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(15.4) Corollary. Suppose char F = 2. The functor P : A1 ’ B1 , which maps
every quaternion algebra Q to the algebra with involution EndF (Q0 ), σs where σs
is the adjoint involution with respect to s, and the functor C : B1 ’ A1 , which maps
every algebra with involution (A, σ) of degree 3 to the quaternion algebra C(A, σ),
de¬ne an equivalence of groupoids:
A1 ≡ B 1 .
In particular, for every central simple algebra with involution (A, σ) of degree 3,
the functor C induces an isomorphism of groups:

O+ (A, σ) = PGO(A, σ) = AutF (A, σ) ’ AutF C(A, σ) = PGSp C(A, σ), σ .

We thus recover the ¬rst case (deg A = 3) of (??).
Indices. Let Q ∈ A1 and (V, q, ζ) ∈ B1 correspond to each other under the
(Q0 , s) and Q
equivalence A1 ≡ B1 , so that (V, q) C0 (V, q). Since Q is
a quaternion algebra, its (Schur) index may be either 1 or 2; for the canonical
involution γ on Q, the index ind(Q, γ) is thus (respectively) either {0, 1} or {0}.
On the other hand, since dim V = 3, the Witt index w(V, q) is 1 whenever q is
isotropic.
The following correspondence between the various cases is well-known:
(15.5) Proposition. The indices of Q, (Q, γ) and (V, q) are related as follows:
ind Q = 2 ⇐’ ind(Q, γ) = {0} ⇐’ w(V, q) = 0;
ind Q = 1 ⇐’ ind(Q, γ) = {0, 1} ⇐’ w(V, q) = 1.
In other words, Q is a division algebra if and only if q is anisotropic, and Q is split
if and only if q is isotropic.
210 IV. ALGEBRAS OF DEGREE FOUR


Proof : If Q is a division algebra, then Q0 does not contain any nonzero nilpotent
elements. Therefore, the quadratic form s, hence also q, is anisotropic. On the
other hand, q is isotropic if Q is split, since M2 (F ) contains nonzero matrices
whose square is 0.
15.B. A2 ≡ D2 . The Cli¬ord algebra construction yields a functor C : D2 ’
1
A2 . In order to show that this functor de¬nes an equivalence of groupoids, we ¬rst
1
describe a functor N: A2 ’ D2 which arises from the norm construction.
1
2
Let Q ∈ A1 be a quaternion algebra over some ´tale quadratic extension K/F .
e
(If K = F — F , then Q should be understood as a direct product of quaternion
F -algebras.) Let ι be the nontrivial automorphism of K/F . Recall from (??) that
NK/F (Q) is the F -subalgebra of ι Q —K Q consisting of elements ¬xed by the switch
map
s : ι Q —K Q ’ ι Q —K Q.
The tensor product ι γ — γ of the canonical involutions on ι Q and Q restricts to an
involution NK/F (γ) of the ¬rst kind on NK/F (Q). By (??), we have
ι
Q —K Q, ι γ — γ ,
NK/F (Q)K , NK/F (γ)K
hence NK/F (γ) has the same type as ι γ — γ. Proposition (??) thus shows that
NK/F (γ) is orthogonal if char F = 2 and symplectic if char F = 2. Corollary (??)
further yields a quadratic pair (ι γ — γ, f— ) on ι Q — Q, which is uniquely determined
by the condition that f— vanishes on Skew(ι Q, ι γ) —K Skew(Q, γ). It is readily seen
that (ι γ —γ, ι—¦f— —¦s) is a quadratic pair with the same property, hence ι—¦f— —¦s = f
and therefore
for all x ∈ Sym(ι Q — Q, ι γ — γ).
f— s(x) = ιf— (x)
It follows that f— (x) ∈ F for all x ∈ Sym NK/F (Q), NK/F (γ) , hence (ι γ — γ, f— )
restricts to a quadratic pair on NK/F (Q). We denote this quadratic pair by
NK/F (γ), fN . The norm thus de¬nes a functor
N : A2 ’ D 2
1

which maps Q ∈ A2 to N(Q) = NK/F (Q), NK/F (γ), fN where K is the center
1
of Q.
On the other hand, for (A, σ, f ) ∈ D2 the Cli¬ord algebra C(A, σ, f ) is a quater-
nion algebra over an ´tale quadratic extension, as the structure theorem (??) shows.
e
Therefore, the Cli¬ord algebra construction yields a functor
C : D2 ’ A 2 .
1

The key tool to show that N and C de¬ne an equivalence of categories is the Lie
algebra isomorphism which we de¬ne next. For Q ∈ A2 , consider the F -linear map
1
n : Q ’ NK/F (Q) de¬ned by

n(q) = ι q — 1 + ι 1 — q
™ for q ∈ Q.
This map is easily checked to be a Lie algebra homomorphism; it is in fact the
di¬erential of the group homomorphism n : Q— ’ NK/F (Q)— which maps q ∈ Q—
to ι q — q. We have the nonsingular F -bilinear form TNK/F (Q) on NK/F (Q) and the
nonsingular F -bilinear form on Q which is the transfer of TQ with respect to the
trace TK/F . Using these, we may form the adjoint linear map
n— : NK/F (Q) ’ Q

§15. EXCEPTIONAL ISOMORPHISMS 211


which is explicitly de¬ned as follows: for x ∈ NK/F (Q), the element n— (x) ∈ Q is

uniquely determined by the condition
TK/F TrdQ n— (x)y
™ = TrdNK/F (Q) xn(y)
™ for all y ∈ Q.
(15.6) Proposition. Let Q = { x ∈ Q | TrdQ (x) ∈ F }. The linear map n— ™
factors through the canonical map c : NK/F (Q) ’ c N(Q) and induces an isomor-
phism of Lie algebras

n— : c N(Q) ’ Q .
™ ’
This isomorphism is the identity on F .
Proof : Suppose ¬rst that K F — F . We may then assume that Q = Q1 — Q2
for some quaternion F -algebras Q1 , Q2 , and NK/F (Q) = Q1 — Q2 . Under this
identi¬cation, the map n is de¬ned by

n(q1 , q2 ) = q1 — 1 + 1 — q2
™ for q1 ∈ Q1 and q2 ∈ Q2 .
It is readily veri¬ed that
n— (q1 — q2 ) = TrdQ2 (q2 )q1 , TrdQ1 (q1 )q2
™ for q1 ∈ Q1 and q2 ∈ Q2 ,
hence n— is the map ˜ of (??). From (??), it follows that n— factors through c and
™ ™
induces an isomorphism of Lie algebras
c N(Q) = c(Q1 — Q2 ) ’ Q = { (q1 , q2 ) ∈ Q1 — Q2 | TrdQ1 (q1 ) = TrdQ2 (q2 ) }.
For i = 1, 2, let i ∈ Qi be such that TrdQi ( i ) = 1. Then TrdQ1 —Q2 ( 1 — 2 ) = 1,
hence f 1 — 2 + γ1 ( 1 ) — γ2 ( 2 ) = 1, and therefore c 1 — 2 + γ1 ( 1 ) — γ2 ( 2 ) = 1.
On the other hand,
n—
™ — + γ1 ( 1 ) — γ2 ( 2 ) = ( 1 , 2) + γ1 ( 1 ), γ2 ( 2 ) = (1, 1),
1 2

hence n— maps 1 ∈ c N(Q) to 1 ∈ Q . The map n— thus restricts to the identity
™ ™
on F , completing the proof in the case where K F — F .
In the general case, it su¬ces to prove the proposition over an extension of the
base ¬eld F . Extending scalars to K, we are reduced to the special case considered
above, since K — K K — K.
(15.7) Theorem. The functors N and C de¬ne an equivalence of groupoids:
A2 ≡ D 2 .
1

Moreover, if Q ∈ A2 and (A, σ, f ) ∈ D2 correspond to each other under this equiv-
1
alence, then the center Z(Q) of Q satis¬es
F disc(A, σ, f ) if char F = 2;
Z(Q)
F „˜’1 disc(A, σ, f ) if char F = 2.
Proof : If the ¬rst assertion holds, then the quaternion algebra Q corresponding
to (A, σ, f ) ∈ D2 is the Cli¬ord algebra C(A, σ, f ), hence the description of Z(Q)
follows from the structure theorem for Cli¬ord algebras (??).
In order to prove the ¬rst statement, we establish natural transformations
N —¦ C ∼ IdD2 and C —¦ N ∼ IdA2 . Thus, for (A, σ, f ) ∈ D2 and for Q ∈ A2 , we
= = 1
1
have to describe canonical isomorphisms
(A, σ, f ) NZ(A,σ,f )/F C(A, σ, f ) , NZ(A,σ,f )/F (σ), fN
212 IV. ALGEBRAS OF DEGREE FOUR


and
Q C NZ(Q)/F (Q), NZ(Q)/F (γ), fN
where Z(A, σ, f ) is the center of C(A, σ, f ).
Observe that the fundamental relation (??) between an algebra with invo-
lution and its Cli¬ord algebra already shows that there is an isomorphism A
NZ(A,σ,f )/F C(A, σ, f ) . However, we need a canonical isomorphism which takes
the quadratic pairs into account.
Our construction is based on (??): we use (??) to de¬ne isomorphisms of Lie
algebras and show that these isomorphisms extend to isomorphisms of associative
algebras over an algebraically closed extension, hence also over the base ¬eld.
Let (A, σ, f ) ∈ D2 and let
C(A, σ, f ) = { x ∈ C(A, σ, f ) | TrdC(A,σ,f ) (x) ∈ F }.
Lemma (??) shows that TrdC(A,σ,f ) c(a) = TrdA (a) for a ∈ A, hence c(A) ‚
C(A, σ, f ) , and dimension count shows that this inclusion is an equality. Propo-
sition (??) then yields a Lie algebra isomorphism n— : c N C(A, σ, f ) ’ c(A)

which is the identity on F . By (??), it follows that this isomorphism induces a Lie
algebra isomorphism

n : Alt N C(A, σ, f ) ’ Alt(A, σ).

To prove that this isomorphism extends to an isomorphism of algebras with quad-
ratic pairs, it su¬ces by (??) to consider the split case. We may thus assume that A
is the endomorphism algebra of a hyperbolic quadratic space H(U ) of dimension 4.
Thus
A = EndF H(U ) = EndF (U — • U )
where U is a 2-dimensional vector space, U — is its dual, and (σ, f ) = (σqU , fqU ) is
the quadratic pair associated with the hyperbolic quadratic form on U — • U :
for • ∈ U — , u ∈ U .
qU (• + u) = •(u)
In that case, the Cli¬ord algebra C(A, σ, f ) can be described as
C(A, σ, f ) = C0 H(U ) = EndF ( U ) — EndF ( U ),
0 1
where 0 U (resp. 1 U ) is the 2-dimensional subspace of even- (resp. odd-) degree
elements in the exterior algebra of U (see (??)):
2
U =F • U, U = U.
0 1
Therefore,
NZ(A,σ,f )/F C(A, σ, f ) = EndF ( U— U ).
0 1
On the vector space 0 U — 1 U , we de¬ne a quadratic form q as follows: pick
2
a nonzero element (hence a basis) e ∈ U ; for x, y ∈ U , we may then de¬ne
q(1 — x + e — y) ∈ F by the equation
eq(1 — x + e — y) = x § y.
The associated quadratic pair (σq , fq ) on EndF ( 0 U — 1 U ) is the canonical
quadratic pair N (σ), fN (see Exercise ?? of Chapter ??). A computation shows
that the map g : H(U ) ’ 0 U — 1 U de¬ned by
g(• + u) = 1 — x + e — u,
§15. EXCEPTIONAL ISOMORPHISMS 213


where x ∈ U is such that x § y = e•(y) for all y ∈ U , is a similitude of quadratic
spaces

g : H(U ) ’ (
’ U— U, q).
0 1
By (??), this similitude induces an isomorphism of algebras with quadratic pair

g— : EndF H(U ) , σqU , fqU ’ EndF (
’ U— U ), σq , fq .
0 1
’1
We leave it to the reader to check that g— extends the Lie algebra homomorphism
n, completing the proof that n induces a natural transformation N —¦ C ∼ IdD2 .
=
We use the same technique to prove that C —¦ N ∼ IdA2 . For Q ∈ A1 , Proposi-
2
= 1
tion (??) yields a Lie algebra isomorphism

n— : c N(Q) ’ Q .
™ ’

To prove that n— extends to an isomorphism of F -algebras C N(Q) ’ Q, we
™ ’
may extend scalars, since N(Q) is generated as an associative algebra by c N(Q) .
Extending scalars to Z(Q) if this algebra is a ¬eld, we may therefore assume that
Z(Q) F — F . In that case Q Q1 — Q2 for some quaternion F -algebras Q1 , Q2 ,
hence NZ(Q)/F (Q) Q1 — Q2 , and n— is the map ˜ of (??), de¬ned by

˜ c(x1 — x2 ) = TrdQ2 (x2 )x1 , TrdQ1 (x1 )x2 for x1 ∈ Q1 , x2 ∈ Q2 .
Since it was proven in (??) that ˜ extends to an isomorphism of F -algebras

C N(Q) = C(Q1 — Q2 , γ1 — γ2 , f— ) ’ Q1 — Q2 , the proof is complete.


(15.8) Remark. For Q ∈ A2 , the Lie isomorphism n— : c N(Q) ’ Q restricts to

1
∼ 0
an isomorphism c N(Q) 0 ’ Q . If char F = 2, the inverse of this isomorphism

1
is 2 c —¦ n (see Exercise ??). Similarly, for (A, σ, f ) ∈ D2 , the inverse of the Lie

™1
isomorphism n : Alt N C(A, σ, f ) ’ Alt(A, σ) is n —¦ 2 c if char F = 2.
(15.9) Corollary. For every central simple algebra A of degree 4 with quadratic
pair (σ, f ), the functor C induces an isomorphism of groups:

PGO(A, σ, f ) = AutF (A, σ, f ) ’ AutF C(A, σ, f ), σ = AutF C(A, σ, f )

which restricts into an isomorphism of groups:

PGO+ (A, σ, f ) ’ AutZ(A,σ,f ) C(A, σ, f ), σ =

PGSp C(A, σ, f ), σ = C(A, σ, f )— /Z(A, σ, f )— .
Proof : The ¬rst isomorphism follows from the fact that C de¬nes an equivalence
of groupoids D2 ’ A2 (see (??)). Under this isomorphism, the proper similitudes
1
correspond to automorphisms of C(A, σ, f ) which restrict to the identity on the
center Z(A, σ, f ), by (??).
We thus recover the second case (deg A = 4) of (??).
Cli¬ord groups. Let Q ∈ A2 and (A, σ, f ) ∈ D2 . Let Z be the center of Q,
1
and assume that Q and (A, σ, f ) correspond to each other under the groupoid
equivalence A2 ≡ D2 , so that we may identify Q = C(A, σ, f ) and (A, σ, f ) =
1
NZ/F (Q), NZ/F (γ), fN .
(15.10) Proposition. The extended Cli¬ord group of (A, σ, f ) is „¦(A, σ, f ) = Q —
and the canonical map χ0 : Q— ’ GO+ (A, σ, f ) of (??) is given by χ0 (q) = ι q —q ∈
NZ/F (Q) = A. For q ∈ Q— , the multiplier of χ0 (q) is µ χ0 (q) = NZ/F NrdQ (q) .
214 IV. ALGEBRAS OF DEGREE FOUR


The Cli¬ord group of (A, σ, f ) is
“(A, σ, f ) = { q ∈ Q— | NrdQ (q) ∈ F — },
and the vector representation map χ : “(A, σ, f ) ’ O+ (A, σ, f ) is given by
χ(q) = NrdQ (q)’1ι q — q = ι q — γ(q)’1 .
The spin group is
Spin(A, σ, f ) = SL1 (Q) = { q ∈ Q— | NrdQ (q) = 1 }.
Proof : We identify „¦(A, σ, f ) by means of (??): the canonical map b : A ’ B(A, σ, f )
maps A onto the subspace of invariant elements under the canonical involution ω.
Therefore, the condition σ(x) — Ab · x = Ab holds for all x ∈ Q— .
It su¬ces to check the description of χ0 in the split case, where it follows
from explicit computations. The Cli¬ord group is characterized in (??) by the
condition µ(q) ∈ F — , which here amounts to NrdQ (q) ∈ F — , and the description of
Spin(A, σ, f ) follows.
(15.11) Corollary. With the same notation as above, the group of multipliers of
proper similitudes of (A, σ, f ) is
G+ (A, σ, f ) = F —2 · NZ/F NrdQ (Q— )
and the group of spinor norms is
Sn(A, σ, f ) = F — © NrdQ (Q— ).
Moreover, G+ (A, σ, f ) = G(A, σ, f ) if and only if A is nonsplit and splits over Z.
Proof : The description of G+ (A, σ, f ) follows from (??) and the proposition above,
since χ (q) = χ0 (q) · F — for all q ∈ „¦(A, σ, f ). By de¬nition, the group of spinor
norms is Sn(A, σ, f ) = µ “(A, σ, f ) , and the preceding proposition shows that
µ “(A, σ, f ) = F — © NrdQ (Q— ).
If G(A, σ, f ) = G+ (A, σ, f ), then (??) shows that A is not split and splits
over Z. In order to prove the converse implication, we use the isomorphism
A NZ/F (Q) proved in (??) (and also in (??), see (??)). If A is split by Z,
scalar extension to Z shows that ι Q —Z Q is split, hence Q is isomorphic to
ι
Q as a Z-algebra. It follows that AutF (Q) = AutZ (Q), hence (??) shows that
PGO(A, σ, f ) = PGO+ (A, σ, f ). By (??), it follows that G(A, σ, f ) = G+ (A, σ, f )
if A is not split.
The case of trivial discriminant. If K is a given ´tale quadratic extension
e
2
of F , the equivalence A1 ≡ D2 set up in (??) associates quaternion algebras with
center K with algebras with quadratic pair (A, σ, f ) such that Z(A, σ, f ) = K. In
the particular case where K = F —F , we are led to consider the full subgroupoid 1A2
1
of A2 whose objects are F -algebras of the form Q1 —Q2 where Q1 , Q2 are quaternion
1
F -algebras, and the full subgroupoid 1 D2 of D2 whose objects are central simple
F -algebras with quadratic pair of trivial discriminant. Theorem (??) specializes to
the following statement:
(15.12) Corollary. The functor N : 1A2 ’ 1 D2 which maps the object Q1 — Q2
1
to (Q1 — Q2 , γ1 — γ2 , f— ) (where γ1 , γ2 are the canonical involutions on Q1 , Q2
respectively, and (γ1 — γ2 , f— ) is the quadratic pair of (??)) and the Cli¬ord algebra
functor C : 1 D2 ’ 1A2 de¬ne an equivalence of groupoids:
1
12
≡ 1 D2 .
A1
§15. EXCEPTIONAL ISOMORPHISMS 215


In particular, every central simple algebra A of degree 4 with quadratic pair (σ, f )
of trivial discriminant decomposes as a tensor product of quaternion algebras:
(A, σ, f ) = (Q1 — Q2 , γ1 — γ2 , f— ).
Proof : For (A, σ, f ) ∈ 1 D2 , we have C(A, σ, f ) = Q1 — Q2 for some quaternion
F -algebras Q1 , Q2 . The isomorphism (A, σ, f ) N —¦ C(A, σ, f ) yields:
(A, σ, f ) (Q1 — Q2 , γ1 — γ2 , f— ).



Note that the algebras Q1 , Q2 are uniquely determined by (A, σ, f ) up to
isomorphism since C(A, σ, f ) = Q1 — Q2 . Actually, they are uniquely determined
as subalgebras of A by the relation (A, σ, f ) = (Q1 — Q2 , γ1 — γ2 , f— ). If char F = 2,
this property follows from the observation that Skew(A, σ) = Skew(Q1 , γ1 ) — 1 +
1 — Skew(Q2 , γ2 ), since Skew(Q1 , γ1 ) — 1 and 1 — Skew(Q2 , γ2 ) are the only simple
Lie ideals of Skew(A, σ). See Exercise ?? for the case where char F = 2.
The results in (??), (??) and (??) can also be specialized to the case where the
discriminant of (σ, f ) is trivial. For instance, one has the following description of
the group of similitudes and their multipliers:
(15.13) Corollary. Let (A, σ, f ) = (Q1 — Q2 , γ1 — γ2 , f— ) ∈ 1 D2 . The functor C
induces isomorphisms of groups:

PGO(A, σ, f ) ’ AutF (Q1 — Q2 )

and

PGO+ (A, σ, f ) ’ AutF (Q1 ) — AutF (Q2 ) = PGL(Q1 ) — PGL(Q2 ).

Similarly, Spin(A, σ, f ) SL1 (Q1 ) — SL1 (Q2 ). Moreover,
G(A, σ, f ) = G+ (A, σ, f ) = NrdQ1 (Q— ) · NrdQ2 (Q— )
1 2

and
Sn(A, σ, f ) = NrdQ1 (Q— ) © NrdQ2 (Q— ).
1 2

Indices. Let Q ∈ A2 and (A, σ, f ) ∈ D2 correspond to each other under the
1
equivalence A2 ≡ D2 . Since deg A = 4, there are four possibilities for ind(A, σ, f ):
1

{0}, {0, 1}, {0, 2}, {0, 1, 2}.
The following proposition describes the corresponding possibilities for the algebra
Q. Let K be the center of Q, so K F disc(A, σ, f ) if char F = 2 and K
’1
F„˜ disc(A, σ, f ) if char F = 2.
(15.14) Proposition. With the notation above,
(1) ind(A, σ, f ) = {0} if and only if either Q is a division algebra (so K is a ¬eld )
or Q Q1 — Q2 for some quaternion division F -algebras Q1 , Q2 (so K F — F );
(2) ind(A, σ, f ) = {0, 1} if and only if K is a ¬eld and Q M2 (K);
(3) ind(A, σ, f ) = {0, 2} if and only if Q M2 (F ) — Q0 for some quaternion
division F -algebra Q0 ;
(4) ind(A, σ, f ) = {0, 1, 2} if and only if Q M2 (F ) — M2 (F ).
216 IV. ALGEBRAS OF DEGREE FOUR


Proof : If 1 ∈ ind(A, σ, f ), then A is split and the quadratic pair (σ, f ) is isotropic.
Thus, A EndF (V ) for some 4-dimensional F -vector space V , and (σ, f ) is
the quadratic pair associated with some isotropic quadratic form q on V . Since
dim V = 4, the quadratic space (V, q) is hyperbolic if and only if its discriminant
is trivial, i.e., K F — F . Therefore, if ind(A, σ, f ) = {0, 1}, then K is a ¬eld;
by (??), the canonical involution on C0 (V, q) Q is hyperbolic, hence Q is split.
If ind(A, σ, f ) = {0, 1, 2}, then (V, q) is hyperbolic and K F — F . By (??), it
follows that Q M2 (F ) — M2 (F ). Conversely, if Q M2 (K) (and K is either a
¬eld or isomorphic to F — F ), then Q contains a nonzero element q which is not
invertible. The element ι q — q ∈ NK/F (Q) A generates an isotropic right ideal of
reduced dimension 1, hence 1 ∈ ind(A, σ, f ). This proves (??) and (??).
If 2 ∈ ind(A, σ, f ), then (σ, f ) is hyperbolic, hence Proposition (??) shows
that Q M2 (F ) — Q0 for some quaternion F -algebra Q0 , since Q C(A, σ, f ).
Conversely, if Q M2 (F ) — Q0 for some quaternion F -algebra Q0 , then
(A, σ, f ) M2 (F ) — Q0 , γM — γ0 , f— ,
where γM and γ0 are the canonical (symplectic) involutions on M2 (F ) and Q0
respectively. If x ∈ M2 (F ) is a nonzero singular matrix, then x — 1 generates an
isotropic right ideal of reduced dimension 2 in A, hence 2 ∈ ind(A, σ, f ). This
proves (??) and yields an alternate proof of (??).

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