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y ? y ? y ?
tible over-determined system of nonlinear partial di?erential equations

fy2 = sin f tan g, fy3 = sin y2 ? ? sin f tan g ? cos y2 cos f tan g,
? ?
? ?
gy2 = cos f, gy3 = sin f cos y2 ? ? cos f,
?
? ?
hy2 = ? sin f sec g, hy3 = (cos f cos y2 + ? sin f ) sec g,
?
? ?

reduces operators (19) to the form (14).
Subcase 2.2. fj = 0, j = 2, . . . , n. Substituting the operators (16) under fj = 0
into the commutation relation [J1 , J2 ] = J3 and equating coe?cients of the linearly-
independent operators ?z1 , . . . , ?zn yield system of algebraic equations

?f 2 = 1, f gj = 0, j = 2, . . . , n.

As the function f is a real-valued one, the system obtained is inconsistent.
Thus we have proved that the formulae (13)–(12) give all possible inequivalent
realizations of the Lie algebra (7) within the class of ?rst-order di?erential opera-
tors (11). The theorem is proved.
If we realize the rotation group as the group of transformations of the space of
spherical functions, then the basis elements of its Lie algebra are exactly of the form
(13) [1]. Hence it follows that the realization space V of the Lie algebra (13) is a di-
rect sum of subspaces V2l+1 of spherical functions of the order l. Furthermore, if we
consider O(3) as the group of transformations of the space of generalized spherical
functions [1], then the operators (14) are the basis elements of the corresponding Lie
algebra.
302 R.Z. Zhdanov, V.I. Lahno, W.I. Fushchych

Realizations of the algebra AE(3)
4
First we will prove an auxiliary assertion giving inequivalent realizations of Lie algeb-
ras of the translation T (3) group within the class of LVFs.
Lemma 1. Let mutually commuting LVFs
(1) (1)
Pa = ?ab (x, u)?xb + ?ai (x, u)?ui ,
where a, b = 1, . . . , N , satisfy the relation
(1) (1) (1) (1)
?11 ... ?1N ?11 ... ?1n
. . . . . .
. . . . . .
rank (20)
= N.
. . . . . .
(1) (1) (1) (1)
?N 1 ... ?N N ?N 1 ... ?N n
Then there exists a transformation of the form (3) reducing operators Pa to become
Pa = ?ya , a = 1, . . . , N .
Proof. To avoid unessential technicalities we will give the detailed proof of the lemma
for the case N = 3.
Given a condition N = 3, relation (20) reduces to the form (8). Due to the latter
Pa = 0 for all a = 1, 2, 3. It is well-known that a non-zero operator
(1) (1)
P1 = ?1b (x, u)?xb + ?1i (x, u)?ui
can always be reduced to the form P1 = ?y1 by a transformation (3) with m = 3.
If we denote by P2 , P3 the operators P2 , P3 written in the new variables y, v, then
owing to the commutation relations (5) they commute with the operator P1 = ?y1 .
Hence, we conclude that their coe?cients are independent of y1 .
(1) (1)
Furthermore due to the condition (8) at least one of the coe?cients ?22 , ?23 ,
(1) (1)
?21 , . . ., ?2n of the operator P2 is not equal to zero.
Summing up, we conclude that the operator P2 is of the form
(1) (1)
P2 = ?2b (y2 , y3 , v)?yb + ?2i (y2 , y3 , v)?vi = 0,
(1) (1) (1) (1)
not all the functions ?22 , ?23 , ?21 , . . ., ?2n being identically equal to zero.
Making a transformation
z1 = y1 + F (y2 , y3 , v),
z2 = G(y2 , y3 , v),
(21)
z3 = ?0 (y2 , y3 , v),
wi = ?i (y2 , y3 , v), i = 1, . . . , n,
where the functions F , G are particular solutions of di?erential equations
(1) (1) (1) (1)
?22 (y2 , y3 , v)Fy2 + ?22 (y2 , y3 , v)Fy3 + ?2i (y2 , y3 , v)Fui + ?21 (y2 , y3 , v) = 0,
(1) (1) (1)
?22 (y2 , y3 , v)Gy2 + ?22 (y2 , y3 , v)Gy3 + ?2i (y2 , y3 , v)Gui = 1
and ?0 , ?1 , . . . , ?n are functionally-independent ?rst integrals of the Euler–Lagrange
system
dy2 dy3 dv1 dvn
= ··· =
= = ,
(1) (1) (1) (1)
?22 ?23 ?21 ?2n
On covariant realizations of the Euclid group 303

which has exactly n + 1 functionally-independent integrals, we reduce the operator P2
to the form P2 = ?z2 . It is easy to check that the transformation (21) does not alter
form of the operator P1 . Being rewritten in the new variables z, w it reads P1 = ?z1 .
As the right-hand sides of (21) are functionally-independent by construction, the
transformation (21) is invertible. Consequently, operators Pa are equivalent to opera-
tors Pa , where P1 = ?z1 , P2 = ?z2 and
(1) (1)
P3 = ?3b (z3 , w)?yb + ?3i (z3 , w)?vi = 0.
(Coe?cients of the above operator are independent of z1 , z2 because of the fact that
it commutes with the operators P1 , P2 .) And what is more, due to (8) at least one
(1) (1) (1)
of the coe?cients ?33 , ?31 , . . ., ?3i of the operator P3 is not identically equal to
zero.
Making a transformation
Z1 = z1 + F (z3 , w),
Z2 = z2 + G(z3 , w),
Z3 = H(z3 , w),
Wi = ?i (z3 , w), i = 1, . . . , n,
where F , G, H are particular solutions of partial di?erential equations
(1) (1) (1)
?33 (z3 , w)Fz3 + ?3i (z3 , w)Fwi = ??31 (z3 , w),
(1) (1) (1)
?33 (z3 , w)Gz3 + ?3i (z3 , w)Gwi = ??32 (z3 , w),
(1) (1)
?33 (z3 , w)Hz3 + ?3i (z3 , w)Hwi = 1,
and ?1 , . . . , ?n are functionally-independent ?rst integrals of the Euler–Lagrange
system
dz3 dw1 dwn
= ··· =
= ,
(1) (1) (1)
?33 ?31 ?3n
we reduce the operators Pa , a = 1, 2, 3 to the form Pa = ?Za , a = 1, 2, 3, the same
as what was to be proved.
Note 1. In the papers [9, 17] mentioned above a classi?cation of realizations of the
groups G2 (1, 1), C(n, m) was carried out under assumption that mutually commuting
LVFs
Qa = ?a? (x)?x? , a = 1, . . . , N
can be simultaneously reduced by the map
(22)
y? = f? (x), ? = 1, . . . , n
to the form Qa = ?ya .
It is not di?cult to become convinced of the fact that this is possible if and only
if the condition
N n
rank ?a? (23)
=N
a=1?=1

holds.
304 R.Z. Zhdanov, V.I. Lahno, W.I. Fushchych

The su?ciency of the above statement is a consequence of Lemma 1. The necessi-
ty follows from the fact that function-rows of coe?cients of operators Q1 , . . . , QN
transformed according to formulae (22) are obtained by multiplying function-rows of
coe?cients of the operators Q1 , . . . , QN by a Jacobi matrix of the map (22), i.e.

?a? = ?a? f?x? , a = 1, . . . , N, ? = 1, . . . , n

which leaves the relation (23) invariant.
Consequently, in [9, 17] only covariant realizations of the corresponding Lie algeb-
ras were considered, which, generally speaking, do not exhaust a set of all possible
realizations.
Now we can prove a principal theorem giving a description of all inequivalent
covariant realizations of the Euclid algebra AE(3).
Theorem 2. Any covariant realization of the algebra AE(3) within the class of ?rst-
order di?erential operators is equivalent to one of the following realizations:

Ja = ??abc xb ?xc ,
1. Pa = ?xa , a = 1, 2, 3; (24)

2. Pa = ?xa , a = 1, 2, 3,
J1 = ?x2 ?x3 + x3 ?x2 + f ?x1 ? fu2 sin u1 ?x3 ?
? sin u1 tan u2 ?u1 ? cos u1 ?u2 ,
(25)
J2 = ?x3 ?x1 + x1 ?x3 + f ?x2 ? fu2 cos u1 ?x3 ?
? cos u1 tan u2 ?u1 + sin u1 ?u2 ,
J3 = ?x1 ?x2 + x2 ?x1 + ?u1 ;

3. Pa = ?xa , a = 1, 2, 3,
J1 = ?x2 ?x3 + x3 ?x2 + g?x1 ? (sin u1 gu2 + cos u1 sec u2 gu3 )?x3 ?
? sin u1 tan u2 ?u1 ? cos u1 ?u2 + sin u1 sec u2 ?u3 ,
(26)
J2 = ?x3 ?x1 + x1 ?x3 + g?x2 ? (cos u1 gu2 ? sin u1 sec u2 gu3 )?x3 ?
? cos u1 tan u2 ?u1 + sin u1 ?u2 + cos u1 sec u2 ?u3 ,
J3 = ?x1 ?x2 + x2 ?x1 + ?u1 .

Here f = f (u2 , . . . , un ) is given by the formula

sin u2 + 1
?1 , (27)
f = ? sin u2 + ? sin u2 ln
cos u2

?, ? are arbitrary smooth functions of u3 , . . . , un and g = g(u2 , . . . , un ) is a solution
of the following linear partial di?erential equation:

cos2 u2 gu2 u2 + gu3 u3 ? sin u2 cos u2 gu2 + 2 cos2 u2 g = 0. (28)

Proof. Due to Lemma 1 operators Pa can always be reduced to the form Pa = ?xa
by means of a properly chosen transformation (3). Inserting the operators

Pa = ?xa , Ja = ?ab (x, u)?xb + ?ai (x, u)?ui
On covariant realizations of the Euclid group 305

into the commutation relations (6) and equating the coe?cients of the linearly-
independent operators ?x1 , ?x2 , ?x3 , ?u1 , . . . , ?un we arrive at the system of partial
di?erential equations for the functions ?ab (x, u), ?ai (x, u),

?acxb = ??abc , ?aixb = 0, a, b, c = 1, 2, 3, i = 1 . . . , n.

Integrating the above system we conclude that the operators Ja have the form

Ja = ??abc xb ?xc + jab (u)?xb + ?ai (u)?ui , (29)
? a = 1, 2, 3,

where jab , ?ab are arbitrary smooth functions.
?
Inserting (29) into the commutation relations (7) and equating coe?cients of
?u1 , . . . , ?un show that the operators Ja = ?ai ?ui , a = 1, 2, 3 have to ful?ll (7) with
?
Ja > Ja . Hence, taking into account Theorem 1 we conclude that any covariant
realization of the algebra AE(3) is equivalent to the following one:

Ja = ??abc xb ?xc + jab (u)?xb + Ja , (30)
Pa = ?xa , a = 1, 2, 3,

operators Ja being given by one of the formulae (12)–(14).
Making a transformation

ya = xa + Fa (u), vi = ui , a = 1, 2, 3, i = 1, . . . , n,

we reduce operators Ja from (30) to be

J1 = ?y2 ?y3 + y3 ?y2 + A?y1 + B?y2 + C?y3 + J1 ,
J2 = ?y3 ?y1 + y1 ?y3 + F ?y2 + G?y3 + J2 , (31)
J3 = ?y1 ?y2 + y2 ?y1 + H?y3 + J3 ,

where A, B, C, F , G, H are arbitrary smooth functions of v1 , . . . , vn .
Substituting the operators (31) into (7) and equating coe?cients of linearly-inde-
pendent operators ?y1 , ?y2 , ?y3 , ?v1 , . . . , ?vn result in the following system of partial
di?erential equations:

J2 A = ?C, J3 C ? J1 H = G,
1) 6)
J3 F = ?B, J1 G ? J2 C = H ? A ? F,
2) 7)
J3 A = B, J3 B = F ? A ? H,
3) 8) (32)
J1 F ? J2 B = G, A ? F ? H = 0.
4) 9)
J2 H ? J3 G = C,
5)

Case 1. All operators J1 , J2 , J3 are equal to zero. Then, (32) reduces to the
system of linear algebraic equations

H ? A ? F = 0, F ? A ? H = 0, A ? F ? H = 0,
B = C = G = 0,

whence it follows immediately that A = F = G = 0. Substituting the above results
into formulae (31) we arrive at the realization (24).
Case 2. Suppose now that not all operators J1 , J2 , J3 vanish. Then, they are given
either by formulae (13) or (14), where one should replace u1 , . . . , un by v1 , . . . , vn . As
for the both cases J3 = ?v1 , a subsystem of equations 2, 3, 8, 9 forms a system of
306 R.Z. Zhdanov, V.I. Lahno, W.I. Fushchych

linear ordinary di?erential equations for functions A, B, F , H with respect to v1 .
Integrating it we have
A = B0 + B1 sin 2v1 ? B2 cos 2v1 , B = 2B1 cos 2v1 + 2B2 sin 2v1 ,
(33)
F = B0 + B2 cos 2v1 ? B1 sin 2v1 , H = 2B1 sin 2v1 ? 2B2 cos 2v1 ,
where B0 , B1 , B2 are arbitrary smooth functions of v2 , . . . , vn .
Subcase 2.1. Let the operators J1 , J2 , J3 be of the form (13). Then, making
a transformation
z1 = y1 + R1 cos v1 + R2 sin v1 ,
z2 = y2 + R2 cos v1 ? R1 sin v1 ,
1 1
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