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154 III. SIMILITUDES


Speci¬c notations for the groups Sim(V, b), Iso(V, b) and PSim(V, b) are used ac-
cording to the type of b. If b is symmetric nonalternating, we set
O(V, b) = Iso(V, b), GO(V, b) = Sim(V, b) and PGO(V, b) = PSim(V, b);
if b is alternating, we let
Sp(V, b) = Iso(V, b), GSp(V, b) = Sim(V, b) and PGSp(V, b) = PSim(V, b).
Note that condition (??), de¬ning a similitude of (V, b) with multiplier ±, can be
rephrased as follows, using the adjoint involution σb :
(12.2) σb (g) —¦ g = ±IdV .
By taking the determinant of both sides, we obtain (det g)2 = ±n where n = dim V .
It follows that the determinant of an isometry is ±1 and that, if n is even,
det g = ±µ(g)n/2 for g ∈ Sim(V, b).
A ¬rst di¬erence between the orthogonal case and the symplectic case shows up in
the following result:
(12.3) Proposition. If b is a nonsingular alternating bilinear form on a vector
space V of dimension n (necessarily even), then
det g = µ(g)n/2 for g ∈ GSp(V, b).
Proof : Let g ∈ GSp(V, b) and let G, B denote the matrices of g and b respectively
with respect to some arbitrary basis of V . The matrix B is alternating and we have
Gt BG = µ(g)B.
By taking the pfa¬an of both sides, we obtain, by known formulas for pfa¬ans (see
Artin [?, Theorem 3.28]; compare with (??)):
det G pf B = µ(g)n/2 pf B,
hence det g = µ(g)n/2 .
By contrast, if b is symmetric and char F = 2, every hyperplane re¬‚ection is an
isometry with determinant ’1 (see (??)), hence it satis¬es det g = ’µ(g)n/2 .
We set
O+ (V, b) = { g ∈ O(V, b) | det g = 1 }.
Of course, O+ (V, b) = O(V, b) if char F = 2.
Similarly, if dim V = n is even, we set
GO+ (V, b) = { g ∈ GO(V, b) | det g = µ(g)n/2 },
and
PGO+ (V, b) = GO+ (V, b)/F — .
The elements in GO+ (V, b), O+ (V, b) are called proper similitudes and proper isome-
tries respectively.
If dim V is odd, there is a close relationship between similitudes and isometries,
as the next proposition shows:
(12.4) Proposition. Suppose that (V, b) is a nonsingular symmetric bilinear space
of odd dimension over an arbitrary ¬eld F ; then
GO(V, b) = O+ (V, b) · F — O+ (V, b) — F — O+ (V, b).
PGO(V, b)
and
§12. GENERAL PROPERTIES 155


Proof : If g is a similitude of (V, b) with multiplier ± ∈ F — , then by taking the
b we get ± ∈ F —2 . If ± = ±2 ,
determinant of both sides of the isometry ± · b 1
’1
then ±1 g is an isometry. Moreover, after changing the sign of ±1 if necessary, we
may assume that det(±’1 g) = 1. The factorization g = (±’1 g) · ±1 shows that
1 1
GO(V, b) = O+ (V, b) · F — , and the other isomorphisms are clear.
Hermitian spaces. Suppose (V, h) is a nonsingular hermitian space over a
quadratic separable ¬eld extension K of F (with respect to the nontrivial automor-
phism of K/F ). A similitude of (V, h) is an invertible linear map g : V ’ V for
which there exists a constant ± ∈ F — , called the multiplier of g, such that
(12.5) h g(v), g(w) = ±h(v, w) for v, w ∈ V .
As in the case of bilinear spaces, we write Sim(V, h) for the group of similitudes of
(V, h); let
µ : Sim(V, h) ’ F —
be the group homomorphism which carries every similitude to its multiplier; write
Iso(V, h) for the kernel of µ, whose elements are called isometries, and let
PSim(V, h) = Sim(V, h)/K — .
We also use the following more speci¬c notation:
U(V, h) = Iso(V, h), GU(V, h) = Sim(V, h), PGU(V, h) = PSim(V, h).
Condition (??) can be rephrased as
σh (g) —¦ g = ±IdV .
By taking the determinant of both sides, we obtain
NK/F (det g) = µ(g)n , where n = dim V .
This relation shows that the determinant of every isometry has norm 1. Set
SU(V, h) = { g ∈ U(V, h) | det g = 1 }.
Quadratic spaces. Let (V, q) be a nonsingular quadratic space over an arbi-
trary ¬eld F . A similitude of (V, q) is an invertible linear map g : V ’ V for which
there exists a constant ± ∈ F — , called the multiplier of g, such that
q g(v) = ±q(v) for v ∈ V .
The groups Sim(V, q), Iso(V, q), PSim(V, q) and the group homomorphism
µ : Sim(V, q) ’ F —
are de¬ned as for nonsingular bilinear forms. We also use the notation
O(V, q) = Iso(V, q), GO(V, q) = Sim(V, q), PGO(V, q) = PSim(V, q).
It is clear from the de¬nitions that every similitude of (V, q) is also a similitude of
its polar bilinear space (V, bq ), with the same multiplier, hence
GO(V, bq ) if char F = 2,
GO(V, q) ‚ Sim(V, b) =
GSp(V, bq ) if char F = 2,
and the reverse inclusion also holds if char F = 2.
For the rest of this section, we assume therefore char F = 2. If dim V is odd,
the same arguments as in (??) yield:
156 III. SIMILITUDES


(12.6) Proposition. Suppose (V, q) is a nonsingular symmetric quadratic space
of odd dimension over a ¬eld F of characteristic 2; then
GO(V, q) = O(V, q) · F — O(V, q) — F — PGO(V, q) O(V, q).
and
We omit the proof, since it is exactly the same as for (??), using the determinant
of q de¬ned in (??).
If dim V is even, we may again distinguish proper and improper similitudes, as
we now show.
By using a basis of V , we may represent the quadratic map q by a quadratic
form, which we denote again q. Let M be a matrix such that
q(X) = X t · M · X.
Since q is nonsingular, the matrix W = M + M t is invertible. Let g be a similitude
of V with multiplier ±, and let G be its matrix with respect to the chosen basis of V .
The equation q(G · X) = ±q(X) shows that the matrices Gt M G and ±M represent
the same quadratic form. Therefore, Gt M G ’ ±M is an alternating matrix. Let
R ∈ Mn (K) be such that
Gt M G ’ ±M = R ’ Rt .
(12.7) Proposition. The element tr(±’1 W ’1 R) ∈ K depends only on the simil-
itude g, and not on the choice of basis of V nor on the choices of matrices M
and R. It equals 0 or 1.
Proof : With a di¬erent choice of basis of V , the matrix G is replaced by G =
P ’1 GP for some invertible matrix P ∈ GLn (K), and the matrix M is replaced by
t
a matrix M = P t M P + U ’ U t for some matrix U . Then W = M + M is related
to W by W = P t W P . Suppose R, R are matrices such that
t t
Gt M G ’ ±M = R ’ Rt
(12.8) and G M G ’ ±M = R ’ R .
By adding each side to its transpose, we derive from these equations:
t
Gt W G = ±W
(12.9) and G W G = ±W .
In order to prove that tr(±’1 W ’1 R) depends only on the similitude g, we have
’1
to show tr(W ’1 R) = tr(W R ). By substituting for M its expression in terms
t t
of M , we derive from (??) that R ’ R = R ’ R , where
R = P t RP + P t Gt (P ’1 )t U P ’1 GP ’ ±U,
(12.10)
’1
hence R = R + S for some symmetric matrix S ∈ Mn (K). Since W =
’1 ’1 ’1 ’1 t ’1
W MW + (W M W ) , it follows that W is alternating. By (??),
alternating matrices are orthogonal to symmetric matrices for the trace bilinear
’1 ’1
form, hence tr(g R ) = tr(g R ). In view of (??) we have
’1
R ) = tr(P ’1 W ’1 RP ) + tr P ’1 W ’1 Gt (P ’1 )t U P ’1 GP
(12.11) tr(W
+ ± tr P ’1 W ’1 (P ’1 )t U .
By (??), W ’1 Gt = ±G’1 W ’1 , hence the second term on the right side of (??)
equals
± tr P ’1 G’1 W ’1 (P ’1 )t U P ’1 GP = ± tr W ’1 (P ’1 )t U P ’1 .
§12. GENERAL PROPERTIES 157


Therefore, the last two terms on the right side of (??) cancel, and we get
’1 ’1
R ) = tr(W ’1 R),
tr(W R ) = tr(W
proving that tr(±’1 W ’1 R) depends only on the similitude g.
In order to prove that this element is 0 or 1, we compute s2 (W ’1 M ), the coef-
¬cient of X n’2 in the characteristic polynomial of W ’1 M (see (??)). By (??), we
have G’1 W ’1 = ±’1 W ’1 Gt , hence G’1 W ’1 M G = ±’1 W ’1 Gt M G, and there-
fore
s2 (W ’1 M ) = s2 (±’1 W ’1 Gt M G).
On the other hand, (??) also yields Gt M G = ±M + R ’ Rt , hence by substituting
this in the right side of the preceding equation we get
s2 (W ’1 M ) = s2 (W ’1 M + ±’1 W ’1 R ’ ±’1 W ’1 Rt ).
By (??), we may expand the right side to get
s2 (W ’1 M ) = s2 (W ’1 M ) + tr(±’1 W ’1 R) + tr(±’1 W ’1 R)2 .
Therefore, tr(±’1 W ’1 R) + tr(±’1 W ’1 R)2 = 0, hence
tr(±’1 W ’1 R) = 0, 1.


(12.12) De¬nition. Let (V, q) be a nonsingular quadratic space of even dimension
over a ¬eld F of characteristic 2. Keep the same notation as above. In view of the
preceding proposition, we set
∆(g) = tr(±’1 W ’1 R) ∈ {0, 1} for g ∈ GO(V, q).
Straightforward veri¬cations show that ∆ is a group homomorphism
∆ : GO(V, q) ’ Z/2Z,
called the Dickson invariant. We write GO+ (V, q) for the kernel of this homomor-
phism. Its elements are called proper similitudes, and the similitudes which are
mapped to 1 under ∆ are called improper. We also let
O+ (V, q) = { g ∈ O(V, q) | ∆(g) = 0 } and PGO+ (V, q) = GO+ (V, q)/F — .
(12.13) Example. Let dim V = n = 2m. For any anisotropic vector v ∈ V , the
hyperplane re¬‚ection ρv : V ’ V is de¬ned in arbitrary characteristic by
ρv (x) = x ’ vq(v)’1 bq (v, x) for x ∈ V .
This map is an isometry of (V, q). We claim that it is improper.
This is clear if char F = 2, since the matrix of ρv with respect to an orthogonal
basis whose ¬rst vector is v is diagonal with diagonal entries (’1, 1, . . . , 1), hence
det ρv = ’1.
If char F = 2, we compute ∆(ρv ) by means of a symplectic basis (e1 , . . . , en )
of (V, bq ) such that e1 = v. With respect to that basis, the quadratic form q is
represented by the matrix
« 
M1 0
q(e2i’1 ) 0
¬ ·
..
M =  where Mi = ,
. 1 q(e2i )
0 Mm
158 III. SIMILITUDES


and the map ρv is represented by
« 
G1 0
1 q(e1 )’1
¬ ·
..
G=  where G1 = , Gi = I, i ≥ 2.
. 0 1
0 Gm
As a matrix R such that Gt M G + M = R + Rt we may take
« 
R1 0
01
¬ ·
..
R=  where R1 = , Ri = I, i ≥ 2.
. 00
0 Rm
It is readily veri¬ed that tr(W ’1 R) = 1, hence ∆(ρv ) = 1, proving the claim.
12.B. Similitudes of algebras with involution. In view of the charac-
terization of similitudes of bilinear or hermitian spaces by means of the adjoint
involution (see (??)), the following de¬nition is natural:
(12.14) De¬nition. Let (A, σ) be a central simple F -algebra with involution. A
similitude of (A, σ) is an element g ∈ A such that
σ(g)g ∈ F — .
The scalar σ(g)g is called the multiplier of g and is denoted µ(g). The set of all
similitudes of (A, σ) is a subgroup of A— which we call Sim(A, σ), and the map µ
is a group homomorphism
µ : Sim(A, σ) ’ F — .
It is then clear that similitudes of bilinear spaces are similitudes of their endo-
morphism algebras:
Sim EndF (V ), σb = Sim(V, b)
if (V, b) is a nonsingular symmetric or alternating bilinear space. There is a corre-
sponding result for hermitian spaces.
Similitudes can also be characterized in terms of automorphisms of the algebra
with involution. Recall that an automorphism of (A, σ) is an F -algebra automor-
phism which commutes with σ:
AutF (A, σ) = { θ ∈ AutF (A) | σ —¦ θ = θ —¦ σ }.
Let K be the center of A, so that K = F if σ is of the ¬rst kind and K is a quadratic
´tale F -algebra if σ is of the second kind. De¬ne Aut K (A, σ) = AutF (A, σ) ©
e
AutK (A).
(12.15) Theorem. With the notation above,
AutK (A, σ) = { Int(g) | g ∈ Sim(A, σ) }.
There is therefore an exact sequence:
Int
1 ’ K — ’ Sim(A, σ) ’’ AutK (A, σ) ’ 1.

Proof : By the Skolem-Noether theorem, every automorphism of A over K has the
form Int(g) for some g ∈ A— . Since
σ —¦ Int(g) = Int σ(g)’1 —¦ σ,
§12. GENERAL PROPERTIES 159


the automorphism Int(g) commutes with σ if and only if σ(g)’1 ≡ g mod K — , i.e.,
σ(g)g ∈ K — . Since σ(g)g is invariant under σ, the latter condition is also equivalent
to σ(g)g ∈ F — .

Let PSim(A, σ) be the group of projective similitudes, de¬ned as
PSim(A, σ) = Sim(A, σ)/K — .
In view of the preceding theorem, the map Int de¬nes a natural isomorphism

PSim(A, σ) ’ AutK (A, σ).

Speci¬c notations for the groups Sim(A, σ) and PSim(A, σ) are used according
to the type of σ, re¬‚ecting the notations for similitudes of bilinear or hermitian
spaces:
±
GO(A, σ) if σ is of orthogonal type,

Sim(A, σ) = GSp(A, σ) if σ is of symplectic type,


GU(A, σ) if σ is of unitary type,

and
±
PGO(A, σ) if σ is of orthogonal type,

PSim(A, σ) = PGSp(A, σ) if σ is of symplectic type,


PGU(A, σ) if σ is of unitary type.
Similitudes with multiplier 1 are isometries; they make up the group Iso(A, σ):
Iso(A, σ) = { g ∈ A— | σ(g) = g ’1 }.
We also use the following notation:
±
O(A, σ) if σ is of orthogonal type,

Iso(A, σ) = Sp(A, σ) if σ is of symplectic type,


U(A, σ) if σ is of unitary type.
For quadratic pairs, the corresponding notions are de¬ned as follows:
(12.16) De¬nition. Let (σ, f ) be a quadratic pair on a central simple F -algebra A.
An automorphism of (A, σ, f ) is an F -algebra automorphism θ of A such that
σ—¦θ =θ—¦σ and f —¦ θ = f.
A similitude of (A, σ, f ) is an element g ∈ A— such that σ(g)g ∈ F — and f (gsg ’1 ) =
f (s) for all s ∈ Sym(A, σ). Let GO(A, σ, f ) be the group of similitudes of (A, σ, f ),
let
PGO(A, σ, f ) = GO(A, σ, f )/F —
and write AutF (A, σ, f ) for the group of automorphisms of (A, σ, f ). The same
arguments as in (??) yield an exact sequence
Int
1 ’ F — ’ GO(A, σ, f ) ’’ AutF (A, σ, f ) ’ 1,

hence also an isomorphism

PGO(A, σ, f ) ’ AutF (A, σ, f ).

160 III. SIMILITUDES


For g ∈ GO(A, σ, f ) we set µ(g) = σ(g)g ∈ F — . The element µ(g) is called the
multiplier of g and the map
µ : GO(A, σ, f ) ’ F —
is a group homomorphism. Its kernel is denoted O(A, σ, f ).
It is clear from the de¬nition that GO(A, σ, f ) ‚ Sim(A, σ). If char F = 2, the
map f is the restriction of 1 TrdA to Sym(A, σ), hence the condition f (gsg ’1 ) =
2
f (s) for all s ∈ Sym(A, σ) holds for all g ∈ GO(A, σ). Therefore, we have in this
case
GO(A, σ, f ) = GO(A, σ), PGO(A, σ, f ) = PGO(A, σ) and O(A, σ, f ) = O(A, σ).
In particular, if (V, q) is a nonsingular quadratic space over F and (σq , fq ) is the
associated quadratic pair on EndF (V ) (see (??)),
GO EndF (V ), σq , fq = GO EndF (V ), σq = GO(V, q).
There is a corresponding result if char F = 2:
(12.17) Example. Let (V, q) be a nonsingular quadratic space of even dimension
over a ¬eld F of characteristic 2, and let (σq , fq ) be the associated quadratic pair
on EndF (V ). We claim that
GO EndF (V ), σq , fq = GO(V, q),
hence also PGO EndF (V ), σq , fq = PGO(V, q) and O EndF (V ), σq , fq = O(V, q).
In order to prove these equalities, observe ¬rst that the standard identi¬cation
•q of (??) associated with the polar of q satis¬es the following property: for all
g ∈ EndF (V ), and for all v, w ∈ V ,
g —¦ •q (v — w) —¦ σq (g) = •q g(v) — g(w) .
Therefore, if g ∈ GO EndF (V, σq , fq ) and ± = µ(g) ∈ F — , the condition
fq g —¦ •q (v — v) —¦ g ’1 = fq —¦ •q (v — v) for v ∈ V
amounts to
q g(v) = ±q(v) for v ∈ V ,
which means that g is a similitude of the quadratic space (V, q), with multiplier ±.
This shows GO EndF (V ), σq , fq ‚ GO(V, q).
For the reverse inclusion, observe that if g is a similitude of (V, q) with multiplier
±, then σq (g)g = ± since g also is a similitude of the associated bilinear space (V, bq ).
Moreover, the same calculation as above shows that
fq g —¦ •q (v — v) —¦ g ’1 = fq —¦ •q (v — v) for v ∈ V .
Since Sym EndF (V ), σq , fq is spanned by elements of the form •q (v —v), it follows
that fq (gsg ’1 ) = fq (s) for all s ∈ Sym(A, σ), hence g ∈ GO EndF (V ), σq , fq . This
proves the claim.
We next determine the groups of similitudes for quaternion algebras.
(12.18) Example. Let Q be a quaternion algebra with canonical (symplectic)
involution γ over an arbitrary ¬eld F . Since γ(q)q ∈ F for all q ∈ Q, we have
Sim(Q, γ) = GSp(Q, γ) = Q— .
Therefore, γ commutes with all the inner automorphisms of Q. (This observation
also follows from the fact that γ is the unique symplectic involution of Q: for
§12. GENERAL PROPERTIES 161


every automorphism θ, the composite θ —¦ γ —¦ θ ’1 is a symplectic involution, hence
θ —¦ γ —¦ θ’1 = γ).
Let σ be an orthogonal involution on Q; by (??) we have
σ = Int(u) —¦ γ
for some invertible quaternion u such that γ(u) = ’u and u ∈ F . Since γ commutes
with all automorphisms of Q, an inner automorphism Int(g) commutes with σ if
and only if it commutes with Int(u), i.e., gu ≡ ug mod F — . If » ∈ F — is such that
gu = »ug, then by taking the reduced norm of both sides of this equation we obtain
»2 = 1, hence gu = ±ug. The group of similitudes of (Q, σ) therefore consists of
the invertible elements which commute or anticommute with u. If char F = 2, we
thus obtain
GO(Q, σ) = F (u)— .
If char F = 2, let v be any invertible element which anticommutes with u; then
GO(Q, σ) = F (u)— ∪ F (u)— · v .
Finally, we consider the case of quadratic pairs on Q. We assume that char F =
2 since, if the characteristic is di¬erent from 2, the similitudes of a quadratic pair
(σ, f ) are exactly the similitudes of the orthogonal involution σ. Since char F =
2, every involution which is part of a quadratic pair is symplectic, hence every
quadratic pair on Q has the form (γ, f ) for some linear map f : Sym(Q, γ) ’ F .
Take any ∈ Q satisfying
f (s) = TrdQ ( s) for s ∈ Sym(Q, γ)
(see (??)). The element is uniquely determined by the quadratic pair (γ, f ) up to
the addition of an element in Alt(Q, γ) = F , and it satis¬es TrdQ ( ) = 1, by (??)
and (??). Therefore, there exists an element v ∈ Q— such that v ’1 v = + 1. We
claim that
GO(Q, γ, f ) = F ( )— ∪ F ( )— · v .
Since GSp(Q, γ) = Q— , an element g ∈ Q— is a similitude of (Q, γ, f ) if and only
if f (gsg ’1 ) = f (s) for all s ∈ Sym(Q, γ). By de¬nition of , this condition can be
rephrased as
TrdQ ( gsg ’1 ) = TrdQ ( s) for s ∈ Sym(Q, γ).
Since the left-hand expression equals TrdQ (g ’1 gs), this condition is also equivalent
to
TrdQ ( ’ g ’1 g)s = 0 for s ∈ Sym(Q, γ);
that is, ’ g ’1 g ∈ F , since F = Alt(Q, γ) is the orthogonal space of Sym(Q, γ)
for the trace bilinear form (see (??)). Suppose that this condition holds and let
» = ’ g ’1 g ∈ F . We proceed to show that » = 0 or 1. Since TrdQ ( ) = 1 we
have NrdQ ( ) = 2 + and NrdQ ( + ») = 2 + + »2 + ». On the other hand, we
must have NrdQ ( ) = NrdQ ( + »), since + » = g ’1 g. Therefore, »2 + » = 0 and
» = 0 or 1. Therefore,
GO(Q, γ, f ) = { g ∈ Q— | g ’1 g = } ∪ { g ∈ Q— | g ’1 g = + 1 },
and the claim is proved.
162 III. SIMILITUDES


(12.19) Example. Let A = Q1 —F Q2 be a tensor product of two quaternion
algebras over a ¬eld F of characteristic di¬erent from 2, and σ = γ1 — γ2 , the
tensor product of the canonical involutions. A direct computation shows that the
Lie algebra Alt(A, σ) decomposes as a direct sum of the (Lie) algebras of pure
quaternions in Q1 and Q2 :
Alt(A, σ) = (Q0 — 1) • (1 — Q0 ).
1 2

Since Lie algebras of pure quaternions are simple and since the decomposition of

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