Study of Lie algebras by using combinatorial structures
Full text
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
S udy o Lie Algeb as by Using
Combina o ial S uc u es
Manuel Ceballos, Juan Núñez and Ángel F. Teno io
Uni e si y o Se ille and Pablo de Ola ide Uni e si y
24 - 28 May 2010
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
Con en s
1
P elimina ies on Lie algeb as
2
Associa ing combina o ial s uc u es wi h Lie algeb as
3
Cycle Dig aphs and Lie Algeb as
4
Implemen ing he algo i hm wi h Maple
5
2-s ep sol able Lie algeb as and combina o ial s uc u es
6
Re e ences
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
P elimina ies
Lie algeb a
ALie algeb a
g
is a ec o space wi h a second bilinea
composi ion law (
[,]
) which sa ises:
[
X
,
X
] =
0
,
∀
X
∈g
and
[[
X
,
Y
],
Z
] + [[
Y
,
Z
],
X
] + [[
Z
,
X
],
Y
] =
0
,
∀
X
,
Y
,
Z
∈g
.
S uc u e cons an s
A basis
{
e
h
}
n
h
=
1
o
g
is cha ac e ized by i s s uc u e
cons an s:
[
e
i
,
e
j
] = P
c
h
i
,
j
e
h
, o
1
≤
i
,
j
≤
n
.
Semisimple and simple Lie algeb as
A Lie algeb a
g
is semisimple i i does no con ain any
p ope abelian ideal. A simple Lie algeb a is a non-abelian
Lie algeb a wi h no non- i ial ideals.
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
P elimina ies
Lie algeb a
ALie algeb a
g
is a ec o space wi h a second bilinea
composi ion law (
[,]
) which sa ises:
[
X
,
X
] =
0
,
∀
X
∈g
and
[[
X
,
Y
],
Z
] + [[
Y
,
Z
],
X
] + [[
Z
,
X
],
Y
] =
0
,
∀
X
,
Y
,
Z
∈g
.
S uc u e cons an s
A basis
{
e
h
}
n
h
=
1
o
g
is cha ac e ized by i s s uc u e
cons an s:
[
e
i
,
e
j
] = P
c
h
i
,
j
e
h
, o
1
≤
i
,
j
≤
n
.
Semisimple and simple Lie algeb as
A Lie algeb a
g
is semisimple i i does no con ain any
p ope abelian ideal. A simple Lie algeb a is a non-abelian
Lie algeb a wi h no non- i ial ideals.
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
P elimina ies
Lie algeb a
ALie algeb a
g
is a ec o space wi h a second bilinea
composi ion law (
[,]
) which sa ises:
[
X
,
X
] =
0
,
∀
X
∈g
and
[[
X
,
Y
],
Z
] + [[
Y
,
Z
],
X
] + [[
Z
,
X
],
Y
] =
0
,
∀
X
,
Y
,
Z
∈g
.
S uc u e cons an s
A basis
{
e
h
}
n
h
=
1
o
g
is cha ac e ized by i s s uc u e
cons an s:
[
e
i
,
e
j
] = P
c
h
i
,
j
e
h
, o
1
≤
i
,
j
≤
n
.
Semisimple and simple Lie algeb as
A Lie algeb a
g
is semisimple i i does no con ain any
p ope abelian ideal. A simple Lie algeb a is a non-abelian
Lie algeb a wi h no non- i ial ideals.
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
P elimina ies
Uppe cen al se ies
The uppe cen al se ies o a Lie algeb a
g
is dened as
C
1
(g) = g,C
2
(g)=[g,g],C
3
(g)=[C
2
(g),C
2
(g)], . . .
,
C
k
(g) = [C
k
−
1
(g),C
k
−
1
(g)], . . .
Sol able Lie algeb a
I he e exis s
m
∈N
such ha
C
m
(g)≡ {
0
}
, he Lie algeb a
g
is sol able. A sol able Lie algeb a is
k
-s ep i
C
k
(g)6={
0
}
and
C
k
+
1
(g)≡ {
0
}
.
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
P elimina ies
Uppe cen al se ies
The uppe cen al se ies o a Lie algeb a
g
is dened as
C
1
(g) = g,C
2
(g)=[g,g],C
3
(g)=[C
2
(g),C
2
(g)], . . .
,
C
k
(g) = [C
k
−
1
(g),C
k
−
1
(g)], . . .
Sol able Lie algeb a
I he e exis s
m
∈N
such ha
C
m
(g)≡ {
0
}
, he Lie algeb a
g
is sol able. A sol able Lie algeb a is
k
-s ep i
C
k
(g)6={
0
}
and
C
k
+
1
(g)≡ {
0
}
.
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
P elimina ies
Lowe cen al se ies
The lowe cen al se ies o a Lie algeb a
g
is dened as:
C
1
(g) = g,C
2
(g)=[g,g],C
3
(g)=[C
2
(g),g], . . . ,
C
k
(g)=[C
k
−
1
(g),g], . . .
Nilpo en Lie algeb a
I he e exis s
m
∈N
such ha
C
m
(g)≡ {
0
}
, he Lie algeb a
g
is nilpo en . A nilpo en Lie algeb a is
k
-s ep i
C
k
(g)6={
0
}
and
C
k
+
1
(g)≡ {
0
}
.
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
P elimina ies
Lowe cen al se ies
The lowe cen al se ies o a Lie algeb a
g
is dened as:
C
1
(g) = g,C
2
(g)=[g,g],C
3
(g)=[C
2
(g),g], . . . ,
C
k
(g)=[C
k
−
1
(g),g], . . .
Nilpo en Lie algeb a
I he e exis s
m
∈N
such ha
C
m
(g)≡ {
0
}
, he Lie algeb a
g
is nilpo en . A nilpo en Lie algeb a is
k
-s ep i
C
k
(g)6={
0
}
and
C
k
+
1
(g)≡ {
0
}
.
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
Combina o ial s uc u es and Lie algeb as
Gi en wo e ices
i
<
j
, i
c
i
i
,
j
6=
0
o
c
j
i
,
j
6=
0
, hen a
di ec ed edge is d awn.
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
Combina o ial s uc u es and Lie algeb as
Gi en wo e ices
i
<
j
, i
c
i
i
,
j
6=
0
o
c
j
i
,
j
6=
0
, hen a
di ec ed edge is d awn.
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
Combina o ial s uc u es and Lie algeb as
Gi en wo e ices
i
<
j
, i
c
i
i
,
j
6=
0
o
c
j
i
,
j
6=
0
, hen a
di ec ed edge is d awn.
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
Combina o ial s uc u es and Lie algeb as
Going-in and going-ou e ex
A e ex
is said o be a going-in ( espec i ely going-ou )
e ex i all he di ec ed inciden edges wi h
a e o ien ed
owa ds
( espec i ely, om
).
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
Combina o ial s uc u es and Lie algeb as
Going-in and going-ou e ex
A e ex
is said o be a going-in ( espec i ely going-ou )
e ex i all he di ec ed inciden edges wi h
a e o ien ed
owa ds
( espec i ely, om
).
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
Combina o ial s uc u es and Lie algeb as
Co olla y
E e y Lie algeb a wi h a selec ed basis is associa ed wi h a
combina o ial s uc u e. This associa ion depends on he
selec ed basis.
Isola ed e ex
An isola ed e ex co esponds o a ec o om he cen e o
g
.
Comple e g aph
Acycle dig aph,
G
, is dened as a cycle g aph wi h di ec ed
edges.
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
Combina o ial s uc u es and Lie algeb as
Co olla y
E e y Lie algeb a wi h a selec ed basis is associa ed wi h a
combina o ial s uc u e. This associa ion depends on he
selec ed basis.
Isola ed e ex
An isola ed e ex co esponds o a ec o om he cen e o
g
.
Comple e g aph
Acycle dig aph,
G
, is dened as a cycle g aph wi h di ec ed
edges.
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
Combina o ial s uc u es and Lie algeb as
Co olla y
E e y Lie algeb a wi h a selec ed basis is associa ed wi h a
combina o ial s uc u e. This associa ion depends on he
selec ed basis.
Isola ed e ex
An isola ed e ex co esponds o a ec o om he cen e o
g
.
Comple e g aph
Acycle dig aph,
G
, is dened as a cycle g aph wi h di ec ed
edges.
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
Cycle Dig aphs and Lie Algeb as
Cycle Dig aphs
ACycle Dig aph is a cycle g aph wi h di ec ed edges. We
conside a well-o ien ed weigh ed cycle dig aph wi h double
edges be ween hei e ices.
Gi en a combina o ial s uc u e,
T
, o
n
e ices:
Label all he e ices by
1
,
2
,...,
n
, ollowing he posi i e
coun e clockwise.
The weigh o he edge
ij
will be deno ed by
c
i
,
j
.
Dene a ec o space
V
wi h basis
{
e
1
,...,
e
n
}
whe e
e
i
co esponds o he e ex
i
o
T
and b acke s
[
e
i
,
e
j
] =
c
i
i
,
j
e
i
+
c
j
i
,
j
e
j
.
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
Cycle Dig aphs and Lie Algeb as
Cycle Dig aphs
ACycle Dig aph is a cycle g aph wi h di ec ed edges. We
conside a well-o ien ed weigh ed cycle dig aph wi h double
edges be ween hei e ices.
Gi en a combina o ial s uc u e,
T
, o
n
e ices:
Label all he e ices by
1
,
2
,...,
n
, ollowing he posi i e
coun e clockwise.
The weigh o he edge
ij
will be deno ed by
c
i
,
j
.
Dene a ec o space
V
wi h basis
{
e
1
,...,
e
n
}
whe e
e
i
co esponds o he e ex
i
o
T
and b acke s
[
e
i
,
e
j
] =
c
i
i
,
j
e
i
+
c
j
i
,
j
e
j
.
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
Case
n
≥
4
We mus sol e he sys em o equa ions gi en by all he
Jacobi iden i ies. When imposing
J
(
e
i
,
e
j
,
e
k
) =
0
, he
ollowing equa ion is ob ained
c
j
j
,
k
c
i
i
,
j
+
c
k
j
,
k
c
i
i
,
k
=
0
,
c
k
i
,
k
c
j
j
,
k
−
c
i
i
,
k
c
j
i
,
j
=
0
,
c
i
i
,
j
c
k
i
,
k
+
c
j
i
,
j
c
k
j
,
k
=
0
.
When imposing all he Jacobi iden i ies and he es ic ions:
c
p
+
1
p
,
p
+
1
=
1
, o all
p
∈ {
1
,...,
n
}
and
c
1
1
,
n
=
1
, we ob ain he
law o a pa icula Lie algeb a, ha can be conside ed o e
Z
/
3
Z
.
[
e
p
,
e
q
] = −
e
p
+
e
q
[
e
p
,
e
n
] =
e
p
+
e
n
,
whe e
1
≤
p
≤
n
−
2
;
p
+
1
≤
q
≤
n
−
1
.
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
Case
n
≥
4
We mus sol e he sys em o equa ions gi en by all he
Jacobi iden i ies. When imposing
J
(
e
i
,
e
j
,
e
k
) =
0
, he
ollowing equa ion is ob ained
c
j
j
,
k
c
i
i
,
j
+
c
k
j
,
k
c
i
i
,
k
=
0
,
c
k
i
,
k
c
j
j
,
k
−
c
i
i
,
k
c
j
i
,
j
=
0
,
c
i
i
,
j
c
k
i
,
k
+
c
j
i
,
j
c
k
j
,
k
=
0
.
When imposing all he Jacobi iden i ies and he es ic ions:
c
p
+
1
p
,
p
+
1
=
1
, o all
p
∈ {
1
,...,
n
}
and
c
1
1
,
n
=
1
, we ob ain he
law o a pa icula Lie algeb a, ha can be conside ed o e
Z
/
3
Z
.
[
e
p
,
e
q
] = −
e
p
+
e
q
[
e
p
,
e
n
] =
e
p
+
e
n
,
whe e
1
≤
p
≤
n
−
2
;
p
+
1
≤
q
≤
n
−
1
.
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
In his way, we can es ablish he ollowing
P oposi ion
Le us conside a well-o ien ed, weigh ed cycle dig aph
G
wi h double edges o
3
e ices. Then,
G
is associa ed wi h a
3
-dimensional Lie algeb a i and only i he weigh s o i s
edges sa is y one o he ollowing cons ain s
(i)
c
1
1
,
3
=
1
,
c
2
1
,
2
=
1
,
c
3
2
,
3
=
1
,
c
2
2
,
3
=
1
,
c
3
1
,
3
=
1
and
c
1
1
,
2
=−
1
.
In his case, he Lie algeb a, deno ed by
g
, is pe ec .
(ii)
c
1
1
,
3
=
1
,
c
2
1
,
2
=
1
,
c
3
2
,
3
=
1
,
c
2
2
,
3
=−
1
,
c
3
1
,
3
=−
1
and
c
1
1
,
2
=
1
. In his case, he Lie algeb a, deno ed by
h
, is
2
-s ep sol able and non-nilpo en .
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
In his way, we can es ablish he ollowing
P oposi ion
Le us conside a well-o ien ed, weigh ed cycle dig aph
G
wi h double edges o
3
e ices. Then,
G
is associa ed wi h a
3
-dimensional Lie algeb a i and only i he weigh s o i s
edges sa is y one o he ollowing cons ain s
(i)
c
1
1
,
3
=
1
,
c
2
1
,
2
=
1
,
c
3
2
,
3
=
1
,
c
2
2
,
3
=
1
,
c
3
1
,
3
=
1
and
c
1
1
,
2
=−
1
.
In his case, he Lie algeb a, deno ed by
g
, is pe ec .
(ii)
c
1
1
,
3
=
1
,
c
2
1
,
2
=
1
,
c
3
2
,
3
=
1
,
c
2
2
,
3
=−
1
,
c
3
1
,
3
=−
1
and
c
1
1
,
2
=
1
. In his case, he Lie algeb a, deno ed by
h
, is
2
-s ep sol able and non-nilpo en .
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
P oposi ion
Le us conside a well-o ien ed, weigh ed cycle dig aph
G
wi h double edges o
n
≥
4
e ices. Then,
G
is associa ed
wi h an
n
-dimensional Lie algeb a i and only i he weigh s
o he edges sa is y
c
p
p
,
q
=−
1
,
c
q
p
,
q
=
1
,
c
p
p
,
n
=
c
n
p
,
n
=
1
,
whe e
1
≤
p
≤
n
−
2
;
p
+
1
≤
q
≤
n
−
1
.
The Lie algeb a associa ed wi h he dig aph is unique and is
deno ed by
g
n
. Mo eo e , he Lie algeb a
g
n
is
2
-s ep
sol able and non-nilpo en .
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
Nex , o he sake o example, we show he dig aph
associa ed wi h a
4
-dimensional Lie algeb a.
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
Implemen ing he algo i hm wi h Maple
We implemen he algo i hm by using he symbolic
compu a ion package MAPLE wi h a sub ou ine and a main
ou ine.
Sub ou ine
sub
>
sub :=p oc(n,i,j,k)
>
S:=c[j,k,j]*c[i,j,i]+c[j,k,k]*c[i,k,i]=0,
>
c[i,k,k]*c[j,k,j]-c[i,k,i]*c[i,j,j]=0,
>
c[i,j,i]*c[i,k,k]+c[i,j,j]*c[j,k,k]=0;
>
o q om 1 o n-1 do
>
S:=op(S),c[q,q+1,q+1]=1;end do;
>
S:=op(S),c[1,n,1]=1;
>
e u n S; end p oc:
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
Implemen ing he algo i hm wi h Maple
We implemen he algo i hm by using he symbolic
compu a ion package MAPLE wi h a sub ou ine and a main
ou ine.
Sub ou ine
sub
>
sub :=p oc(n,i,j,k)
>
S:=c[j,k,j]*c[i,j,i]+c[j,k,k]*c[i,k,i]=0,
>
c[i,k,k]*c[j,k,j]-c[i,k,i]*c[i,j,j]=0,
>
c[i,j,i]*c[i,k,k]+c[i,j,j]*c[j,k,k]=0;
>
o q om 1 o n-1 do
>
S:=op(S),c[q,q+1,q+1]=1;end do;
>
S:=op(S),c[1,n,1]=1;
>
e u n S; end p oc:
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
Implemen ing he algo i hm wi h Maple
Rou ine
main
>
main :=p oc(n)
>
local L,T;
>
L:=choose(n,3); T:=;
>
o p om 1 o nops(L) do
>
T:=op(T),op(sub (n,L[p][1],L[p][2],L[p][3]));
>
end do;
>
e u n sol e(T);
>
end p oc;
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
We ha e p o ed ha se e al dig aphs a e associa ed wi h
2
-s ep sol able non-nilpo en Lie algeb as unde some
es ic ions. E e y
2
-s ep sol able non-nilpo en Lie algeb a
is associa ed wi h a dig aph?
g
Lie b acke s Pa ame e s
2
[
e
1
,
e
2
] =
e
2
3
[
e
1
,
e
2
] =
e
2
,[
e
1
,
e
3
] =
e
2
+
e
3
3
,
p
[
e
1
,
e
2
] =
e
2
,[
e
1
,
e
3
] =
pe
3
p
∈C∗,|
p
≤
1
4
,
q
[
e
1
,
e
2
] =
e
2
,[
e
1
,
e
3
] =
e
3
,[
e
1
,
e
4
] =
qe
4
q
∈C∗
4
,α,β [
e
1
,
e
2
] =
e
3
,[
e
1
,
e
3
] =
e
4
,[
e
1
,
e
4
] = α
e
2
−β
e
3
+
e
4
α∈C∗
,
β∈C
o
α, β =
0
gα
4
,
11
[
e
1
,
e
2
] =
e
3
,[
e
1
,
e
3
] =
e
4
,[
e
1
,
e
4
] = α(
e
2
+
e
3
)α∈C∗
g
4
,
12
[
e
1
,
e
2
] =
e
3
,[
e
1
,
e
3
] =
e
4
,[
e
1
,
e
4
] =
e
2
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es
P elimina ies on Lie algeb as
Associa ing combina o ial s uc u es wi h Lie algeb as
Cycle Dig aphs and Lie Algeb as
Implemen ing he algo i hm wi h Maple
2-s ep sol able Lie algeb as and combina o ial s uc u es
Re e ences
S udy o Lie Algeb as by Using
Combina o ial S uc u es
Manuel Ceballos, Juan Núñez and Ángel F. Teno io
Uni e si y o Se ille and Pablo de Ola ide Uni e si y
24 - 28 May 2010
Manuel Ceballos, Juan Núñez and Ángel F. Teno io S udy o Lie Algeb as by Using Combina o ial S uc u es