µ(g)’m NrdB (g) = ±ι(±)’1 .

We may therefore de¬ne a homomorphism ν : GU(A, σ) ’ K — /F — by

ν(g) = ± · F — .

(14.2)

Let SGU(B, „ ) be the kernel of ν, and let SU(B, „ ) be the intersection SGU(B, „ ) ©

U(B, „ ):

SGU(B, „ ) = { g ∈ GU(B, „ ) | NrdB (g) = µ(g)m }

SU(B, „ ) = { u ∈ GU(B, „ ) | NrdB (u) = µ(u) = 1 }.

We thus have the following diagram, where all the maps are inclusions:

SU(B, „ ) ’ ’ ’ SGU(B, „ )

’’

¦ ¦

¦ ¦

U(B, „ ) ’ ’ ’ GU(B, „ ).

’’

Consider for example the case where K = F —F ; we may then assume B = E —E op

for some central simple F -algebra E of degree 2m, and „ = µ is the exchange

involution. We then have

GU(B, „ ) = { x, ±(x’1 )op | ± ∈ F — , x ∈ E — } E— — F —

and the maps µ and ν are de¬ned by

µ x, ±(x’1 )op = ±, ν x, ±(x’1 )op = NrdE (x), ±m · F — .

Therefore,

{ (x, ±) ∈ E — — F — | NrdE (x) = ±m },

SGU(B, „ )

E—

U(B, „ )

and the group SU(B, „ ) is isomorphic to the group of elements of reduced norm 1

in E, which we write SL(E):

{ x ∈ E — | NrdE (x) = 1 } = SL(E).

SU(B, „ )

14.C. Relation with the discriminant algebra. Our ¬rst results in this

direction do not assume the existence of an involution; we formulate them for an

arbitrary central simple F -algebra A:

§14. UNITARY INVOLUTIONS 195

The canonical map »k .

(14.3) Proposition. Let A be any central simple algebra over a ¬eld F . For all

integers k such that 1 ¤ k ¤ deg A, there is a homogeneous polynomial map of

degree k:

»k : A ’ » k A

which restricts to a group homomorphism A— ’ (»k A)— . If the algebra A is split,

let A = EndF (V ), then under the identi¬cation »k A = EndF ( k V ) the map »k is

de¬ned by

k

»k (f ) = f = f § ···§ f for f ∈ EndF (V ).

Proof : Let gk : Sk ’ (A—k )— be the homomorphism of (??). By (??), it is clear

that for all a ∈ A— the element —k a = a — · · · — a commutes with gk (π) for all

π ∈ Sk , hence also with sk = π∈Sk sgn(π)gk (π). Multiplication on the right by

—k a is therefore an endomorphism of the left A—k -module A—k sk . We denote this

endomorphism by »k a; thus »k a ∈ EndA—k (A—k sk ) = »k A is de¬ned by

»k a

= (a1 — · · · — ak ) · sk · —k a = (a1 a — · · · — ak a) · sk .

(a1 — · · · — ak ) · sk

If A = EndF (V ), there is a natural isomorphism (see (??)):

k

A—k sk = HomF ( V , V —k ),

k

V ’ V —k de¬ned by

under which sk is identi¬ed with the map sk :

sk (v1 § · · · § vk ) = sk (v1 — · · · — vk ) for v1 , . . . , vk ∈ V .

»k f

(v1 § · · · § vk ) = sk —k f (v1 — · · · — vk ) , hence

For f ∈ EndF (V ) we have sk

»k f

sk (v1 § · · · § vk ) = sk f (v1 ) § · · · § f (vk ) .

»k f k

f , which means that »k (f ) ∈ »k EndF (V ) is identi¬ed

Therefore, sk = sk —¦

k k

f ∈ EndF ( V ). It is then clear that »k is a homogeneous polynomial map

with

of degree k, and that its restriction to A— is a group homomorphism to (»k A)— .

For the following result, we assume deg A = 2m, so that »m A has a canonical

involution γ of the ¬rst kind (see (??)).

(14.4) Proposition. If deg A = 2m, then γ(»m a)»m a = NrdA (a) for all a ∈ A.

In particular, if a ∈ A— , then »m a ∈ Sim(»m A, γ) and µ(»m a) = NrdA (a).

Proof : It su¬ces to check the split case. We may thus assume A = EndF (V ),

m

hence »m A = EndF ( V ) and the canonical involution γ is the adjoint involution

m m 2m

with respect to the canonical bilinear map § : V— V’ V . Moreover,

m

m m

» (f ) = f for f ∈ EndF (V ). The statement that » (f ) is a similitude for γ

therefore follows from the following identities

m m

f(v1 § · · · § vm ) § f (w1 § · · · § wm )

= f (v1 ) § · · · § f (vm ) § f (w1 ) § · · · § f (wm )

= det f · v1 § · · · § vm § w1 § · · · § wm

for v1 , . . . , vm , w1 , . . . , wm ∈ V .

196 III. SIMILITUDES

The canonical map D. We now return to the case of central simple algebras

with unitary involution (B, „ ). We postpone until after Proposition (??) the dis-

cussion of the case where the center K of B is isomorphic to F — F ; we thus assume

for now that K is a ¬eld.

(14.5) Lemma. For k = 1, . . . , deg B, let „ §k be the involution on »k B induced

by „ (see (??)). For all k, the canonical map »k : B ’ »k B satis¬es

„ §k —¦ »k = »k —¦ „.

Proof : By extending scalars to a splitting ¬eld of B, we reduce to considering

the split case. We may thus assume B = EndK (V ) and „ = σh is the adjoint

involution with respect to some nonsingular hermitian form h on V . According to

(??), the involution „ §k is the adjoint involution with respect to h§k . Therefore, for

f ∈ EndK (V ), the element „ §k —¦ »k (f ) ∈ EndK ( k V ) is de¬ned by the condition:

h§k „ §k —¦ »k (f )(v1 § · · · § vk ), w1 § · · · § wk =

h§k v1 § · · · § vk , »k (f )(w1 § · · · § wk )

k

for v1 , . . . , vk , w1 , . . . , wk ∈ V . Since »k (f ) = f , the right-hand expression

equals

det h vi , f (wj ) = det h „ (f )(vi ), wj

1¤i,j¤k 1¤i,j¤k

§k k

=h » „ (f ) (v1 § · · · § vk ), w1 § · · · § wk .

Assume now deg B = 2m; we may then de¬ne the discriminant algebra D(B, „ )

as the subalgebra of »m B of elements ¬xed by „ §m —¦ γ, see (??).

(14.6) Lemma. For g ∈ GU(B, „ ), let ± ∈ K — be such that ν(g) = ± · F — ; then

±’1 »m g ∈ D(B, „ ) and „ (±’1 »m g) · ±’1 »m g = NK/F (±)’1 µ(g)m . In particular,

»m g ∈ Sim D(B, „ ), „ for all g ∈ SGU(B, „ ).

Proof : By (??) we have

γ(»m g) = NrdB (g)»m g ’1 = NrdB (g)µ(g)’m »m „ (g) ,

hence, by (??),

„ §m —¦ γ(»m g) = ι NrdB (g) µ(g)’m »m g = ±’1 ι(±)»m g.

Therefore, ±’1 »m g ∈ D(B, „ ). Since „ is the restriction of γ to D(B, „ ), we have

„ (±’1 »m g) · ±’1 »m g = ±’2 γ(»m g)»m g,

and (??) completes the proof.

The lemma shows that the inner automorphism Int(»m g) = Int(±’1 »m g)

of »m B preserves D(B, „ ) and induces an automorphism of D(B, „ ), „ . Since

this automorphism is also induced by the automorphism Int(g) of (B, „ ), by func-

toriality of the discriminant algebra construction, we denote it by D(g). Alternately,

under the identi¬cation Aut D(B, „ ), „ = PSim D(B, „ ), „ of (??), we may set

D(g) = ±’1 »m g · F — , where ± ∈ K — is a representative of ν(g) as above.

The next proposition follows from the de¬nitions and from (??):

§14. UNITARY INVOLUTIONS 197

(14.7) Proposition. The following diagram commutes:

SGU(B, „ ) ’’’

’’ GU(B, „ )

¦ ¦

¦ ¦

m

D

»

Int

Sim D(B, „ ), „ ’ ’ ’ AutF D(B, „ ), „ .

’’

Moreover, for g ∈ SGU(B, „ ), the multipliers of g and »m g are related by

µ(»m g) = µ(g)m = NrdB (g).

Therefore, »m restricts to a group homomorphism SU(B, „ ) ’ Iso D(B, „ ), „ .

We now turn to the case where K F — F , which was put aside for the

preceding discussion. In this case, we may assume B = E — E op for some central

simple F -algebra E of degree 2m and „ = µ is the exchange involution. As observed

in §?? and §??, we may then identify

D(B, „ ), „ = (»m E, γ) and GU(B, „ ) = E — — F — .

The discussion above remains valid without change if we set D(x, ±) = Int(»m x) for

(x, ±) ∈ E — — F — = GU(B, „ ), a de¬nition which is compatible with the de¬nitions

above (in the case where K is a ¬eld) under scalar extension.

™

The canonical Lie homomorphism »k . To conclude this section, we derive

from the map »m a Lie homomorphism from the Lie algebra Skew(B, „ )0 of skew-

symmetric elements of reduced trace zero to the Lie algebra Skew D(B, „ ), „ . This

Lie homomorphism plays a crucial rˆle in §?? (see (??)).

o

As above, we start with an arbitrary central simple F -algebra A. Let t be an

indeterminate over F . For k = 1, . . . , deg A, consider the canonical map

»k : A — F (t) ’ »k A — F (t).

Since this map is polynomial of degree k and »k (1) = 1, there is a linear map

™

»k : A ’ »k A such that for all a ∈ A,

™

»k (t + a) = tk + »k (a)tk’1 + · · · + »k (a).

™

(14.8) Proposition. The map »k is a Lie-algebra homomorphism

™

»k : L(A) ’ L(»k A).

k

If A = EndF (V ), then under the identi¬cation »k A = EndF ( V ) we have

™

»k (f )(v1 § · · · § vk ) =

f (v1 ) § v2 § · · · § vk + v1 § f (v2 ) § · · · § vk + · · · + v1 § v2 § · · · § f (vk )

for all f ∈ EndF (V ) and v1 , . . . , vk ∈ V .

™

Proof : The description of »k in the split case readily follows from that of »k in

™ k

(??). To prove that » is a Lie homomorphism, we may reduce to the split case by

a scalar extension. The property then follows from an explicit computation: for f ,

198 III. SIMILITUDES

g ∈ EndF (V ) and v1 , . . . , vk ∈ V we have

™ ™

»k (f ) —¦ »k (g)(v1 § · · · § vk ) = v1 § · · · § f (vi ) § · · · § g(vj ) § · · · § vk

1¤i<j¤k

+ v1 § · · · § f —¦ g(vi ) § · · · § vk

1¤i¤k

+ v1 § · · · § g(vj ) § · · · § f (vi ) § · · · § vk ,

1¤j<i¤k

™ ™ ™ ™

hence »k (f ) —¦ »k (g) ’ »k (g) —¦ »k (f ) maps v1 § · · · § vk to

™

v1 § · · · § (f —¦ g ’ g —¦ f )(vi ) § · · · § vk = »k [f, g] (v1 § · · · § vk ).

1¤i¤k

™ ™ ™

This shows »k (f ), »k (g) = »k [f, g] .

™

(14.9) Corollary. Suppose k ¤ deg A ’ 1. If a ∈ A satis¬es »k a ∈ F , then a ∈ F

™ ™

and »k a = ka. In particular, ker »k = { a ∈ F | ka = 0 }.

Proof : It su¬ces to consider the split case; we may thus assume that A = EndF (V )

™

for some vector space V . If »k a ∈ F , then for all x1 , . . . , xk ∈ V we have

™

x1 § »k a(x1 § · · · § xk ) = 0, hence x1 § a(x1 ) § x2 § · · · § xk = 0. Since k < dim V ,

this relation shows that a(x1 ) ∈ x1 · F for all x1 ∈ V , hence a ∈ F . The other

statements are then clear.

1

In the particular case where k = deg A, we have:

2

(14.10) Proposition. Suppose deg A = 2m, and let γ be the canonical involution

on »m A. For all a ∈ A,

™ ™

»m a + γ(»m a) = TrdA (a).

Proof : By (??), we have

γ »m (t + a) · »m (t + a) = Nrd(t + a).

The proposition follows by comparing the coe¬cients of t2m’1 on each side.

We now consider a central simple algebra with unitary involution (B, „ ) over F ,

and assume that the center K of B is a ¬eld. Suppose also that the degree of B is

even: deg B = 2m. Since (??) shows that

„ §m —¦ »m (t + b) = »m —¦ „ (t + b) for b ∈ B,

it follows that

™ ™

„ §m —¦ »m = »m —¦ „.

™

It is now easy to determine under which condition »m b ∈ D(B, „ ): this holds if and

™ ™

only if γ(»m b) = „ §m (»m b), which means that

™ ™

TrdB (b) ’ »m b = »m „ (b).

By (??), this equality holds if and only if b + „ (b) ∈ F and TrdB (b) = m b + „ (b) .

Let

(14.11) s(B, „ ) = { b ∈ B | b + „ (b) ∈ F and TrdB (b) = m b + „ (b) },

™

so s(B, „ ) = (»m )’1 D(B, „ ) . The algebra s(B, „ ) is contained in g(B, „ ) = { b ∈

B | b + „ (b) ∈ F } (see §??). If µB : g(B, „ ) ’ F is the map which carries

™

EXERCISES 199

b ∈ g(B, „ ) to b + „ (b), we may describe s(B, „ ) as the kernel of the F -linear

map TrdB ’mµB : g(B, „ ) ’ K. The image of this map lies in K 0 = { x ∈

™

K | TK/F (x) = 0 }, since taking the reduced trace of both sides of the relation

b + „ (b) = µ(b) yields TK/F TrdB (b) = 2mµB (b). On the other hand, this map is

™ ™

not trivial: since TrdB is surjective, we may ¬nd x ∈ B such that TrdB (x) ∈ F ;

/

then x ’ „ (x) ∈ g(B, „ ) is not mapped to 0. Therefore,

dimF s(B, „ ) = dimF g(B, „ ) ’ 1 = 4m2 .

™

(14.12) Proposition. The homomorphism »m restricts to a Lie algebra homo-

morphism

™

»m : s(B, „ ) ’ g D(B, „ ), „ .

By denoting by µD : g D(B, „ ), „ ’ F the map which carries x ∈ D(B, „ ) to

™

x + „ (x), we have

™™

µD (»m b) = mµB (b) = TrdB (b)

™ for b ∈ s(B, „ ).

™

Therefore, »m restricts to a Lie algebra homomorphism

™

»m : Skew(B, „ )0 ’ Skew D(B, „ ), „ ,

where Skew(B, „ )0 is the Lie algebra of skew-symmetric elements of reduced trace

zero in B.

™ ™

Proof : Since s(B, „ ) = (»m )’1 D(B, „ ) , it is clear that »m restricts to a homo-

morphism from s(B, „ ) to L D(B, „ ) . To prove that its image lies in g D(B, „ ), „ ,

it su¬ces to prove

™ ™

»m b + „ (»m b) = TrdB (b) for b ∈ s(B, „ ).

This follows from (??), since „ is the restriction of γ to D(B, „ ).

Suppose ¬nally K F — F ; we may then assume B = E — E op for some central

simple F -algebra E of degree 2m, and „ = µ is the exchange involution. We have

s(B, „ ) = { (x, ± ’ xop ) | x ∈ E, ± ∈ F , and TrdE (x) = m± },

and D(B, „ ), „ may be identi¬ed with (»m E, γ). The Lie algebra homomorphism

™ ™

»m : s(B, „ ) ’ g(»m E, γ) then maps (x, ± ’ xop ) to »m x. Identifying Skew(B, „ )0

with the Lie algebra E 0 of elements of reduced trace zero (see (??)), we may restrict

this homomorphism to a Lie algebra homomorphism:

™

»m : E 0 ’ Skew(»m E, γ).

Exercises

1. Let Q be a quaternion algebra with canonical involution γ over a ¬eld F of

arbitrary characteristic. On the algebra A = Q —F Q, consider the canonical

quadratic pair (γ — γ, f ) (see (??)). Prove that

GO+ (A, γ — γ, f ) = { q1 — q2 | q1 , q2 ∈ Q— }

and determine the group of multipliers µ GO+ (A, γ — γ, f ) .

2. Let (A, σ) be a central simple F -algebra with involution of any kind with cen-

ter K and let ± ∈ AutK (A). Prove that the following statements are equivalent:

200 III. SIMILITUDES

(a) ± ∈ AutK (A, σ).

(b) ± Sym(A, σ) = Sym(A, σ).

(c) ± Alt(A, σ) = Alt(A, σ).

3. Let (A, σ) be a central simple algebra with orthogonal involution and degree

a power of 2 over a ¬eld F of characteristic di¬erent from 2, and let B ‚ A

be a proper subalgebra with center F = B. Prove that every similitude f ∈

GO(A, σ) such that f Bf ’1 = B is proper.

4. (Wonenburger [?]) The aim of this exercise is to give a proof of Wonenburger™s

theorem on the image of GO(V, q) in Aut C0 (V, q) , see (??).

Let q be a nonsingular quadratic form on an even-dimensional vector space

V over a ¬eld F of arbitrary characteristic. Using the canonical identi¬cation

∼

•q : V — V ’ EndF (V ), we identify c EndF (V ) = V · V ‚ C0 (V, q) and

’

2

Alt EndF (V ), σq = V . An element x ∈ V · V is called a regular plane

element if x = v · w for some vectors v, w ∈ V which span a nonsingular 2-

dimensional subspace of V . The ¬rst goal is to show that the regular plane

elements are preserved by the automorphisms of C0 (V, q) which preserve V · V .

2 4

Let ρ : V’ V be the quadratic map which vanishes on elements of

the type v § w and whose polar is the exterior product (compare with (??)),

and let „ be the canonical involution on C(V, q) which is the identity on V .

2

(a) Show that x ∈ V has the form x = v § w for some v, w ∈ V if and only

if ρ(x) = 0.

(b) Show that the Lie homomorphism δ : V · V ’ 2 V maps v · w to v § w.

(c) Let W = { x + „ (x) | x ∈ V · V · V · V } ‚ C0 (V, q). Show that F ‚ W and

that there is a surjective map ω : 4 V ’ W/F such that

ω(v1 § v2 § v3 § v4 ) = v1 · v2 · v3 · v4 + v4 · v3 · v2 · v1 + F.

Show that for all x ∈ V · V ,

’ω —¦ ρ —¦ δ(x) = „ (x) · x + F.

(d) Assume that char F = 2. Show that ω is bijective and use the results

above to show that if x ∈ V · V satis¬es „ (x) · x ∈ F , then δ(x) = v § w

for some v, w ∈ V .

(e) Assume that (V, q) is a 6-dimensional hyperbolic space over a ¬eld F of

characteristic 2, and let (e1 , . . . , e6 ) be a symplectic basis of V consisting

of isotropic vectors. Show that x = e1 · e2 + e3 · e4 + e5 · e6 ∈ V · V satis¬es

„ (x) · x = 0, „ (x) + x = 1, but δ(x) cannot be written in the form v § w

with v, w ∈ V .

(f) For the rest of this exercise, assume that char F = 2. Show that x ∈ V · V

is a regular plane element if and only if „ (x) · x ∈ F , „ (x) + x ∈ F and

2

„ (x) + x = 4„ (x) · x. Conclude that every automorphism of C0 (V, q)

which commutes with „ and preserves V · V maps regular plane elements

to regular plane elements.

(g) Show that regular plane elements x, y ∈ V · V anticommute if and only if

x = u · v and y = u · w for some pairwise orthogonal anisotropic vectors u,

v, w ∈ V .

(h) Let (e1 , . . . , en ) be an orthogonal basis of V . Let θ ∈ Aut C0 (V, q), „ be

an automorphism which preserves V · V . Show that there is an orthogonal

basis (v1 , v2 , . . . , vn ) of V such that θ(e1 · ei ) = v1 · vi for i = 2, . . . , n. Let

EXERCISES 201

± = q(v1 )’1 q(e1 ). Show that the linear transformation of V which maps

e1 to ±v1 and ei to vi for i = 2, . . . , n is a similitude which induces θ.

5. Let D be a central division algebra with involution over a ¬eld F of charac-

teristic di¬erent from 2 and let V be a (¬nite dimensional) right vector space

over D with a nonsingular hermitian form h. Let v ∈ V be an anisotropic

vector and let d ∈ D — be such that

h(v, v) = dh(v, v)d (= h(vd, vd)).