn (A) such that σ(±) = ± for all ± ∈ F can be described as

follows:

(1) If is of the ¬rst kind, then every involution of the ¬rst kind on Mn (A) is of

the form σ = Int(u) —¦ — for some u ∈ GLn (A), uniquely determined up to a factor

in F — , such that u— = ±u. If char F = 2, the involution Int(u) —¦ — is of the same

type as if and only if u— = u. If char F = 2, the involution Int(u) —¦ — is symplectic

if and only if u ∈ Alt Mn (A), — .

(2) If is of the second kind, then every involution of the second kind σ on M n (A)

such that σ(±) = ± for all ± ∈ F is of the form σ = Int(u)—¦ — for some u ∈ GLn (A),

uniquely determined up to a factor in F — invariant under , such that u— = u.

Proof : From the de¬nition of — it follows that ±— = ± for all ± ∈ F . Therefore, the

involutions — and are of the same kind.

Suppose that is of the ¬rst kind. A matrix (aij )1¤i,j¤n is — -symmetric if and

only if its diagonal entries are -symmetric and aji = aij for i > j, hence

n(n’1)

dim Sym Mn (A), — = n dim Sym(A, ) + dim A.

2

If deg A = d and dim Sym(A, ) = d(d + δ)/2, where δ = ±1, we thus get

dim Sym Mn (A), — = nd(nd + δ)/2.

§2. INVOLUTIONS 25

Therefore, if char F = 2 the type of — is the same as the type of .

n

Since TrdMn (A) (aij )1¤i,j¤n = i=1 TrdA (aii ), we have

TrdMn (A) Sym Mn (A), — = {0} if and only if TrdA Sym(A, ) = {0}.

Therefore, when char F = 2 the involution — is symplectic if and only if is sym-

plectic.

We have thus shown that in all cases the involutions — and are of the same

type (orthogonal, symplectic or unitary). The other assertions follow from (??)

and (??).

In §?? below, it is shown how the various involutions on Mn (A) are associated

to hermitian forms on An under the identi¬cation EndA (An ) = Mn (A).

Quaternion algebras. A quaternion algebra over a ¬eld F is a central simple

F -algebra of degree 2. If the characteristic of F is di¬erent from 2, it can be shown

(see Scharlau [?, §8.11]) that every quaternion algebra Q has a basis (1, i, j, k)

subject to the relations

i2 ∈ F — , j 2 ∈ F —, ij = k = ’ji.

Such a basis is called a quaternion basis; if i2 = a and j 2 = b, the quaternion

algebra Q is denoted

Q = (a, b)F .

Conversely, for any a, b ∈ F — the 4-dimensional F -algebra Q with basis (1, i, j, k)

where multiplication is de¬ned through the relations i2 = a, j 2 = b, ij = k = ’ji

is central simple and is therefore a quaternion algebra (a, b)F .

If char F = 2, every quaternion algebra Q has a basis (1, u, v, w) subject to the

relations

u2 + u ∈ F, v2 ∈ F —, uv = w = vu + v

(see Draxl [?, §11]). Such a basis is called a quaternion basis in characteristic 2. If

u2 + u = a and v 2 = b, the quaternion algebra Q is denoted

Q = [a, b)F .

Conversely, for all a ∈ F , b ∈ F — , the relations u2 + u = a, v 2 = b and vu = uv + v

give the span of 1, u, v, uv the structure of a quaternion algebra.

Quaternion algebras in characteristic 2 may alternately be de¬ned as algebras

generated by two elements r, s subject to

r2 ∈ F, s2 ∈ F, rs + sr = 1.

Indeed, if r2 = 0 the algebra thus de¬ned is isomorphic to M2 (F ); if r2 = 0 it

has a quaternion basis (1, sr, r, sr 2 ). Conversely, every quaternion algebra with

quaternion basis (1, u, v, w) as above is generated by r = v and s = uv ’1 satisfying

the required relations. The quaternion algebra Q generated by r, s subject to the

relations r2 = a ∈ F , s2 = b ∈ F and rs + sr = 1 is denoted

Q = a, b .

F

Thus, a, b F M2 (F ) if a (or b) = 0 and a, b F [ab, a)F if a = 0. The

quaternion algebra a, b F is thus the Cli¬ord algebra of the quadratic form [a, b].

For every quaternion algebra Q, an F -linear map γ : Q ’ Q can be de¬ned by

γ(x) = TrdQ (x) ’ x for x ∈ Q

26 I. INVOLUTIONS AND HERMITIAN FORMS

where TrdQ is the reduced trace in Q. Explicitly, for x0 , x1 , x2 , x3 ∈ F ,

γ(x0 + x1 i + x2 j + x3 k) = x0 ’ x1 i ’ x2 j ’ x3 k

if char F = 2 and

γ(x0 + x1 u + x2 v + x3 w) = x0 + x1 (u + 1) + x2 v + x3 w

if char F = 2. For the split quaternion algebra Q = M2 (F ) (in arbitrary character-

istic),

x11 x12 x22 ’x12

γ =

x21 x22 ’x21 x11

Direct computations show that γ is an involution, called the quaternion con-

jugation or the canonical involution. If char F = 2, then Sym(Q, γ) = F and

Skew(Q, γ) has dimension 3. If char F = 2, then Sym(Q, γ) is spanned by 1, v, w

which have reduced trace equal to zero. Therefore, the involution γ is symplectic

in every characteristic.

(2.21) Proposition. The canonical involution γ on a quaternion algebra Q is the

unique symplectic involution on Q. Every orthogonal involution σ on Q is of the

form

σ = Int(u) —¦ γ

where u is an invertible quaternion in Skew(Q, γ) F which is uniquely determined

by σ up to a factor in F — .

Proof : From (??), it follows that every involution of the ¬rst kind σ on Q has the

form σ = Int(u) —¦ γ where u is a unit such that γ(u) = ±u. Suppose that σ is

symplectic. If char F = 2, Proposition (??) shows that u ∈ Alt(Q, γ) = F , hence

σ = γ. Similarly, if char F = 2, Proposition (??) shows that γ(u) = u, hence

u ∈ F — and σ = γ.

Unitary involutions on quaternion algebras also have a very particular type, as

we proceed to show.

(2.22) Proposition (Albert). Let K/F be a separable quadratic ¬eld extension

with nontrivial automorphism ι. Let „ be an involution of the second kind on a

quaternion algebra Q over K, whose restriction to K is ι. There exists a unique

quaternion F -subalgebra Q0 ‚ Q such that

Q = Q 0 —F K „ = γ0 — ι

and

where γ0 is the canonical involution on Q0 . Moreover, the algebra Q0 is uniquely

determined by these conditions.

Proof : Let γ be the canonical involution on Q. Then „ —¦ γ —¦ „ is an involution

of the ¬rst kind and symplectic type on Q, so „ —¦ γ —¦ „ = γ by (??). From this

last relation, it follows that „ —¦ γ is a ι-semilinear automorphism of order 2 of Q.

The F -subalgebra Q0 of invariant elements then satis¬es the required conditions.

Since these conditions imply that every element in Q0 is invariant under „ —¦ γ, the

algebra Q0 is uniquely determined by „ . (It is also the F -subalgebra of Q generated

by „ -skew-symmetric elements of trace zero, see (??).)

The proof holds without change in the case where K F — F , provided that

quaternion algebras over K are de¬ned as direct products of two quaternion F -

algebras.

§2. INVOLUTIONS 27

Symbol algebras. Let n be an arbitrary positive integer and let K be a ¬eld

containing a primitive nth root of unity ζ. For a, b ∈ K — , write (a, b)ζ,K for the

K-algebra generated by two elements x, y subject to the relations

xn = a, y n = b, yx = ζxy.

This algebra is central simple of degree n (see Draxl [?, §11]); it is called a symbol

algebra.5 Clearly, quaternion algebras are the symbol algebras of degree 2.

If K has an automorphism ι of order 2 which leaves a and b invariant and maps

ζ to ζ ’1 , then this automorphism extends to an involution „ on (a, b)ζ,K which

leaves x and y invariant.

Similarly, any automorphism ι of order 2 of K which leaves a and ζ invariant

and maps b to b’1 (if any) extends to an involution σ on (a, b)ζ,K which leaves x

invariant and maps y to y ’1 .

Tensor products.

(2.23) Proposition. (1) Let (A1 , σ1 ), . . . , (An , σn ) be central simple F -algebras

with involution of the ¬rst kind. Then σ1 — · · · — σn is an involution of the ¬rst

kind on A1 —F · · · —F An . If char F = 2, this involution is symplectic if and only if

an odd number of involutions among σ1 , . . . , σn are symplectic. If char F = 2, it

is symplectic if and only if at least one of σ1 , . . . , σn is symplectic.

(2) Let K/F be a separable quadratic ¬eld extension with nontrivial automorphism ι

and let (B1 , „1 ), . . . , (Bn , „n ) be central simple F -algebras with involution of the

second kind with center K. Then „1 — · · · — „n is an involution of the second kind

on B1 —K · · · —K Bn .

(3) Let K/F be a separable quadratic ¬eld extension with nontrivial automorphism ι

and let (A, σ) be a central simple F -algebra with involution of the ¬rst kind. Then

(A —F K, σ — ι) is a central simple F -algebra with involution of the second kind.

The proof, by induction on n for the ¬rst two parts, is straightforward. In

case (??), we denote

(A1 , σ1 ) —F · · · —F (An , σn ) = (A1 —F · · · —F An , σ1 — · · · — σn ),

and use similar notations in the other two cases.

Tensor products of quaternion algebras thus yield examples of central simple

algebras with involution. Merkurjev™s theorem [?] shows that every central simple

algebra with involution is Brauer-equivalent to a tensor product of quaternion alge-

bras. However, there are examples of division algebras with involution of degree 8

which do not decompose into tensor products of quaternion algebras, and there are

examples of involutions σ on tensor products of two quaternion algebras which are

not of the form σ1 — σ2 (see Amitsur-Rowen-Tignol [?]). A necessary and su¬cient

decomposability condition for an involution on a tensor product of two quaternion

algebras has been given by Knus-Parimala-Sridharan [?]; see (??) and (??).

2.D. Lie and Jordan structures. Every associative algebra A over an arbi-

trary ¬eld F is endowed with a Lie algebra structure for the bracket [x, y] = xy’yx.

We denote this Lie algebra by L(A). Similarly, if char F = 2, a Jordan product can

1

be de¬ned on A by x q y = 2 (xy + yx). If A is viewed as a Jordan algebra for the

product q, we denote it by A+ .

5 Draxl calls it a power norm residue algebra.

28 I. INVOLUTIONS AND HERMITIAN FORMS

The relevance of the Lie and Jordan structures for algebras with involution

stems from the observation that for every algebra with involution (A, σ) (of any

kind), the spaces Skew(A, σ) and Alt(A, σ) are Lie subalgebras of L(A), and the

space Sym(A, σ) is a Jordan subalgebra of A+ if char F = 2. Indeed, for x, y ∈

Skew(A, σ) we have

[x, y] = xy ’ σ(xy) ∈ Alt(A, σ) ‚ Skew(A, σ)

hence Alt(A, σ) and Skew(A, σ) are Lie subalgebras of L(A). On the other hand,

for x, y ∈ Sym(A, σ),

1

x qy = xy + σ(xy) ∈ Sym(A, σ),

2

hence Sym(A, σ) is a Jordan subalgebra of A+ . This Jordan subalgebra is usually

denoted by H(A, σ).

The algebra Skew(A, σ) is contained in the Lie algebra

g(A, σ) = { a ∈ A | a + σ(a) ∈ F };

indeed, Skew(A, σ) is the kernel of the Lie algebra homomorphism6

µ : g(A, σ) ’ F

™

de¬ned by µ(a) = a + σ(a), for a ∈ g(A, σ). The map µ is surjective, except when

™ ™

char F = 2 and σ is orthogonal, since the condition 1 ∈ Symd(A, σ) characterizes

symplectic involutions among involutions of the ¬rst kind in characteristic 2, and

Symd(A, σ) = Sym(A, σ) if σ is of the second kind. Thus, g(A, σ) = Skew(A, σ) if

σ is orthogonal and char F = 2, and dim g(A, σ) = dim Skew(A, σ) + 1 in the other

cases.

Another important subalgebra of L(A) is the kernel A0 of the reduced trace

map:

A0 = { a ∈ A | TrdA (a) = 0 }.

If σ is symplectic (in arbitrary characteristic) or if it is orthogonal in characteristic

di¬erent from 2, we have Skew(A, σ) ‚ A0 ; in the other cases, we also consider the

intersection

Skew(A, σ)0 = Skew(A, σ) © A0 .

(2.24) Example. Let E be an arbitrary central simple F -algebra and let µ be

the exchange involution on E — E op . There are canonical isomorphisms of Lie and

Jordan algebras

∼ ∼

L(E) ’ Skew(E — E op , µ), E + ’ H(E — E op , µ)

’ ’

which map x ∈ E respectively to (x, ’xop ) and to (x, xop ). Indeed, these maps are

obviously injective homomorphisms, and they are surjective by dimension count

(see (??)). We also have

g(A, σ) = { (x, ± ’ xop ) | x ∈ E, ± ∈ F } L(E — F ).

Jordan algebras of symmetric elements in central simple algebras with involu-

tion are investigated in Chapter ?? in relation with twisted compositions and the

Tits constructions. Similarly, the Lie algebras of skew-symmetric and alternating

elements play a crucial rˆle in the study of algebraic groups associated to algebras

o

6 The notation µ is motivated by the observation that this map is the di¬erential of the

™

multiplier map µ : Sim(A, σ) ’ F — de¬ned in (??).

§2. INVOLUTIONS 29

with involution in Chapter ??. In this section, we content ourselves with a few basic

observations which will be used in the proofs of some speci¬c results in Chapters

?? and ??.

∼

It is clear that every isomorphism of algebras with involution f : (A, σ) ’ ’

(A , σ ) carries symmetric, skew-symmetric and alternating elements in A to ele-

∼

ments of the same type in A and therefore induces Lie isomorphisms Skew(A, σ) ’ ’

∼ ∼

Skew(A , σ ), Alt(A, σ) ’ Alt(A , σ ) and a Jordan isomorphism H(A, σ) ’

’ ’

H(A , σ ). Conversely, if the degrees of A and A are large enough, every iso-

morphism of Lie or Jordan algebras as above is induced by an automorphism of

algebras with involution: see Jacobson [?, Chapter X, §4] and Jacobson [?, Theo-

rem 11, p. 210]. However, this property does not hold for algebras of low degrees;

the exceptional isomorphisms investigated in Chapter ?? indeed yield examples

of nonisomorphic algebras with involution which have isomorphic Lie algebras of

skew-symmetric elements. Other examples arise from triality, see (??).

The main result of this subsection is the following extension property, which is

much weaker than those referred to above, but holds under weaker degree restric-

tions:

(2.25) Proposition. (1) Let (A, σ) and (A , σ ) be central simple F -algebras with

involution of the ¬rst kind and let L/F be a ¬eld extension. Suppose that deg A > 2

and let

∼

f : Alt(A, σ) ’ Alt(A , σ )

’

be a Lie isomorphism which has the following property: there is an isomorphism

∼

of L-algebras with involution (AL , σL ) ’ (AL , σL ) whose restriction to Alt(A, σ)

’

is f . Then f extends uniquely to an isomorphism of F -algebras with involution

∼

(A, σ) ’ (A , σ ).

’

(2) Let (B, „ ) and (B , „ ) be central simple F -algebras with involution of the second

kind and let L/F be a ¬eld extension. Suppose that deg(B, „ ) > 2 and let

∼

f : Skew(B, „ )0 ’ Skew(B , „ )0

’

be a Lie isomorphism which has the following property: there is an isomorphism of

∼

L-algebras with involution (BL , „L ) ’ (BL , „L ) whose restriction to Skew(B, „ )0

’

is f . Then f extends uniquely to an isomorphism of F -algebras with involution

∼

(B, „ ) ’ (B , „ ).

’

The proof relies on the following crucial lemma:

(2.26) Lemma. (1) Let (A, σ) be a central simple F -algebra with involution of the

¬rst kind. The set Alt(A, σ) generates A as an associative algebra if deg A > 2.

(2) Let (B, „ ) be a central simple F -algebra with involution of the second kind. The

set Skew(B, „ )0 generates B as an associative F -algebra if deg(B, „ ) > 2.

Proof : (??) Let S ‚ A be the associative subalgebra of A generated by Alt(A, σ).

For every ¬eld extension L/F , the subalgebra of AL generated by Alt(AL , σL ) =

Alt(A, σ) —F L is then SL ; therefore, it su¬ces to prove SL = AL for some exten-

sion L/F .

Suppose that L is a splitting ¬eld of A. By (??), we have

(AL , σL ) EndL (V ), σb

for some vector space V over L and some nonsingular symmetric or skew-symmetric

bilinear form b on V .

30 I. INVOLUTIONS AND HERMITIAN FORMS

Suppose ¬rst that σ is symplectic, hence b is alternating. Identifying V with Ln

by means of a symplectic basis, we get

(AL , σL ) = Mn (L), σg

where g is the n — n block-diagonal matrix

01 01

g = diag ,...,

’1 0 ’1 0

and σg (m) = g ’1 · mt · g for all m ∈ Mn (L). The σg -alternating elements in Mn (L)

are of the form

g ’1 · x ’ σg (g ’1 · x) = g ’1 · (x + xt ),

where x ∈ Mn (L). Let (eij )1¤i,j¤n be the standard basis of Mn (L). For i = 1,

. . . , n/2 and j = 2i ’ 1, 2i we have

e2i’1,j = g ’1 · (e2i’1,2i + e2i,2i’1 ) · g ’1 · (e2i,j + ej,2i )

and

e2i,j = g ’1 · (e2i’1,2i + e2i,2i’1 ) · g ’1 · (e2i’1,j + ej,2i’1 ),

hence e2i’1,j and e2i,j are in the subalgebra SL of Mn (L) generated by σg -alterna-

ting elements. Since n ≥ 4 we may ¬nd for all i = 1, . . . , n/2 some j = 2i’1, 2i; the

elements e2i’1,j , e2i,j and their transposes are then in SL , hence also the products

of these elements, among which one can ¬nd e2i’1,2i’1 , e2i’1,2i , e2i,2i’1 and e2i,2i .

Therefore, SL = Mn (L) and the proof is complete if σ is symplectic.

Suppose next that σ is orthogonal, hence that b is symmetric but not alternat-

ing. The vector space V then contains an orthogonal basis (vi )1¤i¤n . (If char F = 2,

this follows from a theorem of Albert, see Kaplansky [?, Theorem 20].) Extending L

further, if necessary, we may assume that b(vi , vi ) = 1 for all i, hence

(AL , σL ) Mn (L), t .

If i, j, k ∈ {1, . . . , n} are pairwise distinct, then we have eij ’ eji , eik ’ eki ∈

Alt Mn (L), t and

eii = (eij ’ eji )2 · (eik ’ eki )2 , eij = eii · (eij ’ eji ),

hence alternating elements generate Mn (L).

(??) The same argument as in (??) shows that it su¬ces to prove the propo-

sition over an arbitrary scalar extension. Extending scalars to the center of B if

(E — E op , µ) for

this center is a ¬eld, we are reduced to the case where (B, „ )

some central simple F -algebra E, by (??). Extending scalars further to a splitting

¬eld of E, we may assume that E is split. Therefore, it su¬ces to consider the case

of Mn (F ) — Mn (F )op , µ . Again, let (eij )1¤i,j¤n be the standard basis of Mn (F ).

For i, j, k ∈ {1, . . . , n} pairwise distinct we have in Mn (F ) — Mn (F )op

(eij , ’eop ) · (ejk , ’eop ) = (eik , 0) and (ejk , ’eop ) · (eij , ’eop ) = (0, eop ).

ij ij

jk jk ik

In each case, both factors on the left side are skew-symmetric of trace zero, hence

0

Skew Mn (F ) — Mn (F )op , µ generates Mn (F ) — Mn (F )op if n ≥ 3.

(2.27) Remarks. (1) Suppose that A is a quaternion algebra over F and that

σ is an involution of the ¬rst kind on A. If char F = 2 or if σ is orthogonal,

the space Alt(A, σ) has dimension 1, hence it generates a commutative subalgebra

of A. However, (??.??) also holds when deg A = 2, provided char F = 2 and σ is

symplectic, since then A = F • Alt(A, σ).

§3. EXISTENCE OF INVOLUTIONS 31

(2) Suppose B is a quaternion algebra over a quadratic ´tale F -algebra K and that

e

„ is an involution of the second kind on B leaving F elementwise invariant. Let

ι be the nontrivial automorphism of K/F . Proposition (??) shows that there is a

quaternion F -algebra Q in B such that

(B, „ ) = (Q, γ) —F (K, ι),

where γ is the canonical involution on Q. It is easily veri¬ed that

Skew(B, „ )0 = Skew(Q, γ).

Therefore, the subalgebra of B generated by Skew(B, „ )0 is Q and not B.

Proof of (??): Since the arguments are the same for both parts, we just give the

∼

proof of (??). Let g : (AL , σL ) ’ (AL , σL ) be an isomorphism of L-algebras with

’

involution whose restriction to Alt(A, σ) is f . In particular, g maps Alt(A, σ)

to Alt(A , σ ). Since deg AL = deg AL and the degree does not change under

scalar extension, A and A have the same degree, which by hypothesis is at least 3.

Lemma (??) shows that Alt(A, σ) generates A and Alt(A , σ ) generates A , hence

g maps A to A and restricts to an isomorphism of F -algebras with involution

∼

(A, σ) ’ (A , σ ). This isomorphism is uniquely determined by f since Alt(A, σ)

’

generates A.

It is not di¬cult to give examples to show that (??) does not hold for alge-

bras of degree 2. The easiest example is obtained from quaternion algebras Q, Q

of characteristic 2 with canonical involutions γ, γ . Then Alt(Q, γ) = L(F ) =

∼

Alt(Q , γ ) and the identity map Alt(Q, γ) ’ Alt(Q , γ ) extends to an isomor-

’

∼

phism (QL , γL ) ’ (QL , γL ) if L is an algebraic closure of F . However, Q and Q

’

may not be isomorphic.

(2.28) Remark. Inspection of the proof of (??) shows that the Lie algebra struc-

tures on Alt(A, σ) or Skew(B, „ )0 are not explicitly used. Therefore, (??) also

holds for any linear map f ; indeed, if f extends to an isomorphism of (associative)

L-algebras with involution, then it necessarily is an isomorphism of Lie algebras.

§3. Existence of Involutions

The aim of this section is to give a proof of the following Brauer-group charac-

terization of central simple algebras with involution:

(3.1) Theorem. (1) (Albert) Let A be a central simple algebra over a ¬eld F .

There is an involution of the ¬rst kind on A if and only if A —F A splits.

(2) (Albert-Riehm-Scharlau) Let K/F be a separable quadratic extension of ¬elds

and let B be a central simple algebra over K. There is an involution of the second

kind on B which leaves F elementwise invariant if and only if the norm7 NK/F (B)

splits.

In particular, if a central simple algebra has an involution, then every Brauer-

equivalent algebra has an involution of the same kind.

We treat each part separately. We follow an approach based on ideas of

T. Tamagawa (oral tradition”see Berele-Saltman [?, §2] and Jacobson [?, §5.2]),

starting with the case of involutions of the ¬rst kind. For involutions of the second

kind, our arguments are very close in spirit to those of Deligne and Sullivan [?,

Appendix B].

7 See (??) below for the de¬nition of the norm (or corestriction) of a central simple algebra.

32 I. INVOLUTIONS AND HERMITIAN FORMS

3.A. Existence of involutions of the ¬rst kind. The fact that A —F A

splits when A has an involution of the ¬rst kind is easy to see (and was already

observed in the proof of Corollary (??)).

(3.2) Proposition. Every F -linear anti-automorphism σ on a central simple al-

gebra A endows A with a right A —F A-module structure de¬ned by

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

The reduced dimension of A as a right A —F A-module is 1, hence A —F A is split.

Proof : It is straightforward to check that the multiplication —σ de¬nes a right

A —F A-module structure on A. Since dimF A = deg(A —F A), we have rdim A = 1,

hence A—F A is split, since the index of a central simple algebra divides the reduced

dimension of every module of ¬nite type.

∼

(3.3) Remark. The isomorphism σ— : A —F A ’ EndF (A) de¬ned in the proof

’

of Corollary (??) endows A with a structure of left A —F A-module, which is less

convenient in view of the discussion below (see (??)).

To prove the converse, we need a special element in A—F A, called the Goldman

element (see Knus-Ojanguren [?, p. 112] or Rowen [?, p. 222]).

The Goldman element. For any central simple algebra A over a ¬eld F we

consider the F -linear sandwich map

Sand : A —F A ’ EndF (A)

de¬ned by

Sand(a — b)(x) = axb for a, b, x ∈ A.

(3.4) Lemma. The map Sand is an isomorphism of F -vector spaces.

Proof : Sand is the composite of the isomorphism A —F A A —F Aop which maps

a — b to a — bop and of the canonical F -algebra isomorphism A —F Aop EndF (A)

of Wedderburn™s theorem (??).

Consider the reduced trace TrdA : A ’ F . Composing this map with the

inclusion F ’ A, we may view TrdA as an element in EndF (A).

(3.5) De¬nition. The Goldman element in A —F A is the unique element g ∈

A —F A such that

Sand(g) = TrdA .

(3.6) Proposition. The Goldman element g ∈ A—F A satis¬es the following prop-

erties:

(1) g 2 = 1.

(2) g · (a — b) = (b — a) · g for all a, b ∈ A.

(3) If A = EndF (V ), then with respect to the canonical identi¬cation A —F A =

EndF (V —F V ) the element g is de¬ned by

g(v1 — v2 ) = v2 — v1 for v1 , v2 ∈ V .

Proof : We ¬rst check (??) by using the canonical isomorphism EndF (V ) = V —F

V — , where V — = HomF (V, F ) is the dual of V . If (ei )1¤i¤n is a basis of V and

(e— )1¤i¤n is the dual basis, consider the element

i

ei — e— — ej — e— ∈ V — V — — V — V — = EndF (V ) —F EndF (V ).

g= j i

i,j

§3. EXISTENCE OF INVOLUTIONS 33

For all f ∈ EndF (V ), we have

— e— ) —¦ f —¦ (ej — e— ) = — e— ) · e— f (ej ) .

Sand(g)(f ) = i,j (ei i,j (ei

j i i j

— e— = IdV and e— f (ej ) = tr(f ), the preceding equation shows

Since i ei i j

j

that

Sand(g)(f ) = tr(f ) for f ∈ EndF (V ),

hence g is the Goldman element in EndF (V ) — EndF (V ). On the other hand, for

v1 , v2 ∈ V we have

— e— )(v1 ) — (ej — e— )(v2 )

g(v1 — v2 ) = i,j (ei j i

· e— (v2 ) — ej · e— (v1 )

= i ei i j

j

= v2 — v1 .

This completes the proof of (??).

In view of (??), parts (??) and (??) are easy to check in the split case A =

EndF (V ), hence they hold in the general case also. Indeed, for any splitting ¬eld L

of A the Goldman element g in A —F A is also the Goldman element in AL —L AL

since the sandwich map and the reduced trace map commute with scalar extensions.

Since AL is split we have g 2 = 1 in AL —L AL , and g · (a — b) = (b — a) · g for all a,

b ∈ AL , hence also for all a, b ∈ A.

Consider the left and right ideals in A —F A generated by 1 ’ g:

J = (A —F A) · (1 ’ g), Jr = (1 ’ g) · (A —F A).

Let

»2 A = EndA—A (J ), s2 A = EndA—A (Jr ).

0

If deg A = 1, then A = F and g = 1, hence J = Jr = {0} and »2 A = {0},

s2 A = F . If deg A > 1, Proposition (??) shows that the algebras »2 A and s2 A are

Brauer-equivalent to A —F A.

(3.7) Proposition. If deg A = n > 1,

n(n’1) n(n+1)

rdim J = rdim Jr = deg »2 A = deg s2 A = .

and

2 2

For any vector space V of dimension n > 1, there are canonical isomorphisms

2

»2 EndF (V ) = EndF ( s2 EndF (V ) = EndF (S 2 V ),

V) and

2

V and S 2 V are the exterior and symmetric squares of V , respectively.

where

Proof : Since the reduced dimension of a module and the degree of a central simple

algebra are invariant under scalar extension, we may assume that A is split. Let

A = EndF (V ) and identify A —F A with EndF (V — V ). Then

J = HomF V — V / ker(Id ’ g), V — V and Jr = HomF V — V, im(Id ’ g) ,

and, by (??),

»2 A = EndF V — V / ker(Id ’ g) and s2 A = EndF V — V / im(Id ’ g) .

Since g(v1 — v2 ) = v2 — v1 for v1 , v2 ∈ V , there are canonical isomorphisms

2 ∼ ∼

V ’ V — V / ker(Id ’ g) and S 2 V ’ V — V / im(Id ’ g)

’ ’

34 I. INVOLUTIONS AND HERMITIAN FORMS

which map v1 § v2 to v1 — v2 + ker(Id ’ g) and v1 · v2 to v1 — v2 + im(Id ’ g)

respectively, for v1 , v2 ∈ V . Therefore, »2 A = EndF ( 2 V ), s2 A = EndF (S 2 V )

and

2

rdim Jr = rdim J = dim V — V / ker(Id ’ g) = dim V.

Involutions of the ¬rst kind and one-sided ideals. For every F -linear

anti-automorphism σ on a central simple algebra A, we de¬ne a map

σ : A —F A ’ A

by

σ (a — b) = σ(a)b for a, b ∈ A.

This map is a homomorphism of right A —F A-modules, if A is endowed with the

right A —F A-module structure of Proposition (??). The kernel ker σ is therefore

a right ideal in A —F A which we write Iσ :

Iσ = ker σ .

Since σ is surjective, we have

dimF Iσ = dimF (A —F A) ’ dimF A.

On the other hand, σ (1 — a) = a for a ∈ A, hence Iσ © (1 — A) = {0}. Therefore,

A —F A = Iσ • (1 — A).

As above, we denote by g the Goldman element in A —F A and by J and Jr the

left and right ideals in A —F A generated by 1 ’ g.

(3.8) Theorem. The map σ ’ Iσ de¬nes a one-to-one correspondence between

the F -linear anti-automorphisms of A and the right ideals I of A —F A such that

A —F A = I • (1 — A). Under this correspondence, involutions of symplectic type

correspond to ideals containing J 0 and involutions of orthogonal type to ideals con-

taining Jr but not J 0 . In the split case A = EndF (V ), the ideal corresponding to the

adjoint anti-automorphism σb with respect to a nonsingular bilinear form b on V is

HomF V — V, ker(b —¦ g) where b is considered as a linear map b : V — V ’ F . (If

b is symmetric or skew-symmetric, then ker(b —¦ g) = ker b.)

Proof : To every right ideal I ‚ A —F A such that A —F A = I • (1 — A), we

associate the map σI : A ’ A de¬ned by projection of A — 1 onto 1 — A parallel

to I; for a ∈ A, we de¬ne σI (a) as the unique element in A such that

a — 1 ’ 1 — σI (a) ∈ I.

This map is clearly F -linear. Moreover, for a, b ∈ A we have

ab — 1 ’ 1 — σI (b)σI (a) = a — 1 ’ 1 — σI (a) · b — 1

+ b — 1 ’ 1 — σI (b) · 1 — σI (a) ∈ I,

hence σI (ab) = σI (b)σI (a), which proves that σI is an anti-automorphism.

For every anti-automorphism σ of A, the de¬nition of Iσ shows that

a — 1 ’ 1 — σ(a) ∈ Iσ for a ∈ A.

§3. EXISTENCE OF INVOLUTIONS 35

Therefore, σIσ = σ. Conversely, suppose that I ‚ A —F A is a right ideal such that

A —F A = I • (1 — A). If x = yi — zi ∈ ker σI , then σI (yi )zi = 0, hence

x= yi — zi ’ 1 — σI (yi )zi = yi — 1 ’ 1 — σI (yi ) · (1 — zi ).

This shows that the right ideal ker σI is generated by elements of the form

a — 1 ’ 1 — σI (a).

Since these elements all lie in I, by de¬nition of σI , it follows that ker σI ‚ I.

However, these ideals have the same dimension, hence ker σI = I and therefore

IσI = I.

We have thus shown that the maps σ ’ Iσ and I ’ σI de¬ne inverse bijections

between anti-automorphisms of A and right ideals I in A —F A such that A —F A =

I • (1 — A).

In order to identify the ideals which correspond to involutions, it su¬ces to

consider the split case. Suppose that A = EndF (V ) and that σ = σb is the adjoint

anti-automorphism with respect to some nonsingular bilinear form b on V . By

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

h (x, y) = h v(x), y for x, y ∈ M .

Therefore, the adjoint involutions σh and σh are related by

σh = Int(v) —¦ σh .

Therefore, if σh = σh , then v ∈ F — and the forms h, h di¬er by a factor in F — .

If θ is of the second kind and h, h are both hermitian, the relation h = v · h

implies that θ(v) = v. We have thus shown injectivity of the map h ’ σh on the

set of equivalence classes modulo factors in F — (invariant under θ) in both cases

(??) and (??).

Let D be a central division algebra Brauer-equivalent to E. We may then

identify E with Ms (D) for some integer s, hence also M with Mr,s (D) and A with

Mr (D), as in the proof of (??). We may thus assume that

A = Mr (D), M = Mr,s (D), E = Ms (D).

Theorem (??) shows that D carries an involution such that ± = θ(±) for all ± ∈ F .

We use the same notation — for the maps A ’ A, E ’ E and M ’ Ms,r (D) de¬ned

by

(dij )— = (dij )t .

i,j i,j

44 I. INVOLUTIONS AND HERMITIAN FORMS

Proposition (??) shows that the maps — on A and E are involutions of the same

type as .

Consider now case (??), where is of the ¬rst kind. According to (??), we may

¬nd u ∈ E — such that u— = ±u and θ = Int(u) —¦ — . Moreover, for any involution of

the ¬rst kind σ on A we may ¬nd some g ∈ A— such that g — = ±g and σ = Int(g)—¦ — .

De¬ne then a map h : M — M ’ E by

h(x, y) = u · x— · g ’1 · y for x, y ∈ M .

This map is clearly bi-additive. Moreover, for ±, β ∈ E and x, y ∈ M we have

h(x±, yβ) = u · ±— · x— · g ’1 · y · β = θ(±) · h(x, y) · β

and

—

h(y, x) = u · u— · x— · (g ’1 )— · y · u’1 = δθ h(x, y) ,

where δ = +1 if u’1 u— = g ’1 g — (= ±1) and δ = ’1 if u’1 u— = ’g ’1 g — (= 1).

Therefore, h is a hermitian or skew-hermitian form on M . For a ∈ A and x, y ∈ M ,

h(x, ay) = u · x— · (ga— g ’1 )— · g ’1 · y = h σ(a)x, y ,

hence σ is the adjoint involution with respect to h. To complete the proof of (??),

it remains to relate the type of σ to properties of h.

Suppose ¬rst that char F = 2. Proposition (??) shows that the type of θ (resp.

of σ) is the same as the type of if and only if u’1 u— = +1 (resp. g ’1 g — = +1).

Therefore, σ and θ are of the same type if and only if u’1 u— = g ’1 g — , and this

condition holds if and only if h is hermitian.

Suppose now that char F = 2. We have to show that h(x, x) ∈ Alt(E, θ) for

all x ∈ M if and only if σ is symplectic. Proposition (??) shows that this last

condition is equivalent to g ∈ Alt(A, — ). If g = a ’ a— for some a ∈ A, then

g ’1 = ’g ’1 g(g ’1 )— = b ’ b— for b = g ’1 a— (g ’1 )— . It follows that for all x ∈ M

h(x, x) = u · x— · b · x ’ θ(u · x— · b · x) ∈ Alt(E, θ).

Conversely, if h is alternating, then x— · g ’1 · x ∈ Alt(E, — ) for all x ∈ M , since (??)

shows that Alt(E, — ) = u’1 · Alt(E, θ). In particular, taking for x the matrix ei1

whose entry with indices (i, 1) is 1 and whose other entries are 0, it follows that the

i-th diagonal entry of g ’1 is in Alt(D, ). Let g ’1 = (gij )1¤i,j¤r and gii = di ’ di

for some di ∈ D. Then g ’1 = b ’ b— where the matrix b = (bij )1¤i,j¤r is de¬ned by

±

gij if i < j,

bij = di if i = j,

0 if i > j.

Therefore, g = ’gg ’1 g — = gb— g — ’ (gb— g — )— ∈ Alt(A, — ), completing the proof

of (??).

The proof of (??) is similar, but easier since there is only one type. Propo-

sition (??) yields an element u ∈ E — such that u— = u and θ = Int(u) —¦ — , and

shows that every involution σ on A such that σ(±) = ± for all ± ∈ F has the form

σ = Int(g) —¦ — for some g ∈ A— such that g — = g. The same computations as

for (??) show that σ is the adjoint involution with respect to the hermitian form h

on M de¬ned by

h(x, y) = u · x— · g ’1 · y for x, y ∈ M .

§4. HERMITIAN FORMS 45

The preceding theorem applies notably in the case where E is a division alge-

bra, to yield a correspondence between involutions on a central simple algebra A

and hermitian and skew-hermitian forms on vector spaces over the division algebra

Brauer-equivalent to A. However, (??) shows that a given central simple algebra

may be represented as A = EndE (M ) for any central simple algebra E Brauer-

equivalent to A (and for a suitable E-module M ). Involutions on A then corre-

spond to hermitian and skew-hermitian forms on M by the preceding theorem. In

particular, if A has an involution of the ¬rst kind, then a theorem of Merkurjev [?]

shows that we may take for E a tensor product of quaternion algebras.

4.B. Extension of involutions and transfer. This section analyzes the

possibility of extending an involution from a simple subalgebra. One type of exten-

sion is based on an analogue of the Scharlau transfer for hermitian forms which is

discussed next. The general extension result, due to Kneser, is given thereafter.

The transfer. Throughout this subsection, we consider the following situa-

tion: 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. Let C be the centralizer of T in E. By

the double centralizer theorem (see (??)) this algebra is central simple over Z and

E = T —F C.

Suppose that θ is an involution on E (of any kind) which preserves T , hence also C.

For simplicity, we also call θ the restriction of θ to T and to C.

(4.3) De¬nition. An F -linear map s : E ’ T is called an involution trace if it

satis¬es the following conditions (see Knus [?, (7.2.4)]):

(1) s(t1 xt2 ) = t1 s(x)t2 for all x ∈ E and t1 , t2 ∈ T ;

(2) s θ(x) = θ s(x) for all x ∈ E;

(3) if x ∈ E is such that s θ(x)y = 0 for all y ∈ E, then x = 0.

In view of (??), condition (??) may equivalently be phrased as follows: the only

element y ∈ E such that s θ(x)y = 0 for all x ∈ E is y = 0. It is also equivalent

to the following:

(?? ) ker s does not contain any nontrivial left or right ideal in E.

Indeed, I is a right (resp. left) ideal in ker s if and only if s θ(x)y = 0 for all

θ(x) ∈ I and all y ∈ E (resp. for all x ∈ E and y ∈ I).

For instance, if T = F = Z, the reduced trace TrdE : E ’ F is an involution

trace. Indeed, condition (??) follows from (??) if θ is of the ¬rst kind and from (??)

if θ is of the second kind, and condition (??) follows from the fact that the bilinear

(reduced) trace form is nonsingular (see (??)).

If E = Z and T = F , every nonzero linear map s : Z ’ F which commutes

with θ is an involution trace. Indeed, if x ∈ Z is such that s θ(x)y = 0 for all

y ∈ Z, then x = 0 since s = 0 and Z = { θ(x)y | y ∈ Z } if x = 0.

The next proposition shows that every involution trace s : E ’ T can be ob-

tained by combining these particular cases.

(4.4) Proposition. Fix a nonzero linear map : Z ’ F which commutes with θ.

For every unit u ∈ Sym(C, θ), the map s : E ’ T de¬ned by

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

is an involution trace. Every involution trace from E to T is of the form above for

some unit u ∈ Sym(C, θ).

46 I. INVOLUTIONS AND HERMITIAN FORMS

Proof : Conditions (??) and (??) are clear. Suppose that x = i ti — ci ∈ E is

such that s θ(x)y = 0 for all y ∈ E. We may assume that the elements ti ∈ T

are linearly independent over F . The relation s θ(x) · 1 — c = 0 for all c ∈ C then

yields TrdC uθ(ci )c = 0 for all i and all c ∈ C. Since is nonzero, it follows

that TrdC uθ(ci )c = 0 for all i and all c ∈ C, hence uθ(ci ) = 0 for all i since the

bilinear reduced trace form is nonsingular. It follows that θ(ci ) = 0 for all i since u

is invertible, hence x = 0.

Let s : E ’ T be an arbitrary involution trace. For t ∈ T and c ∈ C,

t · s(1 — c) = s(t — c) = s(1 — c) · t,

hence the restriction of s to C takes values in F and s = IdT — s0 where s0 : C ’ F

denotes this restriction. Since is nonzero and the bilinear reduced trace form is

nonsingular, the linear map C ’ HomF (C, F ) which carries c ∈ C to the linear map