hurwitz’s matrices, cayley transformation and the cartan

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Hurwitz’s matrices, Cayley transformation and theCartan-Weyl basis for the orthogonal groups

Mehdi Hage-Hassan

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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)


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 .


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


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.


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.


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

7. References

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Sciences Eds. B. Gruber and M. Ramek (Plenum Press, New York, 1977))

[13] L.C. Biedenharn, “J. Math. Phys.2,2,433 (1961)

[14] A. R. Edmonds, Angular Momentum in Quatum Mechanics

Princeton, U.P.,Princeton,N.J.,( 1957)

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[17] L. O’Raifeartaigh, “Lectures on Local Lie Group and their Representation” (1964)

unpublished

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hurwitz’s matrices, cayley transformation and the cartan

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