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where [Qa , Qb ] ? Qa Qb ? Qb Qa is the commutator.

In (2) Cab = const ? R are structure constants which determine uniquely the

c

Lie algebra AG. A ?xed set of Lie vector ?elds (LVFs) Qa satisfying (2) is called

a realization of the Lie algebra AG.

Thus the problem of description of all realizations of a given Lie algebra AG

c

reduces to solving the relations (2) with some ?xed structure constants Cab within

the class of LVFs (1).

It is easy to check that the relations (2) are not altered with an arbitrary invertible

transformation of variables x, u

y? = f? (x, u), ? = 1, . . . , m,

(3)

vi = gi (x, u), i = 1, . . . , n,

where f? , gi are smooth functions. That is why we can introduce on the set of reali-

zations of a Lie algebra AG the following relation: two realizations Q1 , . . ., QN

and Q1 , . . . , QN are called equivalent if they are transformed one into another by

means of an invertible transformation (3). As invertible transformations of the form

(3) form a group (called di?eomorphism group), the relation above is an equivalence

relation. It divides the set of all realizations of a Lie algebra AG into equivalence

classes A1 , . . . , Ar . Consequently, to describe all possible realizations of AG it su?ces

to construct one representative of each equivalence class Aj , j = 1, . . . , r.

De?nition 1. First-order linearly-independent di?erential operators

(1) (1)

Pa = ?ab (x, u)?xb + ?ai (x, u)?ui ,

(4)

(2) (2)

Ja = ?ab (x, u)?xb + ?ai (x, u)?ui ,

where the indices a, b take the values 1, 2, 3 and the index i takes the values 1, 2, . . . , n,

form a realization of the Euclid algebra AE(3) provided the following commutation

relations are ful?lled:

(5)

[Pa , Pb ] = 0,

(6)

[Ja , Pb ] = ?abc Pc ,

(7)

[Ja , Jb ] = ?abc Jc ,

298 R.Z. Zhdanov, V.I. Lahno, W.I. Fushchych

where

?

? (abc) = cycle (123),

1,

?1, (abc) = cycle (213),

?abc =

?

in the remaining cases.

0,

De?nition 2. Realization of the Euclid algebra within the class of LVFs (4) is called

covariant if coe?cients of the basis elements Pa satisfy the following condition:

(1) (1) (1) (1) (1)

?11 ?12 ?13 ?11 ... ?1n

(1) (1) (1) (1) (1)

rank (8)

= 3.

?21 ?22 ?23 ?21 ... ?2n

(1) (1) (1) (1) (1)

?31 ?32 ?33 ?31 ... ?3n

3 Realizations of the Lie algebra

of the rotation group O(3)

It is well-known from the classical representation theory that there are in?nitely many

inequivalent matrix representations of the Lie algebra of the rotation group O(3) [1].

A natural equivalence relation on the set of matrix representations of AO(3) is de?ned

as follows

Ja > V Ja V ?1

with an arbitrary constant nonsingular matrix V . If we represent the matrices Ja as

the ?rst-order di?erential operators (see, e.g. [7])

Ja = ?{Ja u}? ?u? , (9)

where u is a vector-column of the corresponding dimension, then the above equivalence

relation means that the representations of the algebra AO(3) are looked within the

class of LVFs (9) up to invertible linear transformations

u > v = V u.

It is proved below that provided realizations of AO(3) are classi?ed within arbitrary

invertible transformations of variables

(10)

vi = Fi (u), i = 1, . . . , n,

there are only two inequivalent realizations.

Theorem 1. Let ?rst-order di?erential operators

Ja = ?ai (u)?ui , (11)

a = 1, 2, 3

satisfy the commutation relations of the Lie algebra of the rotation group O(3) (7).

Then either all of them are equal to zero, i.e.

Ja = 0, (12)

a = 1, 2, 3

On covariant realizations of the Euclid group 299

or there exists a transformation (10) reducing these operators to one of the following

forms:

J1 = ? sin u1 tan u2 ?u1 ? cos u1 ?u2 ,

1.

J2 = ? cos u1 tan u2 ?u1 + sin u1 ?u2 , (13)

J3 = ?u1 ;

J1 = ? sin u1 tan u2 ?u1 ? cos u1 ?u2 + sin u1 sec u2 ?u3 ,

2.

J2 = ? cos u1 tan u2 ?u1 + sin u1 ?u2 + cos u1 sec u2 ?u3 , (14)

J3 = ?u1 .

Proof. If at least one of the operators Ja (say J3 ) is equal to zero, then due to the

commutation relations (7) two other operators J2 , J3 are also equal to zero and we

arrive at the formulae (12).

Let J3 be a non-zero operator. Then, using a transformation (10) we can always

reduce the operator J3 to the form J3 = ?v1 (we should write J3 but to simplify

the notations we omit hereafter the primes). Next, from the commutation relations

[J3 , J1 ] = J2 , [J3 , J2 ] = ?J1 it follows that coe?cients of the operators J1 , J2 satisfy

the system of ordinary di?erential equations with respect to v1 ,

?3iv1 = ??2i . i = 1, . . . , n.

?2iv1 = ?3i ,

Solving the above system yields

?3i = gi cos v1 ? fi sin v1 , (15)

?2i = fi cos v1 + gi sin v1 ,

where fi , gi are arbitrary smooth functions of v2 , . . . , vn , i = 1, . . . , n.

Case 1. fj = gj = 0, j ? 2. In this case operators J1 , J2 read

J1 = f cos v1 ?v1 , J2 = ?f sin v1 ?v1

with an arbitrary smooth function f = f (v2 , . . . , vn ).

Inserting the above expressions into the remaining commutation relation [J1 , J2 ] =

J3 and computing the commutator on the left-hand side we arrive at the equality

f 2 = ?1 which can not be satis?ed by a real-valued function.

Case 2. Not all fj , gj , j ? 2 are equal to 0. Making a change of variables

w1 = v1 + V (v2 , . . . , vn ), wj = vj , j = 2, . . . , n

we transform operators Ja , a = 1, 2, 3 with coe?cients (15) as follows

n

? ?

J1 = f sin w1 ?w1 + (fj cos w1 + gj sin w1 )?wj ,

?

j=2

n

(16)

? ?

J2 = f cos w1 ?w1 + (?j cos w1 ? fj sin w1 )?wj ,

g

j=2

J3 = ?w1 .

???

Here f , fj , gj are some functions of w2 , . . . , wn .

300 R.Z. Zhdanov, V.I. Lahno, W.I. Fushchych

?

Subcase 2.1. Not all fj are equal to 0. Making a transformation

z1 = w1 , zj = Wj (w2 , . . . , wn ), j = 2, . . . , n,

where W2 is a particular solution of partial di?erential equation

n

?

fj ?wj W2 = 1

j=2

and W3 , . . . , Wn are functionally-independent ?rst integrals of partial di?erential

equation

n

?

fj ?wj W = 0,

j=2

we reduce the operators (16) to be

n

J1 = F sin z1 ?z1 + cos z1 ?z2 + Gj sin z1 ?zj ,

j=2

n

(17)

J2 = F cos z1 ?z1 ? sin z1 ?z2 + Gj cos z1 ?wj ,

j=2

J3 = ?z1 .

Substituting operators (17) into the commutation relation [J1 , J2 ] = J3 and

equating coe?cients of the linearly-independent operators ?z1 , . . . , ?zn we arrive at

the following system of partial di?erential equations for the functions F, G2 , . . . , Gn :

Fz2 ? F 2 = 1, Gjz2 ? F Gj = 0, j = 2, . . . , n.

Integrating the above equations yields

cj

F = tan(z2 + c1 ), Gj = ,

cos(z2 + c1 )

where c1 , . . . , cn are arbitrary smooth functions of z3 , . . . , zn , j = 2, . . . , n.

Changing, if necessary, z2 by z2 + c1 (z3 , . . . , zn ) we may put c1 equal to zero. Next,

making a transformation

ya = za , a = 1, 2, 3,

yk = Zk (z3 , . . . , zn ), k = 4, . . . , n,

where Zk are functionally-independent ?rst integrals of partial di?erential equation

n

Gj ?zj Z = 0,

j=3

we can put Gk = 0, k = 4, . . . , n.

With these remarks the operators (17) take the form

sin y1

J1 = sin y1 tan y2 ?y1 + cos y1 ?y2 + (f ?y2 + g?y3 ),

cos y2

cos y1 (18)

J2 = cos y1 tan y2 ?y1 ? sin y1 ?y2 + (f ?y2 + g?y3 ),

cos y2

J3 = ?y1 ,

where f , g are arbitrary smooth functions of y3 , . . . , yn .

On covariant realizations of the Euclid group 301

If g ? 0, then making a transformation

f sin y2

u1 = y1 ? arctan u2 = ? arctan

? , ? , u k = yk ,

?

cos y2 cos2 y2 + f 2

where k = 3, . . . , n, we reduce the operators (18) to the form (13).

If in (18) g ? 0, then changing y3 to y3 = g ?1 dy3 and y2 to y2 = ?y2 we

? ?

transform the above operators to become

sin y1

? sin y1

?

J1 = ? sin y1 tan y2 ?y1 ? cos y1 ? ?

? ?? ? ?y2 + ?y ,

? ?

cos y2 3

cos y2

? ?

cos y1

? cos y1

? (19)

J2 = ? cos y1 tan y2 ?y1

? ?? + sin y1 + ?

? ?y2 + ?y ,

? ?

cos y2 3

cos y2

? ?

J3 = ?y1 .

?

Here ? is an arbitrary smooth function of y3 , . . . , yn .

? ?

Finally, making the transformation

u2 = g,

u1 = y1 + f,

? ? ?2 u3 = h,

? u k = yk ,

? ?

where k = 3, . . . , n and f (?2 , . . . , yn ), g(?2 , . . . , yn ), h(?2 , . . . , yn ) satisfy the compa-

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