Re . Academia de Ciencias. Za agoza. 48 (1993)
CHARACTERIZATION OF COMPLEX FILIFORM LIE ALGEBRAS OF DIMENSION 8
ACCORDING
TO
WHETHER
THEY ARE
OR
NOT DERIVED FROM OTHERS
By
F.
J.
ECHARTE
REULA
1,. ,R.
GOMEZ
MARTII y
J.
NU n Z VALDEs'" .
Uni e sidad
de
Se illa.
Facul ad
de
Ma emá ica
s.
Dp o
de
Alg
eb
a,
Compu ación,
Geome ía
y
Topología.
C/
Ta
i a
s/n.
41012
Se
illa.
España.
2
Uni e sidad
de
Se illa.
Facul ad
de
In o má
ica
y
Es adí
s i
ca.
Dp
o
de
Ma emá ica
Aplicada.
C/
Ta ia
s/n
.
41012
Se illa
, E
spa
ñ
a.
AMS
(1980)
Subj
ec
Cla
ssi
ic
a
ion:
17
B
30
Aos ac
.-
In
h
is
pape ,
W9
cha ac e ize
hose
Comp Lex
FiLilo m
Lie
AL6eo as
01
dimens
ion
8
which
a e
de i ed
I
o
m o he
SoL aoLe
Lie
AL6eo as
01
hi6he
dimensiono
This
esuL
and
h
e
p e ious
one
6i en
in
({8])
aLLow
us
o
lind
a
compLe e
Lis
01
Cha ac e
is icaLLy
NiLpo en
F
iL
ilo
m
Lie
AL6eo as
01
di
mens
io
n 8 .
1.-
In oduc ion
and
No a ions.
The e
does
no
e
xis ,
a
p
esen
, a ny
cla
s s i
ic
a i on o
compl
ex
Nilpo en
L
ie
Alg
eb as
(NLA)
o
dim
ens
ion
g
e
a e
h
an
7 .
Goze
and
Ancochea,
by
he
in oduc ion
o
a new
in
a ian
wh
ich
he
y c
all
he
"cha ac e is ic
sequence",
which
co
e
sponds
o h
e max i mal
d
imen
sions
o
Jo dan
blocks
o
a ce
a
in
n
ilpo
en
ma ix
,
ob a
ined
he
classi ica io
n o
complex
Fi l i o m
Li
e
Alg
eb
a s (FLA) o
dim
ension
8
([1).
These
FLA,
as
i
is
known,
a
e
a
sub
s e o NLA.
The a
u ho s
a
e su
pp
o
e
d
by
he
p ojec
PAICYT o
he
Ju
n a
de
And
alucía.
E
spaña
(1990).
35
[[
A,B],C)
+
[[B,C],A]
+
[[C,A],B)
=O
and
we
deno
e
by
ma FLA
o
dim
ensio
n
~
([8),
ha
is,
a
complex
NLA
ad
mi ing
a b
asi
s
o
o
ep esen
he
Jacobi
O
36
F om
now
on,
we
w i e
(A,B,C)
( 1)
s
uch
ha
X. ii!
[m
,m
i.
( 2 )
id
en i y
A
co
mplex
FLA may
be
de i ed
om
an
SLA
o
highe
dim
ensiono
In
([8)
we
p o
ed
he
ollowing:
(1
.1)
Theo em:
A
complex
FLA
o
dimens
ion
~
is
ei h
e
de i ed
om
a
SLA
o
dimension
n+1
o
no
d
e i ed
om
any
LA.
The ma
in
pu p
o
se
o
his
pap
e
is,
h
e e o e,
s a ing
om
he
classi ic
a ion
o
FLA
o
dimension
8
by
Goze
and
Ancochea,
o
cha ac e iz
e
wo
g oups
o
hese
LA,
acco d
ing
o
whe he
o
no
hey
a e
de
i ed
om
o he
Sol able
Lie
Algeb as
(SLA)
o
highe
dim
ens
iono
This
cha ac
e
i
za ion
we
p opose
could
be
in e es ing
in
he
sense
o
k
no
wing
which
o
hese
FLA
a e
al
so
Cha ac e is ically
Ni l po
e
n
Li
e
Algeb as
(CNLA)
since,
as
we
p e iously
p o ed
in
an
ea
li
e
pa
p
e
([8),
i a FLA
is
no
de i
ed
om
any
LA
hen
i
i s a
CNLA. Th
is
he
o em
will
allow
us
o
w
i e
he
lis
o
Cha ac e is ic
al
ly
Nilp
o en
Fili o m
Lie
Algeb as
(CNFLA)
o
dimens i on
8.
Th
e e o
e,
al hough
he
classi ica ion
o
Lie
Algeb as
(LA)
o d
ime
ns
ion
g
e
a
e
h
an
7 h
as
no
be
en
ob
ained
ye ,
excep
o
FLA,
he
main
p oblem
which
is
now
consid
e ed
is
he
sea ch
o
new
esul s,
a he
han
he
cla
ssi ica ion
i s
el ,
such
as
he
a ainmen
o
new
in a ian s
o
he
desc ip ion
o
he
i educible
componen s
o
a ie ies
o
LA.
Fo
his
pu pose,
Vale
i a
s
p o ed
ecen ly
ha
h
e
a i
e y
o
NLA
o
dimen
sion
8
has
8
i educibl
e
componen s
and
ha
only
h
e
wo
i s
o
h
em
mee
h
e
open
s
e
o FLA
([Q).
lIow
e
e ,
he
impo
ance
o
hese
au ho s'
wo k
is
no
only
o
ha e ob
ai
n
ed
hi s
cl
a s
si
ica ion,
bu ,
abo e
all,
o
ha e
de ised
echn
i
ques
wh
ich
c
an
be
appli
ed
in
any
dim
ension,
which,
apa
om
h
e
ac
o
ha
ing
p
e m
i
ed
Gómez
Ma ín
o
clasSi y
FLA
o
dimens
ion
9
([5])
,
o a
lly
so
l
ed
he
p obl
em
o
h
is
cl
assi ica ion
o
FLA
o
g
ea
e
d
imens
ion
,
al
hough,
o
cou se,
his
equi es
ha d
and
comp
li
ca
e
d
calcula ions
which
a e
imp
ossible
wi hou
he
use
o
a
compu
e .
Mo eo e ,
~
will
deno e
a
complex
SLA
o
dim
en
sion
n+1
be
aXz
Sn
Sn
3 S
X.
J-'
IX
.x.:
, J
(1 Sj'S
n-1)
[X
',X.'
I=O
(1
<jS
• J
sui able
way
e i ying
1<j
A
necessa y
and
su icien
condi ion
o
a FLA ID o
be
de i ed
om
he
SLA ~
is
ha
a "" O.
u
A
complex
FLA
ID
is
a
CNLA
i
and
only
i
ID
i s
no
de i ed
om
any
LA.
(3)
(1. 3)
Theo em
As we
men ioned
aboye,
in
([8])
he
ollowing
wo
h
eo em
s
a e
p o ed:
(1 . 2)
Theo em
So,
o
sepa a
e
FLA
o
dimension
8
in o
wo
g
oups
acco
din
g
o
whe he
o
no
hey
a e
de i ed
om
ano he
SLA
(Theo em
(1.1)
we
will
use
a
me hod
p e iou
sly
indica ed
by
us
o
he
ca se o
NLA
o
dimension
7
([4]).
Thi
s
p oceedu e,
which
we
call
ed
he
"m
e
hod o
de e mina ion
o
he
anishing
o
he
co
e
ici
en
a,,"
, c
an
be
applie
d
o
FLA.
I
consis s
o
a
se
o
i e a i e
calc
ula ion
s
ba
s
ed
on
all
(6
)
37
wi h
(Xj,
~
,
U) = O, 1 S
Lh
S n ,
The
FLA
ID
is
said
o
be
de i ed
om
he
SLA
~
i
[~,~I
=ID,
whe e
[~,~]
ep esen s
any
linea
combina ion
o
all
b acke s
among
ields
o
he
basis
(1).
Cha ac e is ically
Nilpo en
Lie
Algeb as
a e
hos
e LA i n
which
all
hei
de i a ions
a e
nilpo en
.
The
i s
examples
o
CNLA
gi en
in
he
li e a u e
we e
o
LA
ha
a e
no
de i
ed
om
a ny LA,
un il
Luks
ga e
an
example
o
a CNLA
which
is
d
e i ed
om a LA o
dimension
18
([7])
.
Howe e ,
i
is
e y
easy
o
gi e
ex
ampl
es o
Nilpo en
Lie
Algeb as
(NLA)
which
a e
de i ed
and
no
CNLA.
X·
=X
jj
hen
i
esul s
ha
Mo
eo
e ,
s i n
ce
[
X.,X
jl
O 1 <j<nand [X.,X
"J
ma E
C),
i
we
con
sid
e
h
e
change
o
ba s i s
gi
e
n
by
X'
= X + a X
" " ,
n) . S
o,
a ba s i s
(1)
c
an
always
chosen
in
'a
(4)
such
ha
(5 ) (X"
••
, X" ' U}
is
a
ba
s
í.s
,
whe e
(X"X
z, .
..
, X,, }
is
he
basis
(1}
men ion
ed
aboye,
and
U a
de i a i
on
o
ID
such
ha
.
and
These
i e a i e
s eps
a e
he
ollowing:
38
wi h
>3
hei
~
,.
hs
a,.
wi h
'J
,
4.-
Repea ing
he
o me
h ee
s eps
wi h
U,
X,.
and
h,. >2
in
he
i s
place.
Secondly,
wi h
U, Xs
and
l),
wi h
s
In
he
case
ha
he
FLA m
is
de i ed
om
ano he
SLA
o
dimension
one
g ea e
han
he
dimension
o
m,
we
can
ob ain
he
simples
SLA
~
by
aking
a..
=1
and
he
es
o
he
coe icien s
aLj O
(i,
j ..
1)
.
Mo eo e ,
we
can
check
ha
in
FLA
o
dimension
S
9,
ei he
di ec ly
o
by
sui able
changes
o
base,
i
is
always
possible
o
ge
(7 ) [l), , U1=ah
~
wi
h
h=
1,
2,
...
, n
(al hough
his
is
no
p o ed
in
he
gene al
case
o
dimension
mnI .
5.-
Pinally,
obse ing
i
in
he
new
exp essions
(6)
ob ained
in
his
way
he
coe icien
a..
appea s
explici y
o
no ;
his
will
indica e,
acco ding
o
heo em
(1.2)
whe he
he
PLA m
which
we
a e
s udying
is
de i ed
om
ano he
SLA
~
o
dimension
one
mo e
han
he
dimension
o
m.
and
so
on,
un il
e mina ing
his
p oceeding
wi h
U, Xn_.
and
Xn'
2.-
Di ision
o
complex
FLA
o
dimension
~
in o
wo
g oups
depending
on
whe
he
hey
~
o
no
de i ed
om
o he LA.
3.-
Replacing
in
(6)
he
coe icien s
co esponding
alues
ob ained
p e iously
in
s ep
2.
2.-
Using
each
equa ion
ob ained
in
his
way
o
exp ess
in
e ms
o
he
o he s
he
coe icien
~
,
whose
pai
o
subindices
is
'J
he
g ea e
(in
lexicog aphic
o de
) . (Po
ins ance,
om
he
equa ion
aaaa=Owe
will
ha e
a a a
a.
7) •
,.,.
••
44
'7
44
,.,.
••
1.-
Using
he
Jacobi
iden i y
(X.'
l), , U) =O
wi h
h.
>
1,
i
de i ing
om
such
iden i y
he
se
o
equa ions
ob ained
by
se ing
o
ze o
he
coe icien
o
each
ield
X..
J
pos
sible
J a
cobi
iden i ie
s
o
he
kind
(~,
~,
U) =o
wi h
1 S
i,j
S
8 , om
which
we
can
d
e
e mine
whe he
he
co
e i
cien
a••
anishes
o
no o
This
in
u n
de
e mines,
by
heo em
(1.2),
whe he
he
PLA m
is
de i ed
om
he
SLA
~
w
i h
{X.'
..
,X.,U}
a s a
ba
sis.
By
appliying
he
me hod
men ioned
abo e
o
each
o
hem
we
ob ain
he
ollowing
e
sul s:
By
he
me hod
m
en ioned
abo
e,
we i nd
ha
6o
he
13
FLA
and
2
o
he
7
amilies
a e
de i ed
om
SLA
o
d
imens
io
n8w
he
eas
he
o he
7FLA
and
he
5
amilies
a e
no o
• Z 8
law
s
¡Jo'
¡Jo'
,
¡JB
'
a e
no
de
i
ed
The
FLA
o
d
imens
ion
8 wi
h
la
w ¡J: i s no de
i
ed
o
m
an
y
LA,
sinc
e:
39
a
X+a
X + a
X+a
X+
a
X+
a X
u z
.0
o
.....
i""
.,,"
.7
7
O
aX
oz z
(a."
+
a.
7)Xz+
(a
oz +
a.?)
Xo
a"z
Xz+
3/5
a.?
Xa+
(a
oz +
a.?)
X..
(3/5
a"z
+ a
...
+
3/5
a."
+
6/25
ai
,,
)Xz+
+
(a"z
+
a.,,)
X
o+
1/5
a.?
X.. + ( ao
z+
a
.?)
X"
a?Z Xz+
(3/5
a"z
+
a."
+
6/25
a.,,
) +
+
(a"z
+
a."
+
2/5
a.,,)
X.. +
1/5
aoX" + (aoz +
a.
? ) X"
aoz Xz
+(a?Z-
a
...
)X
o+
(3/ 5 a" z - a
...
+
2/
5a . ,,+
6/25
a.
,, ) X.. +
+
(a"z
+
1/5
a . ,, ) X"- a
."
X" + a. zX?
The
FLA
o
dimension
8
wi h
law
¡.¡~
is
no
de
i ed
om a ny
LA,
since
:
[X
o'
Uj
[X.
,U]
[Xz
,U]
[X
o'
U]
[X..
,U]
[X",U]
[X",U]
(
2.
2)
(2.1)
So we
ob ain
he
ollowing
P oo :
We
sepa a e
FLA
o
dim
ens
io
n 8 i
n
o
wo
g
ou
p s d
ependin
g
on
whe
h
e
hey
a e
de
i
ed
om
an
o he SLA,
s a ing
om
he
clas
si ica ion
o
hes
e FLA by G
oz
e
an
d Anc o
chea
([1
]).
This
cla
s
si ic
a
ion
(which
i s
indica ed
in
an A
pp
end
i x a
he
end
o
his
pap
e )
con ains
a
lis
o
13 i
sola
ed
FLA
and
7
on
e-p
a ame e
a
milies
o
FLA.
Theo em
1.-
The
complex
PLA
o
dimension
8
wi h
...
c:S,O
8 P,Ol
:l0,0I
:ll
S.8,o .
u:J
,a
and
17
¡Jo, ¡Jo ' ¡JB' ¡Jo
,¡Jo
,¡Jo'
¡Jo
,¡Jo
"«
om
any
o he
LA.

The
FLA
o
di o
ension
8
wi h
law
~o
is
no
de i ed
o o
any
•
X7
X7
X7
a
oz
+a
oz
X7
) Xcs
X+a
CS
02
+a
17
aX
.2
"
(a
+
oz
(a
+
oz
X - a
"
1CS
X+aX
es
17?
X+aX
8
82"
+o ) a17 X¿ +
40
a
X+a
X
+a
X
+a
X+a
X
iZ
2
18
81.'" '" 115 i 1.es d
O
aX
az z
ex a X + a X
1 7 2
82
8
(a 1
CS
+a 1
7)
X2+
(1
+ ex) a 17
acsz Xz+
(a
1
CS
+ a 1
7)
X. +
(2
a1
CS
X2+
(a
02 +a1
7)
X.
a"2
Xz+ (
a.
2+ a 17) X¿
(a
1
CS+
a
1¿)
X2+
(a"2
+ a
1"
+ a17) Xo+ ( a 0 2 +
a72 X2+ (
a"
2 + a
1"
+ a17) X¿ +
(a.
2+ a1
7)
. Xcs
a.
2·X2+ a 7 2 X. -
(a
1¿
+ a1
d)
X¿ +
a"2
X" - a 1
CS
Xcs
LA, s
ince
:
a X + a
X+a
X + a
X+a
X
+a
X
12
2
18
9
14"
la
5
les
es
17
7
O
aX
.2
2
The
FLA
o
di o
ension
8
wi h
l aw
~¿
is
no
de i ed
o o
any
o
T
he
FLA o
di
o
ens
ion
8 wi
h
law
~=.ex
i s
no
de i
ed
o o
any
LA, s i n
ce
:
LA, s
inc
e:
a
X+
a X
+a
X+a
X
+a
X+a
X
12
Zsa 8
14"
115
es
id
eS
1.7
7
O
aX
.z
z
a1
CS
Xz+
(a.
z+ a 1
7)
X.
a"z
Xz+
(a
02 + a 1
7)
X¿
(a
17+ a
..
) X2+
(a"2
+ a
1,,)
X.+
(a
0
2+
a1
7)
X"
a72 X2+ a 17 X. +
(a"2
+ a
1,,)
X¿ +
(a
0 2 +a17
a.
2X2+
(a
72 -a1
CS)
X. - a
1¿
X¿ +
a"2
X" - a 1
CS
aX
.z
z
a1
CS
Xz+
(a
oz + a 1
7)
X.
a"
z Xz+
(a
oz + a 17) X¿
(a
17+a1
CS
+ a 1
¿)
Xz+
(a"z
+ a 1
7+
a
1,,)
Xo+
a7Z Xz+ a17 Xo+
(a"z
+ a
1"
+ a 1
7)
X¿ +
aoz X
Z+
(a 7 Z- a 1
CS
) X. -
(a
1CS-a
1¿)
X¿ +
a"z
a
X+
a
X+a
X+
a X + a
12
2 1 8 a
1.
'"
llS
a
id
O
(
2.5)
[Xi , U )
[XZ
,U]
[X.
,U)
[
X¿,U)
[
X",U)
[XCS,U]
[X
7,
U)
[XO'U)
[X
i'
U)
[XZ
,U]
[X. , U)
[X¿ , U)
[X
",U)
[
XCS
, U]
[Xi
,U)
[XZ
,U)
[X
O'
U]
[X¿,U
)
[X",U)
[XCS,U)
[X
7,U]
[X.
,U)
(2 . 4)
[Xi
,U]
[X
z
,U)
[X. , U]
[X¿ ,
U]
[
X",U)
[XCS,U)
[X7
,U]
[XO
,U]
(2.3)
(2
.8)
Th
eFLA
o
di
mension
8
wi b
l a w
¡.¡
~O,OI
is
n
o
de i
ed
om
any
LA, s
inc
e:
[X" Uj a X + a X +a X + a X + aX + a X
'2
2
,a
a, ¿ ¿
'"
",., a
,?
7
[X2
,U]
O
[X
a'
U] aX
82 2
[X¿'
U] a X + a X
¿2 2 82 8
[
X",U]
a X + a X + aX
"2 2¿ 2 8 82 ¿
[
X."U]
aX + a X +a X + aX
.,2 2
"2
8¿ 2 ¿82 "
[
X?'
Uj a X + a X + a X+a X + aX
?2 2
"2
8"2 ¿¿ 2 "82
es
[X
8,Uj
aX + (a72 -
ex
a - a -a ,
¿)
X + (a - a- a..,, ) X+
82 2
'7
,.,
8" 2 ' 7 ¿
+(
a"2
- a , ., ) X+(a¿2 -a)X + aX
"
,?
.,
82 ?
41
Tbe
FLA o
dimension
8
wi h
law
¡.¡: i
S'
no d c
i
ed
om
any
LA,
si
nce
:
a
X+
a X
+a
X+a
X+a
X
+a
X
i 2 2 sa 8
1'"
'"
115
!S 1 d es
17
7
O
any
X+
¿
aX+aX
1
7!5
92
d
( 2+01)
a,
,,-
al
CS
)X¿
aX
82 2
-aX+ a X
, ? 2 82 a
aX+aX
,., 2 82 ¿
a X +
(a
+
a,?
)X+a X + a X
.,2 2,., 8..? ¿8 2 e
a X +
(a"2
-a,
,,
-
a,
.,)
X + aX+a X + aX
?2
28,? ¿, ? "82
.,
aX+
(a?2
+ a , ¿ ) X.+
(a"2
-
a,,,
- 2 a , ., ) X
.,..
aX + aX
82 28¿,., "82 7
The
FLA o
dimen
sion
8w
i h
law
¡.¡p,
CJ.
is
no d e
i
ed
om
8
LA,
s
ince
:
a X. +aX+ a X + a X+a X + a X
'2
2 ..a 8
,¿
¿'" ",.,
.,
,?
7
O
a ? 2 X2+ (a" 2-
a,,,)
Xo+a, ? X¿ + ( 2+01)
a02 X2+ (a?2- 01 a , ¿ - a, ,, ) X8+(
a.,2
-
-
(2
+01)
a,
., X" +
aa
2 X7
aX
82 2
a X + a X
,?
282 a
aX + a X + a X
"2
2 ..? 882 ¿
a X +
(a"2
+ a..
?)
X+ a X+a X
.,2 28
,?
¿82 "
aX+
(a"2
-
a,
., ) X + (a"2 +a
,?
)X + aX + a X
?2 28¿
,?
"82 a
aX+
(a?2
-
a,
¿ - 01
a,.,)
X + (a - a-2a
ex
a'7
)
82 28
"2
'"
,.,
+
(a
-a..
.,)
X + a X
"2 "82 7
[X" U]
[X2
,U]
[Xa
,U]
[X
¿,U]
[X
",U]
[
X."U]
[X?
,U]
[XO
,U]
[X" U]
[X2'U]
[Xa'U]
[X
¿'
U]
[ X"
,U]
[
X."U]
[X?
,U]
[XO'U]
(
2.
6)
(2
.7)
NOTE:
In
h
i s
alg
eb a
i
is
e i ied
ha
a=
3/2
(o. -1)(o. -
2/
3)
42
(
2.11
)
The
FLA
o
di oension
8
wi h
law
¡.I~"
'
o.
is
no
de i ed
o o
any
LA, s
ince:
[
X,
, U] a X + a X + aX + a X + a X+ a X
iZ
2
..
S
'4
4
'"
"
'"
"
,?
?
[X
2,
UI O
[Xs
,U
Ia X
S2 2
[X
4,U
I a X + a X
42 2 S2 S
[X" , U] a X + a X + a X
"2 242 S S2 4
[X" , UI a X + a X + a X + a X
csz 2
"2
S 42 4S2 "
[ X?
,U
]a X +a X + a X + a X + a X
?2 2
"2
S"2 4
42
"S2 "
[Xs
,U
]a X .+
(a
- o.
a,?-
a,,,-
a,,,)
X +
(a
-a
a,,,)
X +
e2 2 ?2 e
"2
,?
4
+ a X + a X + a X
"2
"42 "S2 ?
FLA o
di o
cnsion
8
wi h
l aw
i
i s
no
de i
cd
o o
Th
c¡.le
any
LA,
si
nc c :
aX+aX+ a X + a X+aX + a X
'2
2
..
S
'4
4
'"
"
'"
"
,?
?
O
aX
e2 2
aX + a X
42 2 e2 s
a X + a X +aX
"2 242 9 S2 4
aX+a X + a X + a X
csz
2
"2
S 42 4 92 "
a X + a X +aX+aX + a X
?2 2
"2
S"2 4 42 "S2 "
aX +
(a
- a -
a,
4
~)
X+
(a"2
-a,,,)
X+
(a
-
a,,,)
X +
e2 2 ?2
,?
94
"2
"
+
(a
42 - a )X+aX
,?
"92 7
The
FLA o
di oen
sion
8 wi h
law
¡.I
~
?
i s no d e i e d
o o
any
LA,
since:
42
The
FLA
o
di oension
8
wi
h
law
¡.I~s,o.
is
no
de i
ed
o o
any
LA,
since:
a
X+a
X+a
X
.+a
X+a
X+a
X+a
X
i2
2
iU
Ss.....
i!5
e
id
di
i7
7
iB
B
O
aX
92 2
aX+aX
""'z
Z
82
a
a"2
X2+
(a
4
2+
o.
ale)
X9+ a 92 X4
acsz X2+
(a"2
+
a,?
+
a'9)
Xs+
(a
42+
(1
+20.) a
..
) X4+ a 92x"
.
a?2
X2+
(a"2
-
a,,,)
X9+
(a"2
+
a,?
· + 2
a,e)
X4+
+
(a
42 +
(2+30.)
a.
e)
X" + a S2 x"
aez
X2+
(a?2-
o.
a,,,-
a,,,)
Xs+
(a"2-
(2+0.)
a,,,-
a,?)
X4+
+
(a"2
+ 2
a'
9- o.
a,?
) X" +
(a
42 +
(2+
30.) a
u)
X" + a 92 X?
( 2 . 1
2)
(
2.10)
[X"
U]
[X
2,
U]
[Xs
,U]
[X
4,U
]
[
X",U
I
[X" , U]
[X
?'
UI
(
2.
9)
[ X" UI
[X2,UI
[Xs
,U]
[X4,UI
[X" , U]
[X",Uj
[X?'
UI
[Xe,UI
[Xl
,U]
aX+aX+aX+ a X+ a X+aX+aX
12
2
18
8
14
4 115
15
ld
d 1 7 7
18
8
[X
2,
U] O
[X
8,U]
aX
92 2
[X
4,U]
aX+aX
42
2
82
9
[X
15,U]
aX+aX+ a X
152 2
42
9
92
4
[Xd,U]
aX+
(a
15
2+a
18
)X+aX+aX
dZ
2942 4
92
15
[X
7
,U]
aX+
(a
+ a 19 )X+
(a
15
2+2 a
19
)X+ a X+ a X
72
2
d2
94
42
15
92
a
[X
9
,U]
aX+
(a
72 - a a1d )X+
(a
+ a -
aH)
X+
92 2
17
9
d2
19
4
(a
15
2+2au)X+aX+ a X•
15
42
d
92
7
Co olla y
1.-
The e
exis
exae ly
12
CNFLA
o
dimension
8 .
These
a
e
he
ollowing:
1294
d,o.
9P,o.
lO
,OI.
/-18'
/-18
'/-lo' /-l. , "« '
/-l.,
/-1
9'/-lo '
11
18,a
l l,a
and
/-1:7
/-l. ' /-l. ' /-l.
P oo :
I
is
an
immedia e
eonsequenee
o
Theo ems
1
and
(1.
3)
••
Theo em
2.-
The
eomplex
FLA o
dimension
8
wi h
laws
/-115,
7,0.
U
8/-lo '
/-1
8'
14,0l
ld
18
lP
and
/-120
a e
de i ed
om
SLA
o
dimension
9.
/-1
9'/-l. ,/-lo '
/-1
9•
P oo :
By
applying
he
me hod
men ioned
aboye
o
ea
eh
o
hem
we
ind
ha
eaeh
o
hem
is
espee i ely
de i ed
om
a SLA
o
d
imension
9,
wi h
basis
{Xl'
. ....
,X
a,
U},
e i ying
[X"
U] =a. X. (1 :S i:S 8 )
e e
whe e
he
eoe ieien s
a.
o
eaeh
o
hem
a e
he
ollow
ing:
e
FLA aaaaaa a a
obs.
12o4
15
d7•
15
17654321
/-l.
7,0. 1 8 765432
/-l.
12
1 8 765432
/-l.
:14
,0.
19876543
/-lo
ld
327 24 21 18 15 12 9(
*)
/-lo
18
110 987654
/-lo
u'
1
11
10 98765
/-l.
20 18765432
/-l.