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