Differential Invariants of SL(2) and SL(3)-ACTIONS on R^2

0.13 MB·English

by  Mehdi Nadjafikhah

Mathematical Sciences
Vol. 1, No. 1,2 (2007) 01-12
Differential Invariants of SL(2) and SL(3)-actions
8
0
0 on R2 Differential Invariants of
2
n SL(2) and SL(3)-actions on R2
a
J Mehdi Nadjafikhaha,1, Seyed Reza Hejazib

6
2
aFact. ofMath.,Dept. ofPureMath.,IranUniversityofScienceandTechnology,Narmak,Tehran,I.R.Iran.

]
G bSameaddress.
D
Abstract
.
h
at The main purpose of this paper is calculation of differential invariants which
m
arise from prolonged actions of two Lie groups SL(2) and SL(3) on the nth jet
[
space of R2. It is necessary to calculate nth prolonged infinitesimal generators of

2
v the action.
1
1 Keywords: Differentialinvariant,infinitesimalgenerator,genericorbit,prolonga-
5
tion.
0
.
1 (cid:13)c 200x Published by Islamic AZAD University-KarajBranch.
1
7
0
: Introduction
v
i
X
Differential invariants theory is one of the most important concept in differential equa-
r
a
tions theory and differential geometry. The study of this theory will help us to analyze
applications of geometry in differential equations. In this paper we will study some
properties of two Lie group actions SL(2) and SL(3). After finding the Lie algebras of
each groups, we will calculate all infinitesimal generators and their nth prolongations.
Next wearegoing to findthenumberof functionally independentdifferential invariants
up to order n (i ), the number of strictly functionally independent differential invari-
n
ants of order n (j ), generic orbit dimensions of nth prolonged action (s ) and isotropic
n n
1Corresponding Author. E-mail Address: m nadjafi[email protected]
2 Mathematical Sciences Vol. x, No. x (200x)
group dimensions of prolonged action (h ). It is necessary to realize that the symbol
n
u means kth derivative of u with respect to x.
(k)
1 Jet and Prolongation
This section is devoted to study of proper geometric context for ”jet spaces” or ”jet
bundles”,well knownto 19th century practitioners, butfirstformally definedby Ehres-
mann. Prolongation will defined after study of jet to find the differential invariants.
1.1 Introduction
In this part we are going to define some fundamental concepts of differential equations
theory. Thetwo most important are Jet and Prolongation butit needs to analyze some
definitions of differential equations.
Definition 1.1 Suppose M and N are p-dimensional and q-dimensional
Euclidean manifolds respectively, i.e. M ≃ Rp and N ≃ Rq. The total space will
be the Euclidean space E = M × N ≃ Rp+q coordinatized by the independent and
dependent variables (x1,...,xp,u1,...,uq).
Let f is a scalar-valued function with p independent and q dependent variables then
it has p = p+k−1 different kth order partial derivatives and suppose the function
k k
characterized(cid:0)by u=(cid:1) f(x).
Definition 1.2 Let N(n) is an appropriate space for representation of the nth Tailor
expansion of f, thus N(n) has the following decomposition
N(n) ≃ Rq ×Rp1q ×Rp2q ×···×Rpnq,
then N(n) is a q p+n dimensional manifold. Define
n
(cid:0) (cid:1)
J(n) = J(n)E = M ×N(n),
Author1 et al. 3
then J(n) is called the nth jet space of the total space E, which is a p + q p+n di-
n
mensional Euclidean manifold with vector bundle structure whose fibers hav(cid:0)e q (cid:1)p+n
n
dimension. (cid:0) (cid:1)
According to the last definition, the nth jet space of R2 has dimension n+2 since
we can write it in the form of R2 = R×R, (M ≃R and N ≃ R).
Theorem 1.3 Suppose G be a Lie group acting on E and v is a member of Lie
algebra. Let Φ : G → E be a smooth map defined byΦ (g) = g.x, then the infinitesimal
x x
generator v corresponding v is given by
e
Φx∗(v |g) = v |g.x,
e
where Φx∗ is the push forward map of Φx.
See [2] for a proof.
LetGbeaLiegroupacting onthetotal spaceE, thenallof infinitesimalgenerators
of the group action have the following form
p q
∂ ∂
v = ξi(x,u) + ϕα(x,u) .
∂xi ∂uα
i=1 α=1
X X
Definition 1.4 Nth prolonged of v denoted by v(n) is a vector field on the J(n)E,
which is an infinitesimal generator of nth prolonged action of G on E.
Nth prolonged action of f represented by u(n) = f(n)(x), is a function from M
to N(n) (also known as the n−jet and denoted by j f), defined by evaluating all the
n
partial derivatives of f up to order n.
Now it’s time to give a formula to calculate nth prolongations. The following theorem
gives an explicit formula for the prolonged vector field.
First of all we need two important definition.
4 Mathematical Sciences Vol. x, No. x (200x)
Definition 1.5 The characteristic of a vector field v on E is a q-tuple of functions
Q(x,u(1)), depending on x and u and first order derivatives of u, defined by
p ∂uα
Qα(x,u(1))= ϕα(x,u(1))− ξi(x,u) , α = 1,...q.
∂xi
i=1
X
Definition 1.6 Let F(x,u(n)) be a differential function of order n. (A smooth, real
valued function F : J(n) → R , defined on an open subset of the nth jet space is called
a differential function of order n. ) The total derivative F with respect to xi is the
(n+1)st order differential function D F satisfying
i
∂
D [F(x,f(n+1)(x))] = F(x,f(n)(x)).
i ∂xi
Theorem 1.7 Let v be an infinitesimal generator on E, and let Q =(Q1,...,Qq) be
its characteristic. The nth prolongation of v is given explicitly by
p q n
∂ ∂
v(n) = ξi(x,u) + ϕJ(x,u(j)) ,
∂xi α ∂uα
i=1 α=1♯J=j=0 J
X X X
with coefficients
p
ϕα = D Qα+ ξiuα ,
J J J,i
i=1
X
where J = (j ,...,j ) is a multi-indices where 1 ≤ j ≤ p and ♯J = k.
1 k k
See [3; Theorem 2.36] for a proof.
Theorem 1.8 (Infinitesimal Method, [2])A function I : J(n) → R is a
differential invariant for a connected transformation group G if and only if it is
annihilated by all prolonged infinitesimal generators:
v(n)(I(x,u(n))) = 0, v ∈ Lie(G) .
Here x= (x1,...,xp) is independent variable and u(n) is a coordinate chart on N(n).
Author1 et al. 5
Definition 1.9 Thecollection {I ,...,I }ofarbitrary differential invariantsiscalled
1 k
functionally independent if
dI ∧...∧dI 6= 0,
1 k
and strictly functionally independent if
dx∧du(n−1) ∧dI ∧...∧dI 6= 0 .
1 k
Now we are ready to give some formulas for calculating i , j , s and h .
n n n n
According to the prolonged action of G on J(n) define V(n) denoted a subset of J(n)
which consists of all points contained in theorbitof maximal dimension(generic orbit).
Then on V(n)
i = dimJ(n) −s = dimJ(n)−dimG+h ,
n n n
thus the number of independentdifferential invariants of order less or equal to n, forms
a nondecreasing sequence
i ≤ i ≤ i ≤ ···.
0 1 2
The difference
jn = in −in−1,
is the number of strictly functionally independent differential invariants of order n.
Note that j cannot exceed the number of independent derivative coordinate of
n
order n, so if
p+n p−n−1 p+n−1
q = dimJ(n)−dimJ(n−1) = q −q = q ,
n
n ! n−1 ! n !
is the number of derivative coordinate of order n, so j ≤ q , which implies that the
n n
elementary inequalities
in−1 ≤ in ≤ in−1+qn .
The maximal orbit dimension s is also a nondecreasing function of n, bounded by
n
r, the dimension of G itself:
s ≤ s ≤ s ≤ ··· ≤ r .
0 1 2
6 Mathematical Sciences Vol. x, No. x (200x)
On the other hand, since the orbit cannot increase in dimension any more than the
increase in dimension of the jet spaces themselves, we have the elementary inequalities
sn−1 ≤ sn ≤ sn−1+qn ,
governing the orbit dimension.
2 ACTION OF LIE GROUP SL(2) AND SL(3)
In this section we are going to define an action of each Lie group on R2, then we will
give the infinitesimal generators arise from Lie algebra and their prolongations, next
we will calculate i ,j ,···.
n n
For more details see [2], [3] and [6].
2.1 Action of SL(2)
Define an action of SL(2) on R2,
ax+b a b
(x,u) 7−→ ,u where ∈ SL(2) and (x,u) ∈ R2 .
 
cx+d
c d
(cid:16) (cid:17)
 
 
The Lie algebra of SL(2) generates by the following matrices:
0 1 1 0 0 0
A = A = A = ,
1   2   3  
0 0 0 −1 1 0
     
     
and the infinitesimal generators corresponding to the above matrices are:
∂ ∂ ∂
v = v = x v = x2 .
1 2 3
∂x ∂x ∂x
According to thetheorem 1.1.7, thethree following vector fieldsare thecorrespond-
ing prolonged infinitesimal generators:
∂
(n)
v =
1 ∂x
Author1 et al. 7
∂ ∂
(n)
v = x −···−nu
2 ∂x (n)∂u
(n)
∂ ∂
v3(n) = x2∂x −···− n(n−1)u(n−1) +2nxu(n) ∂u .
(n)
h i
According to the Theorem 1.1.8, the first three differential invariants of order 0,1,2
are functions of u, thus i = i = i = 1, moreover the third one depends both on u
0 1 2
and
2u u −3u2
(1) (3) (1)
,
2u4
(1)
thus i = 2, so the calculations show that i = 3,···,i = n − 1 and also
3 4 n
s = 1,s = 2 and s = s = ··· = s = 3, but j = j = 0 and j = j = ··· = j = 1,
0 1 2 3 n 1 2 0 3 n
so h = 2,h = 1 and h = ··· = h = 0.
0 1 2 n
2.2 Action of SL(3)
Similarly, we define an action of SL(3) on R2,
ax+bu+c dx+eu+f
(x,u) 7−→ ,
hx+ju+k hx+ju+k
(cid:16) (cid:17)
a b c
where  d e f  ∈ SL(3) and (x,u) ∈ R2. The Lie algebra of SL(3) generates by
 
 h j k 
 
 
the following matrices:
0 0 1 0 0 0
   
A1 = 0 0 0 A2 = 0 0 1
   
 0 0 0   0 0 0 
   
   
   
1 0 0 0 0 0
   
A3 = 0 −1 0 A4 = 0 1 0
   
 0 0 0   0 0 −1 
   
   
   
8 Mathematical Sciences Vol. x, No. x (200x)
0 0 0 0 1 0
   
A5 = 1 0 0 A6 = 0 0 0
   
 0 0 0   0 0 0 
   
   
   
0 0 0 0 0 0
   
A7 = 0 0 0 A8 = 0 0 0 ,
   
 1 0 0   0 1 0 
   
   
   
therefore the infinitesimal generators are:
∂
v =
1
∂x
∂
v =
2
∂u
∂
v = x
3
∂x
∂
v = u
4
∂u
∂
v = x
5
∂u
∂
v = u
6
∂x
∂ ∂
v = x2 +xu
7
∂x ∂u
∂ ∂
v = xu +u2 ,
8
∂x ∂u
and by calculating the prolonged infinitesimal generators, we have,
∂
(n)
v =
1 ∂x
∂
(n)
v =
2 ∂u
∂ ∂
(n)
v = x −···−nu
3 ∂x ∂u
(n)
∂ ∂
(n)
v = u +···+u
4 ∂u (n)∂u
(n)
∂ ∂
(n)
v = x +
5 ∂u ∂u
(1)
Author1 et al. 9
∂ ∂
v(1) = u −u2
6 ∂x (1)∂u
(1)
(n−1)/2
∂ n+1
(n)
v6 = u∂x −···−( i !u(i)u(n−i+1)
i=2
X
1 n+1 2 ∂ ∂
+ u +(n+1)u (n,odd)
2 (n+1)/2! ((n+1)/2) (n)∂u(1))∂u(n)
h i
n/2
∂ n+1
(n)
v6 = u∂x −···− i !u(i)u(n+i−1)
hXi=2
∂
+(n+1)u u (n,even)
(n) (1)
∂u
(n)
i
∂ ∂ ∂
v(n) = x2 +xu −···− n(n−2)+(2n−1)xu
7 ∂x ∂u (n) ∂u
(n)
h i
∂ ∂ ∂
v(1) = xu +u2 −(xu −u)u
8 ∂x ∂u (1) (1)∂u
(1)
∂ ∂ ∂ ∂
v(2) = xu +u2 −(xu −u)u −3xu u
8 ∂x ∂x (1) (1)∂u (1) (2)∂u
(1) (2)
∂ ∂
v(n) = xu +u2 −···
8 ∂x ∂u
n
− (n−2)u((n−1)/2) +u((n+1)/2) u((n+1)/2)
( (n+1)/2!
h i
n
n n+1
+ " j −1!(n−2)u(j−1) + j !xu(j)#u(i))
i=1+X(n+1)/2
n−1
(j = 1,..., and n,odd)
2
∂ ∂ n n
v(n) = xu +u2 −···− ( ) [u ]2
8 ∂x ∂u ( 2 n/2! (n/2)
n
n+1 n ∂
+ xu(j)+ (n−2)u(j−1) u(i)
" j ! j−1! # )∂u(n)
i=1X+n/2
n
(j = 1,..., and n,even),
2
corresponding to minimum value of i in v (n), j takes its maximum value, i.e. when i
8
increases j decreases.
According to the Theorem 1.1.8 with a tedious calculation we have, i = i = i =
0 1 2
i = i = i = i = 0 and i = 1,i = 2,···,i = n−6, consequently j = j = j =
3 4 5 6 7 8 n 0 1 2
10 Mathematical Sciences Vol. x, No. x (200x)
j = j = j = j = 0 and j = ···j = 1, but s = 2,s = 3,s = 4,s = 5,s =
3 4 5 6 7 n 0 1 2 3 4
6,s = 7, thus h = 6,h = 5,h = 4,h = 3,h = 2,h = 1, and h = ···h = 0. The
5 0 1 2 3 4 5 6 n
differential invariants have so complicated forms in this action. For example the first
nonconstant differential invariant (the invariant of 7th prolonged action)is a function
of
1
11200u8
18(40u2 +9u2 u −45u u u ) (3)
(3) (2) (5) (2) (3) (4) (cid:16)
−33600u u6 u +6720u2 u5 u
(2) (3) (4) (2) (3) (5)
+31500u2 u4 u2 −12600u3 u3 u u
(2) (3) (4) (2) (3) (4) (5)
+720u4 u3 u −756u4 u2 u2 +···
(2) (3) (7) (2) (3) (5)
−2835u5 u u2 −189u6 u2 ,
(2) (4) (5) (2) (6)
(cid:17)
and the second differential invariant arise from 8th prolonged action depends on the
above phrase an the following one
1
(20412u3 +6561u2 u −26244u u u )u9
207360000 (6) (5) (8) (5) (6) (7) (2)
(cid:16)
+(((131220u u −65610u u )u +104976u2 u
(6) (7) (5) (8) (4) (5) (7)
.
.
.
more than 10 lines calculation
.
.
.
−201600000u u10u +4480000u12 )/(u3
(2) (3) (4) (3) (3)
9 9
u2 u − u u u )4 .
40 (2) (5) 8 (2) (3) (4)
(cid:17)
3 MORE DETAILS
In this section we will give some details where in special cases we can classify the
differential invariants.