Lie point symmetries of algebraic systems Lie point symmetry

1 lie point symmetries of algebraic systems

1.1 algebraic systems
1.2 definition of lie point symmetries
1.3 example

lie point symmetries of algebraic systems

let define algebraic systems used in forthcoming symmetry definition.

algebraic systems

let



f
=
(

f

1


,
…
,

f

k


)
=
(

p

1



/


q

1


,
…
,

p

k



/


q

k


)


{\displaystyle f=(f_{1},\dots ,f_{k})=(p_{1}/q_{1},\dots ,p_{k}/q_{k})}

finite set of rational functions on field




r



{\displaystyle \mathbb {r} }






p

i




{\displaystyle p_{i}}

,




q

i




{\displaystyle q_{i}}

polynomials in




r

[
z
]


{\displaystyle \mathbb {r} [z]}

i.e. in variables



z
=
(

z

1


,
…
,

z

n


)


{\displaystyle z=(z_{1},\dots ,z_{n})}

coefficients in




r



{\displaystyle \mathbb {r} }

. algebraic system associated



f


{\displaystyle f}

defined following equalities , inequalities:

{





p

1


(
z
)
=
0
,




⋮





p

k


(
z
)
=
0









and




{





q

1


(
z
)
≠
0
,




⋮





q

k


(
z
)
≠
0.












{\displaystyle {\begin{array}{ccc}\left\{{\begin{array}{l}p_{1}(z)=0,\\\vdots \\p_{k}(z)=0\end{array}}\right.&{\mbox{and}}&\left\{{\begin{array}{l}q_{1}(z)\neq 0,\\\vdots \\q_{k}(z)\neq 0.\end{array}}\right.\end{array}}}

an algebraic system defined



f
=
(

f

1


,
…
,

f

k


)


{\displaystyle f=(f_{1},\dots ,f_{k})}

regular (a.k.a. smooth) if system



f


{\displaystyle f}

of maximal rank



k


{\displaystyle k}

, meaning jacobian matrix



(
∂

f

i



/

∂

z

j


)


{\displaystyle (\partial f_{i}/\partial z_{j})}

of rank



k


{\displaystyle k}

@ every solution



z


{\displaystyle z}

of associated semi-algebraic variety.

definition of lie point symmetries

the following theorem (see th. 2.8 in ch.2 of ) gives necessary , sufficient conditions local lie group



g


{\displaystyle g}

symmetry group of algebraic system.

theorem. let



g


{\displaystyle g}

connected local lie group of continuous dynamical system acting in n-dimensional space





r


n




{\displaystyle \mathbb {r} ^{n}}

. let



f
:


r


n


→


r


k




{\displaystyle f:\mathbb {r} ^{n}\rightarrow \mathbb {r} ^{k}}





k
≤
n


{\displaystyle k\leq n}

define regular system of algebraic equations:

f

i


(
z
)
=
0

∀
i
∈
{
1
,
…
,
k
}
.


{\displaystyle f_{i}(z)=0\quad \forall i\in \{1,\dots ,k\}.}

then



g


{\displaystyle g}

symmetry group of algebraic system if, , if,

δ

f

i


(
z
)
=
0

∀
i
∈
{
1
,
…
,
k
}

 whenever 


f

1


(
z
)
=
⋯
=

f

k


(
z
)
=
0


{\displaystyle \delta f_{i}(z)=0\quad \forall i\in \{1,\dots ,k\}{\mbox{ whenever }}f_{1}(z)=\dots =f_{k}(z)=0}

for every infinitesimal generator



δ


{\displaystyle \delta }

in lie algebra





g




{\displaystyle {\mathfrak {g}}}

of



g


{\displaystyle g}

.

example

let consider algebraic system defined on space of 6 variables, namely



z
=
(
p
,
q
,
a
,
b
,
c
,
l
)


{\displaystyle z=(p,q,a,b,c,l)}

with:

{





f

1


(
z
)
=
(
1
−
c
p
)
+
b
q
+
1
,





f

2


(
z
)
=
a
p
−
l
q
+
1.








{\displaystyle \left\{{\begin{array}{l}f_{1}(z)=(1-cp)+bq+1,\\f_{2}(z)=ap-lq+1.\end{array}}\right.}

the infinitesimal generator

δ
=
a
(
a
−
1
)



∂

∂
a




+
(
l
+
b
)



∂

∂
b




+
(
2
a
c
−
c
)



∂

∂
c




+
(
−
a
p
+
p
)



∂

∂
p






{\displaystyle \delta =a(a-1){\dfrac {\partial }{\partial a}}+(l+b){\dfrac {\partial }{\partial b}}+(2ac-c){\dfrac {\partial }{\partial c}}+(-ap+p){\dfrac {\partial }{\partial p}}}

is associated 1 of one-parameter symmetry groups. acts on 4 variables, namely



a
,
b
,
c


{\displaystyle a,b,c}

,



p


{\displaystyle p}

. 1 can verify



δ

f

1


=

f

1


−

f

2




{\displaystyle \delta f_{1}=f_{1}-f_{2}}

,



δ

f

2


=
0


{\displaystyle \delta f_{2}=0}

. relations



δ

f

1


=
δ

f

2


=
0


{\displaystyle \delta f_{1}=\delta f_{2}=0}

satisfied



z


{\displaystyle z}

in





r


6




{\displaystyle \mathbb {r} ^{6}}

vanishes algebraic system.