Transcript No Slide Title

CT08 28/06/2008
Borromean Objects, as examplified by the group G168
of Klein’s Quartic, linked with Moving Logic
René Guitart
Université Paris Diderot Paris 7
PLAN
1- Borromean object
1.2- Borromean link
1.3- Aristotle-Appule square and Sesmart-Blanché hexagram
1.4- A « borromean » object in rings: Mat2(GF(2)), in fields : GF(8)
1.5- An observation on Abel.
2- The Klein’s Quartic X(7) and its group G168
3- Moving Logic
3.1. Every function is a (moving) boolean function
3.2. Parametrization of the family of boolean structures
3.3. Logical differentials or cohomological theory of meaning
REFERENCES
[1] F. Klein, Über die Transformationen siebenter Ordnung der
elliptischen Funktionen, Math. Ann. 14 (1879), 428-471. Translate in [2]
[2] S. Levy, The Eightfold Way, Cambridge U. Press, 1999.
[3] R. Guitart, Théorie cohomologique du sens, SIC Amiens,
8 novembre 2003, compte-rendu 2004-10/Mars 2004,
LAMFA UMR 6140, 39-47.
[4] R. Guitart, Moving Logic, from Boole to Galois, Colloque
International `Charles Ehrersmann : 100 ans’, 7-9 octobre 2005,
Cahiers Top. Géo. Diff. Cat. XVLI-3, 196-198.
1- Borromean Object in a category
RS
R
B/I
R = F(r)
S = F(s)
I = F(i)
B = F({r, s, i})/
with:
-  invariant by the cycle
rsir
-B/F(r) = F’({s, i})
-B/F(s) = F’({i, r})
-B/F(i) = F’({r, s})
B
RI = B/S
S
B/R
I
IS
Examples :
XY = X+Y
(of « trivial » binary composition) X Y = 1
1.2-Borromean links
John Robinson’sculptures
In chemistry
Borromeans links
Notations ?
r
i
From the psychoanalyst
Jacques Lacan:
r = real
s = symbolic
i = imaginary
s
(with a joke in french :
rsi = hérésie)
A clin d’œil
To the old
Theological use
of borromean
Borromeans links : the fundamental group
(Computation à la Dehn)
rir-1sr = *
ri
r
srs-1is = *
i
isi-1ri = *
*
sr
1
is
s
cb-1a
a
c
b
Now, if we add i = 1, the system degenerates
in: sr = sr = sr = * : that is to say : no condition.
a
c
b
Associated borromean object in the category of groups
Z*Z= 1(E3\R)*1(E3\S)
Z= 1(E3\R)
Z*Z= 1(E3\R)*1(E3\I)
Z= 1(E3\I)
1(E3\B)
Z= 1(E3\S)
Z*Z= 1(E3\I)*1(E3\S)
Borromeans links: Tait’s serie
and so on.
So we get different 1(E3\B),
1.3-Aristotle-Appulé square and Sesmat-Blanché hexagone
Predeterminated
R S
Impossible
Necessary
R
S
R S
I
R
Possible
S
I
I
I
Eventual
Contingent
Associated borromean object in the category of boolean algebras
P(R)*P(S)
P(R)
P(E)
P(R)*P(I)
P(S)
P(I)
P(I)*P(S)
1.4.1- A borromean object in rings: Mat2(GF(2))
r = 01
10
,
s=
10
11
,
i=
11
01
.
Mat2(GF(2)) is generated by r, s, i freely with the relations:
r+r = s+s =i+i = 0
r2 = s2 = i2 = 1, rs = si = ir , sr = is = ri, r + s + i = 0.
If we do i = 0 we get:
r2 = s2 =1, rs = 0 , sr = 0, r + s = 0,
So: r = s = 0 = 1.
1.4.2 - Borromean objects in fields: GF(8)
GF(8) = GF(2)[X]/(X3+X2+1)
GF(8) = {0, 1, r, s, i, r-1, s-1, i-1}
r, s, i roots of X3+X2+1= 0:
rsi = 1 rs+si+ir = 0 r+s+i = 1
i-1 = s+1 = rs
s-1 = r+1 = ir
r-1 = i+1 = si
r2 = s
i2 = r
s2 = i
r+r-1 = s
s+s-1 = i
i+i-1 = r
If we do i = 0, we get r = s = i = 0 = 1
1.5- An observation on Abel
1) There is a borromean flavor in Abel approach of elliptics functions:



0
dx
2
2
2
The three Abel functions , f, F
can be normalized in this way:
2
(1 c x )(1 e x )
f  1 c 2 2
F  1 e 2 2
R = ice, S = ef, I = ieF.
Then we have: R = SI, S = IR, I = RS.
2) The 5th degree problem is related to a « borromean » problem:
x = a+yz, y = b+zx, z = c+xy.
x5-ax4-2x3+(2a-bc)x2 +(1-b2-c2)x-(a+bc) = 0
y5-by4-2y3+(2b-ca)y2+(1-c2-a2)y-(b+ca) = 0
z5-cz4-2z3 +(2c-ab)z2+(1-a2-b2)z -(c+ab) = 0
Here Galois group = S5
Hermite, Klein : resolution of the fifth degree equation by elliptic functions
2-The Klein’s Quartic X(7) and its group G168
Equation :
X(7) ={[x:y:z] \in P2(C) ; x3y+y3z+z3x = 0}
A smooth algebraic curve, riemannian, of genus 3
Its group of homographic symetries is G168,
the only simple group of order 168. So we
get the maximal od symetries in genus 3.
QuickTime™ et un
décompresseur TIFF (non compressé)
sont requis pour visionner cette image.
QuickTime™ et un
décompresseur TIFF (non compressé)
sont requis pour visionner cette image.
Relation to the borromean link
Relation to the borromean field
In fact G168 = GL3(GF(2)) = GL(GF(8)/GF(2))
A borromean group: G168
Equation :
X(7) ={[x:y:z]  P2(C) ; x3y+y3z+z3x = 0}
Group of automorphisms = GL3(GF(2)) = G168
Generators:
Relations:
R
+
I
S
R=
1 1 1
1 0 1
S=
0 1 1
1 0 1
1 1 1
1 1 0
(SRIR-1)2 = 1
S7 = 1


((IS3I-1)(SRIR-1))3 = 1
((IS3I-1)4(SRIR-1))4 = 1
R, S, I as permutations
on {1, 2, 3, 4, 5, 6, 7} :
R=(1746325),
S = (1647235),
I= (1564327).
T=(142)(356)
TRT-1 = S
TST-1 = I
TIT-1 = R

I=
0 1 1
1 1 0
1 1 1
A borromean group: G168
Equation :
X(7) ={[x:y:z]  P2(C) ; x3y+y3z+z3x = 0}
Group of automorphisms = GL3(GF(2)) = G168
R, S, I as permutations
on {1, 2, 3, 4, 5, 6, 7} :
R=(1746325),
S = (1647235),
I= (1564327).
A = (46)(57)
B = (23)(67)
C= (15)(37)
IR =: At
RS =: Bt
SI =: Ct
T=(142)(356) TRT-1 = S
TST-1 = I
TIT-1 = R
TAT-1 = B
TBT-1 = C
TCT-1 = A
A = RtIt
B = StRt
C = ItSt
R = ACB
S = BAC
I = CBA
R
Bt =RS
IR = At
S
I
SI = Ct
3. Moving Logic
Theory of Meaning
Theory of True
=
Galois
Boole
3. Moving Logic (continuation)
2n as a (unique) boolean algebra, 2n as a (unique) field ;
so what is the relation between these two unique structures on 2n ?
Let e = (e1, …, en) be a basis of GF(2n) over GF(2)
We get on 2n a conjonction e and a negation e :
( xiei ) e ( x j e j )  ( xi yiei )
e x  x  te
with
,
te  e1  ... en .
x e y  (e x)e y
Could we recovered the field multiplication from this boolean structure ?
No, but…

…We have to thing to the system of all the e
where e moves in the set of basis, as a moving logic,
i.e. a logical analoguous of the moving frame* :
x me y = m(m-1x e m-1y )
and to use of 2n equipped with the system of all these e
as a logical manifold.
*: cf. Serret-Frenet, Darboux, Ribaucour, Cartan, Ehresmann
3.1. Every function is a moving boolean function
Key starting observation:
In boolean algebra we have x2 := xx = x
In a Galois field of car. 2 we have x2  x (indiscernible), x2  x ex
It can be proved (see « Moving Logic, from Boole to Galois », in Cahiers)
that:
1)In GF(2n) every function f: GF(2n)m  GF(2n) is a composition
of constants, ,  (associated to a normal basis) and the Frobenius (-)2.
2) In GF(2n) there exist 4 basis p, q, r,s such that
every function f: GF(2n)m  GF(2n) is a composition
of constants, p, p, q, q, r, r, s, s.
3) In particular of course the multiplication of GF(2n) could be written
in this way.
The idea of this general theorem in fact comes from computations with
borromean things and Klein’quartic hereover.
Let look at the case n = 2, i.e. GF(4)
In this case we can compute that:
x e y  x 2 y 2  t e (x 2 y  xy 2 )
If u and v are the two roots of X2+X+1, we consider the three basis
k = (u,v), a = (1, v), b = (u, 1), and we have:
x 2  x  1 
x  1 x  1
k
a
b
xy  x 2 k y  x k y 2  x 2 k y 2
Remark :
2  x (indiscernible), the product of the Galois Field xy is such that
So,
as
x

xy  xky
Case n = 3, i.e. GF(8) = GF(2)[X]/(X3+X2+1)
r, s, i roots of X3+X2+1= 0: GF(8) = {0, 1, r, s, i, r-1, s-1, i-1}
We have K = (r, s, i) as the unique normal basis of GF(8)
We consider the three matrices R, S, I which generates G168 ;
They provide three auto-dual basis, denoted again by
R = (r-1, i-1, 1)
S = (1, s-1, r-1)
We have tR = r, tS = s, tI = i,
I = (s-1, 1, i-1)
1 2
1
x R y  ix  s x  i x
4
x S y  rx 4  i1 x 2  r1 x
1 2
1
x I y  sx  r x  s x
4
x  x R i  x S r  x I s
xy  x 2 K y 2  x K y 4  x 4 K y  x 4 K y 2  x 2 K y 4
2
Remark : So, as x2  x (indiscernible), the product of the Galois Field xy is
such that

xy  xky
3.2. Parametrization of the family of boolean structures
x t y  x y  t(x y  xy )
2
In GF(4):
2
2
2
with t = e1+e2
4 4
5
4 2
2 4
x

y

x
y

l
m[x
y

x
y ]
In GF(8):
e

l4 (t  1)[ x 4 y  xy 4 ]  l2 t[x 2 y  xy 2 ]
with : q = (e1+e2+e3)(e1e2+e2e3 +e3e1)(e1e2e3)-1
4
2 3 4 5 6 7
t = (q+1) , m = 1+t +t +t +t +t +t ,
l = (e1+e2+e3)t-1.

3.3. Logical differentials or cohomological theory of meaning
« - Do you like this cake ?
- It is very good, but I dont like it »
G but Not G
Paradoxal ?
(G) (G) = 0
Logical expression
!
Speculation of the Meaning :
From the common general point of view C it is very good to eat it,
and
From my individual point of view I it is not good to eat it.
(E C G) ?(E I ??G)
Moving Logic expression
So, in this sentence, « but » is interpreted as:
but = (E C (-)) ?(E I ??(-))
and the meaning lives in the move of indices (C, ?, I, ??) = (ae, e+be, ce, de) :
Meaning(G but Not G) = {(a, b, c, d) ; (E ae G) ce (E bedeG) = 1}
You can compute in GF(4) with
X tY = X2Y2+t(X2Y+XY2+1).
tX = X+t
X tY = X2Y2+t(X2Y+XY2).
There you can use of additive notations for indices
In fact the pure meaning is the Logical Difference Equation
(E e+a G) e+c(E e+b e+dG) - (E e G) e(E e eG) = 1
which can be expressed through logical derivative
like the derivative of implication
[X t+hY - X tY]/h.
END