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Hurwitz’s matrices, Cayley transformation and theCartan-Weyl basis for the orthogonal groups
Mehdi Hage-Hassan
To cite this version:Mehdi Hage-Hassan. Hurwitz’s matrices, Cayley transformation and the Cartan-Weyl basis for theorthogonal groups. 2006. �hal-00105003�
1
Hurwitz’s matrices, Cayley transformation
and the Cartan-Weyl basis for the orthogonal groups
M. Hage Hassan
Université Libanaise, Faculté des Sciences Section (1)
Hadath-Beyrouth
Abstract
We find the transformations from the basis of the hydrogen atom of n-dimensions to the
basis of the harmonic oscillator of N=2(n-1) dimensions using the Cayley transformation
and the Hurwitz matrices. We prove that the eigenfunctions of the Laplacian n are also
eigenfunctions of the Laplacien N for n=1, 3, 5 and 9. A new parameterization of the
transformation 58 RR is derived.
This research leads us first to a new class of spherical functions of the classical groups we
call it the bispherical harmonic functions. Secondly: the development of Hurwitz’s matrix
in terms of adjoint representation of the Cartan-Weyl basis for the orthogonal groups
SO(n) leads to what we call the generating matrices of the Cartan-Weyl basis and then we
establish it for )2,1(2 mn m .
Résumé
Nous trouvons les transformations de la base de l'atome d'hydrogène de dimension n à
la base de l'oscillateur harmonique de dimension N=2 (n-1) en utilisant la transformation
de Cayley et les matrices de Hurwitz . Nous montrons que les fonctions propres du
Laplacian n sont également fonctions propres du Laplacien N pour n=1,3, 5 et 9.
Une nouvelle paramétrisation de la transformation est dérivée. Cette recherche nous
mène d'abord à une nouvelle classe des fonctions sphériques des groupes classiques que
nous l'appelons les fonctions harmoniques bisphériques. Deuxièmement : le
développement de la matrice de Hurwitz en termes de la représentation adjointe de la
base de Cartan-Weyl pour les groupes orthogonaux SO(n) mène à ce que nous appelons
les matrices génératrices de la base de Cartan-Weyl et nous l'établissons
pour )2,1(2 mn m .
1. Introduction
The link between the Kepler problem and the oscillation is introduced by Binet in
classical mechanic. In Quantum mechanic the link between the N=2(n-1) dimensional
harmonic oscillator and the n-dimensional hydrogen atom was known to the physists
since Schrödinger. The transformation (K-S) introduced by Kustaanheimo-Steifel [1] in
celestial mechanics was used by many authors [2-3] for the connection of 3R hydrogen
atom and 4R harmonic oscillator. After that many papers [4-5] were devoted to the
generalization of this transformation and the conformal transformation of Levi-Civita [6],
using Hurwitz’s matrix, which was introduced for the solution of the problem: the
2
sum of squares.
These transformations continue to be interesting for their potential relevance to physics.
However, its relation with the orthogonal groups was not emphasized, in spite of the
well-known work of Weyl [7] on the transformation of Cayley and the group SO(3).
I have not found any extension of this work; it seems to me natural to investigate in this
direction. I noticed that the multiplication of the orthogonal matrix n
n
nSI
SIO
by a reel
number 0r , choosing the anti-symmetric matrix nrS , as the principal minor of Hurwitz
matrix, leads to components of the last row or the last column function of
N=2(n-1) parameters. But we know [8, 9] that last row or the last column of the matrix of
the orthogonal groups are the spherical coordinates on the unitary sphere 1nS therefore
the number of parameters is n -1. We deduce consequently that there is a transformation,
the Hurwitz transformation, which corresponds well to our criterion. We obtain these
transformations by two methods: the first one is a direct calculus using a symbolic
computer program the second is an analytic method for the calculation of the left column
and right row.
One of the goals of these transformations is to find the harmonic spherical functions
eigenfunction of the Laplacian n which are also eigenfunction of the Laplacian N . In
particular cases we prove a generalization of these harmonic spherical functions not just
for n=1, 3 and 5 but also for n=9. We derive then a new parameterization of the
transformation 58 RR using the representation matrix of the group SO (4). We will
prove the existence of the generalizations of the spherical functions left and right [8] for
the classical groups .we call these functions the bispherical harmonic functions.
Considering the importance of these functions in mathematical physics we will study it in
another paper.
The adjoint representations of the orthogonal groups are anti-symmetric and the
number of elements is n (n-1)/2. The matrix nS is anti-symmetric and function of (n-1)
parameters, {u}, and develops in terms of the adjoint representation of SO (n) [10]. To
generate the Cartan-Weyl basis we need consequently n/2 matrices, this number is in
agreement with the number of the simple roots of the orthogonal groups [10].By analogy
with the generating functions we call these matrices by the generating matrices of the
Cartan-Weyl basis and we build it for the cases ).,2,1(2 mn m .
The first part is devoted for the derivation of Hurwitz transformations. The
bispherical harmonic functions and the parameterization of the transformation 58 RR are the subject of the second part. The introduction of the generating matrices of Cartan-
Weyl basis will be treated in third part. The appendix is reserved to the tables of the
generators of SO (5). We emphasize that this work is the continuation of our preceding
paper [11].
2. Hurwitz’s matrix and the Cayley transformation.
For the derivation of the orthogonal matrices we take the miners of Hurwitz matrix and
we use a computer symbolic program. Then we expose in analytic method for the
calculation of the last row and column of these matrices.
3
2.1 The Hurwitz matrix
Let us consider one of the orthogonal matrices [12] of the 8x8 real matrices
)(8 ijhH :
12345678
21436587
34127856
43218765
56781234
65872143
78563412
87654321
8
uuuuuuuu
uuuuuuuu
uuuuuuuu
uuuuuuuu
uuuuuuuu
uuuuuuuu
uuuuuuuu
uuuuuuuu
H (1)
Where iu (i =1, 1, 8) are real numbers. In order to generate Cayley transformation
we assign nH to the formed matrix of first n line and n column, and nS is the same
matrix with .01 u It can be shown that the matrix 8H can be developed as a linear
combination of Clifford matrices. Indeed, we have
8
218 i
t
iiuIuH (2)
Where I the identity matrix, t is the transpose and the matrices i satisfy
)8,,2,(,2 jiIijji (3)
2.2 Cayley Transformation and Hurwitz’s matrices
The Cayley transformation for the orthogonal groups nO is:
n
n
nSI
SIO
(4)
nS Is a skew symmetric matrix of order n.
In order to obtain nO in terms of the variables {u}, we multiply the numerator
and denominator by 1u and nO by nr .
N
i in uur1
22
To simplify the notation we replace nn SbySu1 in the expression of )(uOn we obtain
n
n
nSIu
SIuuuO
1
12)(
(5)
2.2.1 Transformation 22 RR
For n=2 we have
0
u0
2
2
2u
S
4
And simple calculation gives
2
2
2
121
21
2
2
2
1
2
22
21
uuuu
uuuu
rO (6)
2.2.2 Transformation 34 RR
Using a computer symbolic program we find the Weyl’s expression
2
4
2
3
2
2
2
132414213
3241
2
4
2
3
2
2
2
13412
42134312
2
4
2
3
2
2
2
1
2
33133
)(2)(2
)(2)(2
)(2)(2
)22()(
uuuuuuuuuuuu
uuuuuuuuuuuu
uuuuuuuuuuuu
SSuIruO
(7)
In the space of 4-dimensions we derive also the expression
2
3
33
13
33
13
33
0
0)()()(
u
uO
uV
VH
uV
VH ttttt
(8)
With )( 2343 uuuV and )000(03 .
2.2.3 Transformation 78 RR
We also find, using the symbolic program, an analogue expression as above-mentioned:
)22(1
)( 2
7715
5
7 SSuIrr
O (9)
And in the 8-dimensions space we derive the expression
2
7
77
17
77
17
77
0
0)()()(
u
uO
uV
VH
uV
VH ttttt
(10)
With )( 23456787 uuuuuuuV And )0000000(07 .
2.3 Calculation of the last column and row of58 RR and
916 RR We give the method for the calculation of the last column and row using the system
of equations
)SI(uu(u))OSI(u n1
2
nn1 (11)
)())(( 1
2
1 nnn SIuuSIuuO (12)
Put )()( ijn xuO for n= 3, 5, 9, and
1
0
22
1
22
2 ,n
i i
N
ni i uu (13)
We can say left and right with respect to the diagonal of the matrix On instead of line and
row.
2.3.1 Expression of the last column
After doing the identification of the two sides of the first (n-1) equations of the
5
(11) System and replacing nnx by 2
1
2
2 in last raw we obtain:
)1(,
1,
),1(
,1
)1( 2
nn
n
nn
n
n
h
h
x
x
H
The matrix )1( nH is orthogonal therefore we find:
)1(,
1,
)1(
),1(
,1
1
1
2
nn
n
t
n
nn
n
R
n
R
h
h
H
x
x
x
x
(14)
2.3.2 Expression of the last column
After doing the identification of the two sides of the first (n-1) equations of the
(12) System and replacing nnx by 2
1
2
2 in last raw we obtain:
nn
n
nn
n
t
n
h
h
x
x
H
),1(
,1
)1(,
1,
)1( 2
The matrix 1)(nH is orthogonal therefore we deduce:
nn
n
t
n
nn
n
L
n
L
h
h
H
x
x
x
x
),1(
,1
)1(
)1(,
1,
1
1
2 (15)
3. The bispherical harmonic functions
We prove that the Laplacien Nu ,
generate the harmonic functions left and right, and
then we treat the parameterization of the transformation 58 RR . We show also the
existence of bispherical functions of the classical groups.
3.1 The Laplacien of bispherical functions for SO(n), n=2, 3, 5 and 9.
We want to prove that the Laplacien Nu ,
generate the harmonic functions left and
right or simply the bispherical functions not simply for n=3, 5 as well known [4] but also
for the case n=9.
We have R
n
R
i
R
nn
l R
l
R
i
R
l
i
R
x
xf
u
x
x
xf
u
x
u
xf
1
1
0
)()()(
With the help of the relations
niifu
niifu
u
xandnl
u
x
i
i
i
R
n
i
R
l
2
2{)(,0 1
2
2
We can write
6
R
n
R
i
R
n
R
n
R
i
R
n
R
n
R
l
R
i
R
nn
lk
n
li
R
l
R
k
R
l
R
i
R
l
i
R
k
i
R
x
xf
u
x
x
xf
u
x
xx
xf
u
x
u
x
xx
xf
u
x
u
x
u
xf
1
21
1
2
1
2
1
21
0,
2
2
2
)()(
)(
)()(
)(
)()(
)()(
From 2
1
2
2 R
nx
We deduce that ,00 2
1
2
N
ii
R
n
u
xand 2N
0i
2
i
R
1n u4)u
x(
R
nx 1 Is a homogenous function in terms of iu we derive that
022
222
1
11
1
R
l
R
l
N
nii
R
li
n
ii
R
li
N
ii
R
n
i
R
l
xx
u
xu
u
xu
u
x
u
x
The matrix nH is orthogonal then we deduce
lk
n
nii
R
l
i
R
k
lk
n
ii
R
l
i
R
k
u
x
u
x
u
x
u
x
,
2
2
2
1
,
2
11
4
4
We find finally that
)(,4)()(, 2 R
nx
R
Nu xuxfu R (19)
We can also prove )(,4)()(, 2 L
nx
L
Nu xuxfu L (20)
We conclude that the solutions of the Laplacian )(, uNu in the particular cases where
N=2, 4, 8 and 16 are the generalization of the elements of the matrix representation of the
group SU(2), or the bispherical harmonic function of this group. Moreover the research
of the solutions of these Laplacians imposes a suitable parameterization of the variables
{u}, what we will do in the following paragraph.
3.2. Parameterization of the transformation58 RR
It is well known that the transformation of Cayley-Klein [3]
)sin(2
cos),cos(2
cos
),sin(2
sin),cos(2
sin
43
21
ruru
ruru
(21)
Is the parameterization of the application 34 RR with ψ, θ and φ are Euler’s angles.
This parameterization is due to the fact that the measure of integration on 3S must also
be valid on 2S .
The transformations of Hurwitz are quadratic transformations and noninvertible what
causes the difficulty for the parameterization. Knowing that the last column in the matrix
7
of rotation of SO(5) is the component of the unit vectors of 5R we can overcome this
difficulty, thus from the expression (11) of [11] we find that
34
43
12
212
vv
vv
vv
vv
xy
yx
With
)()(
,
,
,,
443322115
874653
432211
4321
vvvvvvvvx
iuuviuuv
iuuviuuv
ixxyixxx
L
LLLL
(22)
We obtain .
),(2),(2
2
4
2
3
2
2
2
15
324143423121
L
LLLL
x
ixxixx
If we put
.2
cos,2
cos
,2
sin,2
sin
4433
2211
wrwr
wrwr
We deduce
cos
),(sin),(sin
5
324143423121
rx
wwwwrixxwwwwrixx
L
LLLL
(23)
For the determination of }{ iw we must find a transformation of which the number of
parameters is 6. We will consider for that the representation matrix of SO (4) [9, 13]:
34
43
12
21
34
43
34
43
5
5
5
5
12
21
12
21
4
00
00
00
00
00
00
00
00
000
000
000
000
00
00
00
00
ww
ww
ww
ww
tt
tt
tt
tt
t
t
t
t
tt
tt
tt
tt
R
34
43
5
5
12
21
34
43
34
43
5
5
12
21
12
21
0
0
,0
0
tt
tt
t
t
tt
tt
ww
ww
tt
tt
t
t
tt
tt
ww
ww
(24)
8
.0,
)2
'sin(),
2
'cos(
)2
sin(),2
cos(
25
)''(2
4
)''(2
3
)(2
2
)(2
1
et
etet
etet
ii
ii
We have for the left side
31424132
32414231
34
43
12
21
wwwwwwww
wwwwwwww
ww
ww
ww
ww (25)
And for the right side [5]
31424132
32414231
34
43
12
21
wwwwwwww
wwwwwwww
ww
ww
ww
ww
Introduce (23) in the expression (24) we find
'cossincos4231 iwwww 'sin'sinsin'cos'sinsin3241 iwwww (26)
Finally we find the expression of last row:
cos
'cos'sinsinsin,'sin'sinsinsin
'cossinsin,cossin
5
43
21
rx
rxrx
rxrx
L
LL
LL
Therefore we deduce from (23) and (24) the parameterization
),)(2
sin(),)(2
sin(
),)(2
cos(),)(2
cos(
232
241874
242
231653
232
241432
242
231211
iiii
iiii
ettettriuuzettettriuuz
ettettriuuzettettriuuz
3.3 Existence of the bispherical harmonic functions
For the clearness of the presentation we consider first the case of rotation and we adopt
the general notations used in group theory [10, 13] for the classical groups what makes
generalization apparent. It’s well known from the theory of angular momentum [14] that
there are two spherical harmonics the left and the right:
)(12
4
)0(
][)(
)(
][)(
12
4)1(
)(
][)(
)0(
][ 2
1
lmlm
m Yl
lR
m
landY
lm
lR
l
9
The product may be written as
)0(
][)(
)0(
][
)0(
][)(
)(
][
)(
][)(
)0(
][
1
21
12
2
21
2
2
2
2
1
1
1
1
m
llR
m
ll
lR
m
l
m
lR
l
(16)
With 12R is the product rotation 21RR .
The coupling of angular momentum of )0(
][
1
21
m
llis given by
)(
])[(}
)0(
][
)(
])[{
)0(
][
1
21
1
21
1
21
1
21
m
lll
m
ll
m
lll
m
ll
lWith
)(
])[(
)0(
])[(
2
21
2
21
m
lll
m
lll is the Clebsh-Gordan coefficient.
Using (16) we deduce that
.)(
])[()(
)(
])[(
})(
])[(
)0(
])[({}
)(
])[(
)0(
])[({
)0(
][)(
)0(
][
1
21
12
2
21
2
21
1
21
2
21
2
21
1
21
12
2
21
m
lllR
m
lll
m
lll
m
lll
m
lll
m
lll
m
llR
m
ll
l
(17)
The element of the matrix of rotation
...,1,0),()(
])[()(
)(
])[(),(
1
21
12
2
21
12 lD
m
lllR
m
llll
mm , (18)
is the bispherical harmonic function of SO(3) which can be easily generalized to the
classical groups.
3.4 The coordinates of the bispherical harmonic functions
We notice that the spherical coordinates on the sphere ‘left’ L
nS 1 is ),,,( 121 nn .
The spherical coordinates on the sphere ‘right’ is R
nS 1 ),,,( 1
'
2
'
1 nn and on the
sphere )1(2 nS is ),,,,,,( '
1
'
2121 nnn .
The points:
L
nnn Sxxxx 121 ),,,( and R
nnn Sxxxx 121 )',',,'(' ,
can be represented as
1
'
111
'
21
'
2212
1
'
21
'
21212
1
'
21
'
11211
cos,cos
cossin,cossin
.............................................,........................................
cossinsin,cossinsin
sinsinsin,sinsinsin
nnnn
nnnnnn
nnnn
nnnn
xx
xx
xx
xx
Owing to the fact that the spherical functions and the bispherical functions are elements
of the matrix of rotation of SO(N), it results from that the invariant measure on the
sphere )1(2 nS must be the integration measure on L
nS 1 and R
nS 1 . This result and the
10
generalization of (21-23) allows us to deduce the coordinates of a point on
the sphere )1(2 nS
1
'
21
)1(221
1
1
'
21
12
12
1
2
'
21
1121
1
sinsin2
cos,cos2
sin
cossin2
cos..,........................................
...........................................,cossin2
sin
cos2
cos,sinsin2
sin
nn
nnn
n
nn
n
n
n
nn
nnn
xx
x
x
xx
4. The generating matrices and the Cartan-Weyl basis
We start with the link of the Cartan-Weyl basis for the group SO(3),SO(4) and
SO(5) with Hurwitz matrices. We use the traditional notations of adjoint representations
of these groups (Appendix). In the general case it is easier to use the notations of the
adjoint representations [10, 16] of the group SO(n) in term of the matrix
nlkjiiljkjlikij ,,1,,,),(
4.1. The Hurwitz’s matrix and the generating matrices
4.1.1 Generating matrices of SO(3)
For n= 2 we have
3212 SuIuH
For n= 3 we must add to the case n =2 21 SandS
.
14233213 SuSuSuIuH
4.1.2 Generating matrices of SO(4)
For n= 4 we obtain by Cayley transformation two orthogonal matrices,
the left and the right:
1423321
2
4
1423321
1
4
TuTuTuIuH
SuSuSuIuH
(27)
S and T are two commuting spins which generate SO(4) transformations.
4.1.3 Generating matrices of SO(5)
For n= 5 we must add only 2121 ,,, VVUU
to the above-mentioned matrices and we
write:
11
18765
81234
72143
63412
54321
281726151423321
1
5
uuuuu
uuuuu
uuuuu
uuuuu
uuuuu
VuVuUuUuSuSuSuIuH
(28)
We must change S by T to obtain the other matrix 2
5H .
4.2. Generating matrices of SO( n2 )
We treat first n= 8 and then the general case.
4.2.1 Generating matrices of SO(8)
For n= 8 we obtain by Cayley transformation two matrix from 1
5H and two matrix
from 2
5H and after calculations we find that three of these matrices are orthogonal.
Moreover these matrices do not generate the Cartan basis what we will do in what
follows.
The number of generating matrices is four thus we must group the elements of the
adjoint representation in four groups, work already carry out by many authors [15-18].
We start from the orthogonal Hurwitz matrix 1
8H writing in term of adjoint
representation }{ ij :
12345678
21436587
34127856
43218765
56781234
65872143
78563412
87654321
1
8
uuuuuuuu
uuuuuuuu
uuuuuuuu
uuuuuuuu
uuuuuuuu
uuuuuuuu
uuuuuuuu
uuuuuuuu
H (29)
][][
][][
][][
][][
453627188586723144
463528177685724133
473825166785634122
483726155443322111
uu
uu
uu
uu
(30)
We remark that the matrices (4, 4)
1111
1111
1111
1111
,
1111
1111
1111
1111
12
formed from the coefficients of Σ are orthogonal therefore we find the matrices
82,8 iH i by cyclic permutation of the row of these coefficients in (30).
4.2.1 Generating matrices for the general case
We determine the anti-symmetric matrices ,,5,4),(2
nH n by recurrence using the
method exposed in [11] part 6.3. We write this matrix in the form (29) like above.
Knowing that the matrices of coefficients are not obligatorily different from zero and
owing to the fact that the generating matrix is not unique we can solve this problem by
changing the coefficients by the coefficients of the well known Hadamard’s matrix which
is defined by:
11
111
SH
And
S
n
S
n
S
n
S
nS
n
SS
nHH
HHHHH
11
11
11
By cyclic permutation of the row of the matrix of coefficients we obtain the generating
matrices.
5. Appendix: The generators of SO (5) groups
The ten generators of SO (5), the group of rotations in five dimensions [19-20],
may be taken as ijjiij LjijiLL .5,,1,;; what generates rotations in the ij plane.
The commutations rules for the L’s are
)(i
j
j
iijx
xx
xiL
)(],[ iljkjkilikjkjlikklij LLLLiLL We define
kjililjlklij EEEE ],[
)4,,1,(,2
1
jiaaEp
p
j
p
iij
ii aanda are the operators of creation and destruction of the harmonic oscillator.
2121
4525452351
2515252151
341232431214231
341232431214231
,
,,
,,
,,
,,
TTTSSS
iLLVLVLV
iLLULULU
LLTLLTLLT
LLSLLSLLS
Then S and T are two commuting spins which generate SO(4) transformations. We put a
hat to the adjoint representations.
13
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