Table Of ContentQuasigroups and Related Systems 10 (2003), 65 (cid:21) 93
O
tonions, simple Moufang loops and triality
7 Gábor P. Nagy and Petr Vojt¥
hovský
0
0
2
n Abstra
t
a
J
4 Nonasso
iative (cid:28)nite simple Moufang loops are exa
tly the loops
onstru
ted by Paige
2
from Zorn ve
tor matrix algebras. We prove this result anew, using geometri
loop
] theory. Inordertomakethepapera
essible toabroaderaudien
e,we
arefullydis
u3ss
R
the
onne
tionsbetween
ompositionalgebras,simpleMoufangloops,simpleMoufang -
G S
nets, -simplegroupsandgroupswithtriality. Relatedresults onmultipli
ationgroups,
.
h automorphismsgroups and generators of Paige loops are provided.
t
a
m
[
1
v Contents
7
0 1 Introdu
tion 2
7
1
2 Loops and nets 3
0
7 2.1 Quasigroupsand loops . . . . . . . . . . . . . . . . . . . . . . . . . 3
0 2.2 Isotopisms versus isomorphisms . . . . . . . . . . . . . . . . . . . . 4
3
/
h 2.3 Loops and -nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
t 2.4 Collineations and autotopisms . . . . . . . . . . . . . . . . . . . . . 6
a
m 2.5 Bol re(cid:29)e
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
:
v 3 Composition algebras 8
i
X 3.1 The Cayley-Di
ksonpro
ess . . . . . . . . . . . . . . . . . . . . . . 9
r 3.2 Split o
tonion algebras . . . . . . . . . . . . . . . . . . . . . . . . . 10
a
2000 Mathemati
s Subje
t Classi(cid:28)
ation: 20N05, 20D05
Keywords: simpleMoufang loop,Paigeloop,o
tonion,
ompositionalgebra,
lassi
al
group, group with triality, net 0063/2001
The (cid:28)rst author was supported by the FKFP grant of the Hungarian Min-
istry for Edu
ation and the OTKA grants nos. F042959 and T043758. The se
ond
a2u69th/2o0r0p1a/rtially supported by the Grant Agen
y of Charles University, grant number
B-MAT/MFF.
2 G. P. Nagy and P. Vojt¥
hovský
4 A
lass of
lassi
al simple Moufang loops 11
4.1 Paige loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2 Orthogonal groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.3 Multipli
ation groups of Paige loops . . . . . . . . . . . . . . . . . 13
5 Groups with triality 16
5.1 Triality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.2 Triality of Moufang nets . . . . . . . . . . . . . . . . . . . . . . . . 17
5.3 Triality
ollineations in
oordinates . . . . . . . . . . . . . . . . . . 19
6 The
lassi(cid:28)
ation of nonasso
iative (cid:28)nite simple Moufang loops 20
3
6.1 Simple -nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
S
6.2 -simple groups with triality . . . . . . . . . . . . . . . . . . . . . 21
6.3 The
lassi(cid:28)
ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
7 Automorphism groups of Paige loops over perfe
t (cid:28)elds 24
7.1 The automorphisms of the split o
tonion algebras. . . . . . . . . . 24
7.2 Geometri
des
ription of loop automorphisms . . . . . . . . . . . . 25
7.3 The automorphisms of Paige loops . . . . . . . . . . . . . . . . . . 26
8 Related results, prospe
ts and open problems 27
8.1 Generators for (cid:28)nite Paige loops . . . . . . . . . . . . . . . . . . . 27
8.2 Generators for integral Cayley numbers of norm one . . . . . . . . 27
8.3 Problems and Conje
tures . . . . . . . . . . . . . . . . . . . . . . . 28
1 Introdu
tion
Thegoalofthispaperistopresentthe
lassi(cid:28)
ationof(cid:28)nitesimpleMoufang
loopsinan a
essibleanduniform way to abroadaudien
e ofresear
hers in
nonasso
iativealgebra. Theresultsarenotnewbuttheargumentsoftenare.
Although not all proofs are in
luded, our intention was to leave out only
those proofs that are standard (that is those that
an be found in many
sour
es), those that are purely group-theoreti
al, and those that require
only basi
knowledge of loop theory. We have rewritten many proofs using
geometri
looptheory(cid:22)amoresuitablesettingforthiskindofreasoning. To
emphasize the links to other areas of loop theory and algebra, we
omment
onde(cid:28)nitionsandresultsgenerously, althoughmostoftheremarkswemake
are not essential later in the text.
Here is a brief des
ription of the
ontent of this paper. After reviewing
some basi
properties of loops, nets and
omposition algebras, we
onstru
t
afamily of simple Moufangloops from the Zorn alternative algebras. These
O
tonions and simple Moufang loops 3
loops are also known as Paige loops. We then brie(cid:29)y dis
uss the multipli
a-
tion groups of Paige loops, be
ause these are essential in the
lassi(cid:28)
ation.
3
With every Moufang loop we asso
iate a Moufang -net, and with this
3 S
-net we asso
iate a group with triality. An -homomorphism is a homo-
morphism between two groups with triality that preserves the respe
tive
S
triality automorphisms. This leads us to the
on
ept of -simple groups
G
with triality, whi
h we
lassify. The group with triality asso
iated with a
L S L
simple Moufang loop must be -simple. Moreover, when is nonasso
ia-
G
tive must be simple. This is the moment when we use results of Liebe
k
on
erning the
lassi(cid:28)
ation of (cid:28)nite simple groups with triality. His work
is based on the
lassi(cid:28)
ation of (cid:28)nite simple groups. The fa
t that there
are no other nonasso
iative (cid:28)nite simple Moufang loops besides (cid:28)nite Paige
loops then follows easily.
Building on the geometri
understanding we have obtained so far, we
determine the automorphism groups of all Paige loops
onstru
ted over
perfe
t (cid:28)elds. We
on
lude the paper with several results
on
erning the
generators of (cid:28)nite Paige loops and integral Cayley numbers. All these re-
sults are mentioned be
ause they point on
e again towards
lassi
al groups.
Several problems and
onje
tures are pondered in the last se
tion.
A few words
on
erning the notation: As is the habit among many
loop theorists, we write maps to the right of their arguments, and therefore
ompose maps from left to right. The only ex
eption to this rule are some
det S
traditional maps, su
h as the determinant . A subloop generated by
hSi n S
n
will be denoted by . The symmetri
group on points is denoted by .
2 Loops and nets
We now give a brief overview of de(cid:28)nitions and results
on
erning loops
and nets. Nets (also
alled webs in the literature) form the foundations of
the geometri
loop theory. All material
overed in 2.1(cid:21)2.3
an be found in
[4℄ and [25℄, with proofs. We refer the reader to [25, Ch. II℄ and [8, Ch.
VIII,X℄ for further study of nets.
2.1 Quasigroups and loops
Q = (Q,·) Q
Let be a groupoid. Then is a quasigroup if the equation
x·y = z Q
has a unique solution in whenever two of the three elements
x y z ∈ Q
, , are spe
i(cid:28)ed. Quasigroups are interesting in their own right,
but also appear in
ombinatori
s under the name latin squares (more pre-
isely, multipli
ation tables of (cid:28)nite quasigroups are exa
tly latin squares),
4 G. P. Nagy and P. Vojt¥
hovský
and in universal algebra, where subvarieties of quasigroups are often used
to provide an instan
e of some universal algebrai
notion that
annot be
demonstrated in groups or other rigid obje
ts. We ought to point out that
in order to de(cid:28)ne the variety of quasigroups equationally, one must intro-
\ /
du
e additional operations and for left and right division, respe
tively.
Q e e·x = x·e = x
A quasigroup that possesses an element satisfying
x∈ Q e
for every is
alled a loop with neutral element . The vastness of the
variety of loops di
tates to fo
us on some subvariety, usually de(cid:28)ned by an
identity approximating the asso
iative law. (Asso
iative loops are exa
tly
groups.) In this paper, we will be
on
erned with Moufang loops, whi
h are
loops satisfying any one of the three equivalent Moufang identities
((xy)x)z = x(y(xz)), ((xy)z)y = x(y(zy)), (xy)(zx) = (x(yz))x,
(1)
xand in parti
ular with simple Moufang loops (see below). Evxer−y1element
ofxax−M1o=ufaxn−g1xloo=p eis a
ompanied by its two-sided inverse satisfy-
ing (xy.)−A1n=yytw−1oxe−l1ements of a Moufang loop generate a
subgroup, and thus .
x Q Q
Ea
h element of a loop gives rise to two permutations on , the
L : y 7→ xy R : y 7→ yx
x x
left translation and the right translation . The
MltQ
group generated by all left and right translations is known as the
Q InnQ MltQ
multipLli
aLtioLn−1groRupRofR−.1 TheRsubLl−o1op x yof∈ Q generated by all
x y yx x y xy x x
maps , and , for , , is
alled the inner
Q ϕ ∈MltQ eϕ = e
mapping group of . It
onsists of all su
h that .
S Q Q Sϕ = S ϕ ∈ InnQ
A subloop of is normal in if for every . The
Q Q Q {e}
loop is said to be simple if the only normal subloops of are and .
Q x y ∈ Q
In any loop , the
ommutator of , is the unique element
[x,y] ∈ Q xy = (yx)[x,y] x y z ∈ Q
satisfying , and the asso
iator of , ,
[x,y,z] ∈ Q (xy)z = (x(yz))[x,y,z]
is the unique element satisfying . We
C(Q) Q x
prefer to
all the subloop of
onsisting of all elements su
h that
[x,y] = [y,x] = e y ∈ Q Q
for every the
ommutant of . (Some authors use
N(Q)
the name
entrum or Moufang
enter.) The subloop
onsisting of all
x ∈ Q [x,y,z] = [y,x,z] = [y,z,x] = e y z ∈ Q
su
h that holds for every ,
Q Z(Q) = C(Q)∩N(Q)
is known as the nu
leus of . Then is the
enter of
Q Q
, whi
h is always a normal subloop of .
2.2 Isotopisms versus isomorphisms
Quasigroups and loops
an be
lassi(cid:28)ed up to isomorphism or up to iso-
Q Q (α β γ)
1 2
topism. When , arequasigroups,thenthetriple , , ofbije
tions
Q Q Q Q xα·yβ = (x·y)γ
1 2 1 2
from onto is an isotopism of onto if holds
O
tonions and simple Moufang loops 5
x y ∈ Q Q = Q
1 1 2
for every , . An isotopism with is
alled an autotopism.
α (α α α)
Every isomorphism gives rise to an isotopism , , . The notion of
isotopismissuper(cid:29)uousingrouptheory, asany two groupsthatareisotopi
are already isomorphi
.
Q Q
1 2
In terms of multipli
ation tables, and are isotopi
if the multi-
Q Q
2 1
pli
ation table of
an be obtained from the multipli
ation table of
α β
by permuting the rows (by ), the
olumns (by ), and by renaming the
γ
elements (by ). Isotopisms are therefore appropriate morphisms for the
study of quasigroups and loops. On the other hand, every quasigroup is
isotopi
to a loop, whi
h shows that the algebrai
properties of isotopi
quasigroups
an di(cid:27)er substantially. Fortunately, the
lassi(cid:28)
ation of (cid:28)nite
simple Moufang loops is the same no matter whi
h kind of equivalen
e (iso-
topism or isomorphism) we use. This is be
ause (as we shall see) there is
at most one nonasso
iative (cid:28)nite simple Moufang loop of a given order, up
to isomorphism.
L G L L
A loop is a -loop if every loop isotopi
to is isomorphi
to . So,
G
(cid:28)nite simple Moufang loops are -loops.
3
2.3 Loops and -nets
k > 2 P L ,...,L
1 k
Let be an integer, a set, and disjoint sets of subsets
P L = L P L
i
of . Put . We
all the elements of and points and lines,
respe
tively, andSuse the
ommon geometri
terminology, su
h as (cid:16)all lines
P ℓ ∈ L i
i
through the point (cid:17), et
. For , we also speak of a line of type or
i
an -line. Lines of the same type are
alled parallel.
(P,L) k
The pair is a -net if the following axioms hold:
1)
Distin
t lines of the same type are disjoint.
2)
Two lines of di(cid:27)erent types have pre
isely one point in
ommon.
3)
Through any point, there is pre
isely one line of ea
h type.
k
Upon inter
hanging the roles of points and lines, we obtain dual -nets.
k
In that
ase, the points
an be partitioned into
lasses so that:
′
1)
Distin
t points of the same type are not
onne
ted by a line.
′
2)
Two points of di(cid:27)erent types are
onne
ted by a unique line.
′
3) k
Every line
onsists of points of pairwise di(cid:27)erent types.
6 G. P. Nagy and P. Vojt¥
hovský
3
There is a natural relation between loops and -nets. Let us (cid:28)rst start
L P = L×L
from a loop and put . De(cid:28)ne the line
lasses
L = {{(x,c) |x∈ L} |c ∈ L},
1
L = {{(c,y) |y ∈ L} |c ∈ L},
2
L = {{(x,y) | x,y ∈ L, xy = c} | c ∈L}.
3
(P,L = L ∪L ∪L ) 3
1 2 3
Then, is a -net. The lines of these
lasses are
also
alled horizontal, verti
al and transversal lines, respe
tively. The point
O = (e,e)
is the origin of the net.
3 (P,L = L ∪ L ∪ L ) O ∈ P
1 2 3
Let us now
onsider a -net . Let be
ℓ k
an arbitrary point, and let , be the unique horizontal and verti
al lines
O
through , respe
tively. Then the
onstru
tion of Figure 1 de(cid:28)nes a loop
ℓ O
operation on with neutral element .
k
y s(x,y)
b b
b b
b b
b b
b b
b b
s bb bb ℓ
x y x·y
(e,e) = O
Figure 1: The geometri
de(cid:28)nition of the
oordinate loop.
Sin
e the parallel proje
tions are bije
tions between lines of di(cid:27)erent
k ℓ
type, we
an index the points of by points of , thus obtaining a bije
tion
P ℓ×ℓ
between and . Thethree line
lassesare determined by the equations
X = c Y = c XY = c c
, , , respe
tively, where is a
onstant. We say that
(ℓ,O) 3 (P L)
is a
oordinate loop of the -net , .
2.4 Collineations and autotopisms
N = (P,L) 3
Let be a -net. Collineations are line preserving bije
tive
P → P N CollN
maps . The group of
ollineations of is denoted by . A
ollineation indu
es a permutation of the line
lasses. There is therefore a
CollN S
3
group homomorphism from to the symmetri
group . The kernel
of this homomorphism
onsists of the dire
tion preserving
ollineations.
L N = (P, L)
Let be the
oordinate loop of with respe
t to some origin
O ∈ P ϕ : P → P ϕ
. Let be a bije
tion. Then preserves the line
lasses
1 2 (x,y) 7→ (xα,yβ)
and if and only if it has the form for some bije
tions
α β : L → L ϕ 1 2 ϕ
, . Moreover, if preserves the line
lasses and then also
γ : L → L
preserves the third
lass if and only if there is a bije
tion su
h
O
tonions and simple Moufang loops 7
(α,β,γ) L L
that the triple is an autotopism of . Automorphisms of
an be
hara
terized in a similar way (see Lemma 7.2).
2.5 Bol re(cid:29)e
tions
N 3 ℓ ∈ L i
i i
Let be a -net and , for some . We de(cid:28)ne a
ertain permutation
σ P P ∈ P a a
ℓi on the point set (
f. Figure 2). For , let j and k be the
P a ∈ L a ∈ L {i, j, k} = {1, 2, 3}
j j k k
lines through su
h that , , and .
Q = a ∩ℓ Q = a ∩ℓ
j j i k k i
Then there are unique interse
tion points , . We
Pσ = b ∩ b b j Q b
de(cid:28)ne ℓi j k, where j is the unique -line through k, and k
k Q σ
the unique -line through j. The permutation ℓi is
learly an involution
L σ = L L σ = L σ
satisfying j ℓi k, k ℓi j. If it happens to be the
ase that ℓi is
ℓ
i
a
ollineation, we
all it the Bol re(cid:29)e
tion with axis .
""sQj bk "s P′ = Pσℓi
" "
a " "
"j "
" " bj
P"s" ak ""s"
Q
k
ℓ
i
ℓ
i
Figure 2: The Bol re(cid:29)e
tion with axis .
Obviously, every Bol re(cid:29)e
tion (cid:28)xes a line pointwise (namely its axis)
andinter
hanges theothertwoline
lasses. Infa
t, itiseasytoseethatany
γ ∈ CollN
ollinℓea∈tiLon with this propγe−rt1yσisγa=Bσol re(cid:29)e
γti−o1nσ. Tγhen for any
ℓ ℓγ ℓ
and we must have , as is a
ollineation (cid:28)xing
ℓγ N
the line pointwise. In words, the set of Bol re(cid:29)e
tions of is invariant
N
under
onjugations by elements of the
ollineation group of .
ℓ ∈ L i = 1,2,3 P N
i i
Let , , be the lines through some point of . As we
σ σ σ = σ ℓ σ = ℓ (σ σ )3 = id
have just seen, ℓ1 ℓ2 ℓ1 ℓ3, sin
e 3 ℓ1 2. Therefore ℓ1 ℓ2
hσ σ σ i S
and ℓ1, ℓ2, ℓ3 isisomorphi
to 3. Thisfa
twillbe ofimportan
e later.
3 N 3 σ
ℓ
A -net is
alled a Moufang -net if is a Bol re(cid:29)e
tion for every
ℓ N
line . Theterminology isjusti(cid:28)ed by Bol, whoproved that isaMoufang
3 N
-net if and only if all
oordinate loops of are Moufang [4, p. 120℄.
8 G. P. Nagy and P. Vojt¥
hovský
s s
(cid:8)(cid:8)(cid:8) (cid:8)(cid:8)(cid:8)Rσℓ
(cid:8) (cid:8)
(cid:8) (cid:8)
(cid:8) (cid:8)
s(cid:8) s(cid:8)
R
s s
(cid:8)(cid:8)(cid:8) (cid:8)(cid:8)(cid:8)Pσℓ
(cid:8) (cid:8)
(cid:8) (cid:8)
(cid:8) (cid:8)
s(cid:8) s(cid:8)
P
ℓ
2
Figure 3: The -Bol
on(cid:28)guration.
2
The
on(cid:28)guration in Figure 3 is
alled the -Bol
on(cid:28)guration. Using
1 3
the other two dire
tions of axes, we obtain - and -Bol
on(cid:28)gurations.
N
With these
on(cid:28)gurations at hand, we see that the net is Moufang if and
Rσ Pσ
ℓ ℓ
only if all its Bol
on(cid:28)gurations
lose (i.e., and are
ollinear). See
[25, Se
. II.3℄ for more on
losures of net
on(cid:28)gurations.
3 Composition algebras
The most famous nonasso
iative Moufang loop is the multipli
ative loop of
real o
tonions. Re
all that o
tonions are built up from quaternions in a
way analogous to the
onstru
tion of quaternions from
omplex numbers,
or
omplex numbers from real numbers. Following Springer and Veldkamp
[22℄, we will imitate this pro
edure over any (cid:28)eld. We then
onstru
t a
ountable family of (cid:28)nite simple Moufang loops, one for every (cid:28)nite (cid:28)eld
GF(q)
.
F V F N : V → F
Let be a (cid:28)eld and a ve
tor spa
e over . A map is a
h , i :V ×V → F hu,vi = (u+v)N−uN−vN
quadrati
form if de(cid:28)nedby
(λu)N = λ2(uN) u ∈V λ ∈ F
isa bilinear form, and if holdsfor every and .
f : V ×V → F u v ∈ V
When is a bilinear form, then , are orthogo-
f (u,v)f = 0 u ⊥ v
nal (with respe⊥
t to ) if . We write . The orthogonal
W W ≤ V {v ∈ V; v ⊥ w
omplement of a subspa
e is the subspa
e ⊥ for
w ∈ W} f V = 0
every . The bilinear form is said to be non-degenerate if .
N h , i
A quadrati
form is non-degenerate if the bilinear form asso
iated
N N V
with is non-degenerate. When is non-degenerate, the ve
tor spa
e
W (V,N)
is said to be nonsingular. A subspa
e of is totally isotropi
if
uN = 0 u ∈ W (V,N)
for every . All maximal totally isotropi
subspa
es of
N
have the same dimension,
alled the Witt index. If isnon-degenerate and
dimV ≤ ∞ dimV/2
then the Witt index
annot ex
eed .
F F
In this paper, an algebra over is a ve
tor spa
e over with bilinear
multipli
ation. Spe
i(cid:28)
ally, we do not assume that multipli
ation in an
algebra is asso
iative.
O
tonions and simple Moufang loops 9
C = (C,N) F
A
omposition algebra over is an algebra with a mul-
e N : C → F
tipli
ative neutral element su
h that the quadrati
form is
non-degenerate and
(uv)N = uNvN
(2)
u v ∈ C N
holds for every , . In this
ontext, the quadrati
form is
alled a
norm.
h , i N
When is the bilinear form asso
iated with the norm , the
on-
x ∈ C x = hx,eie − x x ∈ C
jugate of is the element . Every element
satis(cid:28)es
x2−hx,eix+(xN)e =0
xx = xx = (xN)e
(
f. [22, Prop. 1.2.3℄), and thxus alxs−o1 = (xN)−1x . In pxaNrti
6=ula0r,
the multipli
ative inverse of is , as long as .
0 6= x∈ C xN = 0
Furthermore, is a zero divisor if and only if .
3.1 The Cayley-Di
kson pro
ess
∗
C = (C,N) F λ ∈ F = F \{0}
Let be a
omposition algebra over and .
D = C ×C
De(cid:28)ne a new produ
t on by
(x, y)(u, v) = (xu+λvy, vx+yu),
x y u v C M D
where , , , are elements of . Also de(cid:28)ne the norm on by
(x, y)M = xN −λ(yN),
x y ∈ C C D = (D,M)
where , . By [22, Prop. 1.5.3℄, if is asso
iative then
D C
is a
omposition algebra. Moreover, is asso
iative if and only if is
ommutative and asso
iative. Theabove pro
edure isknown astheCayley-
Di
kson pro
ess.
Wewouldnowliketo
onstru
tall
ompositionalgebrasbyiteratingthe
F F
Cayley-Di
kson pro
ess starting with . However, there is a twist when
2 charF = 2 F
isof
hara
teristi
. Namely, when then isnot a
omposition
hx,xi = (x + x)N − xN − xN = 0 x ∈ F
algebra sin
e for every , thus
hx,yi = hx,λxi = λhx,xi = 0 x y ∈ F N
for every , , and is therefore
degenerate. The situation looks as follows:
F
Theorem 3.1 (Thm. 1.6.2. [22℄). Every
omposition algebra over is
F charF
obtained by iterating the Cayley-Di
kson pro
ess, starting from if
2 2 charF
isnotequal to , and froma -dimensional
omposition algebra when
2 1 2
is equal to . The possible dimensions of a
omposition algebra are , ,
10 G. P. Nagy and P. Vojt¥
hovský
4 8 1 2
and . Composition algebras of dimension or are
ommutative and
4
asso
iative, those of dimension are asso
iative but not
ommutative, and
8
those of dimension are neither
ommutative nor asso
iative.
2 F
A
omposition algebra of dimension over is either a quadrati
(cid:28)eld
F F ⊕F
extension of or is isomorphi
to .
16
For a generalization of
omposition algebras into dimension we refer
the reader to [26℄.
3.2 Split o
tonion algebras
8
Composition algebras of dimension are known as o
tonion algebras. Sin
e
λ
there is a parameter in the Cayley-Di
kson pro
ess, it is
on
eivable (and
F
sometimes true) that there exist two o
tonion algebras over that are not
isomorphi
.
(C,N) 0 6= x ∈ C
A
omposition algebra is
alled split if there is su
h
xN = 0 F
that . By[22, Thm. 1.8.1℄, over any (cid:28)eld there isexa
tly onesplit
2 4 8
omposition algebra in dimension , and , up to isomorphism. As we
have already noti
ed, split
omposition algebras are pre
isely
omposition
F
algebras with zeOro(Fd)ivisors. The unique split o
tonion algeFbra over will
be denoted by . (It isFworth mentioning Oth(aFt)when is (cid:28)nite then
every o
tonion algebra over is isomorphi
to ,
f. [22, p. 22℄.)
O(F)
All split o
tonion algebras were known already to Zorn, who
on-
stru
ted them using the ve
tor matri
es
a α
x= ,
β b (3)
(cid:18) (cid:19)
a b ∈ F α β F3 N
where , and , are ve
tors in . The norm is given as the
detx= ab−α·β α·β
(cid:16)determinant(cid:17) , where is the usual dot produ
t
(α ,α ,α )·(β ,β ,β )= α β +α β +α β .
1 2 3 1 2 3 1 1 2 2 3 3
x
The
onjugate of is
b −α
x = ,
−β a (4)
(cid:18) (cid:19)
and two ve
tor matri
es are multiplied a
ording to
a α c γ ac+α·δ aγ +dα−β ×δ
= ,
β b δ d cβ +bδ+α×γ β ·γ+bd (5)
(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
