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z3 = y3 + (R2v2 + tan v2 R2 ) cos 2v1 ? (R1v2 + tan v2 R1 ) sin 2v1 +
2 2
1
+ (tan v2 R2 ? R2v2 ),
2
where the functions R1 , R2 are solutions of the system of partial di?erential equations
1 1
tan v2 R1 = ?2B2 ,
R1v2 + R2v2 + tan v2 R2 = 2B1 ,
2 2
we reduce the operators (31) with A, B, F , H given by (33) to the form

J1 = ?z2 ?z3 + z3 ?z2 + A?z1 + C?z3 + J1 ,
J2 = ?z3 ?z1 + z1 ?z3 + A?z2 + G?z3 + J2 , (34)
J3 = ?z1 ?z2 + z2 ?z1 + J3 .

Here A, C, G are arbitrary smooth functions of v1 , . . . , vn , and what is more, A does
not depend on v1 .
Given such a form of operators Ja , system (32) reduces to three di?erential equa-
tions

J2 A = ?C, J1 A = G, J1 G ? J2 C = ?2A. (35)

Inserting expressions for the operators J1 , J2 from (13) into the ?rst two equations
we have

C = ? sin v1 Av2 , G = ? cos v1 Av2 .

Substituting the above formulae into the third equation of the system (35) we
conclude that it is equivalent to the di?erential equation

Av2 v2 ? tan v2 Av2 + 2A = 0,

whose general solution is given by (27). At last, inserting the results obtained into
(34) we get the formulae (25).
Subcase 2.2. Let the operators J1 , J2 , J3 be of the form (14). Then, making
a transformation
z1 = y1 + R1 cos v1 + R2 sin v1 ,
z2 = y2 + R2 cos v1 ? R1 sin v1 ,
On covariant realizations of the Euclid group 307

1
(R2v2 ? sec v2 R1v3 + tan v2 R2 ) cos 2v1 ?
z3 = y 3 +
2
1
? (R1v2 + sec v2 R2v3 + tan v2 R1 ) sin 2v1 +
2
1
+ (tan v2 R2 ? sec v2 R1v3 ? R2v2 ),
2
where the functions R1 , R2 are solutions of the system of partial di?erential equations
2B1 = R2v2 ? sec v2 R1v3 + tan v2 R2 ,
2B2 = ?R1v2 ? sec v2 R2v3 ? tan v2 R1 ,
we reduce the operators (31) with A, B, F , H given by (33) to the form (34), where
A, C, G are arbitrary smooth functions, and what is more, A does not depend on v1 .
Given such a form of the operators Ja , system (32) reduces to three di?erential
equations (35). Inserting expressions for the operators J1 , J2 from (13) into the ?rst
two equations of (35) we have

C = ? cos v1 Av2 + sin v1 sec v2 Av3 ,
(36)
G = ? sin v1 Av2 ? cos v1 sec v2 Av3 .
Substituting the above formulae into the third equation of (35) after some algebra
we arrive at the conclusion that it is equivalent to equation (28). Inserting (36) into
(34) yields formulae (26).
Thus we have proved that if LVFs Pa , Ja realize a covariant realization of the
Euclid algebra AE(3), then they can be reduced to one of the forms (24)–(26) by
means of an invertible transformation (3). The theorem is proved.
While proving Theorem 1, we have established, in particular, that any realization
of the Euclid algebra satisfying the condition (8) can be transformed to become
Ja = ??abc xb ?xc + jab (u)?xb + ?ai (u)?ui ,
Pa = ?xa , ? a = 1, 2, 3.
If we choose in the above formulae
?ai (u) = ??aij uj ,
jab (u) = 0, a, b = 1, 2, 3, i = 1, . . . , n,
where ?aij = const, then the following realization
Ja = ??abc xb ?xc + Ja , (37)
Pa = ?xa , a = 1, 2, 3
with Ja = ??aij uj ?ui is obtained.
A realization of the Euclid algebra with generators of the form (37) is called in the
classical linear representation theory a covariant realization. That is why it is natural
to preserve for a realization of the algebra AE(3) within the class of LVFs obeying
(8) the same terminology.
As an illustration to Theorem 2 we will demonstrate how to reduce realizations of
the Euclid algebras realized on sets of solutions of the heat, wave, Laplace, Navier–
Stokes, Lam`, Weyl, Dirac and Maxwell equations to one of the three canonical forms
e
(24)–(26). First of all, we note that the realization (24) is exactly the one realized
on the sets of solutions of the linear and nonlinear heat (Schr?dinger), wave, Laplace
o
equations.
308 R.Z. Zhdanov, V.I. Lahno, W.I. Fushchych

Symmetry algebras of the Navier–Stokes and Lam` equations contain as a subal-
e
gebra the Euclid algebra having basis elements (37), where (see, e.g. [6])

Ja = ??abc vb ?vc , (38)
a = 1, 2, 3.

The change of variables

v1 = u3 sin u1 cos u2 , v2 = u3 cos u1 cos u2 , v3 = u3 sin u2

reduce these LVFs to the form (25) with f = 0.
Next, if we consider the Weyl equation as the system of four real equations for
four real-valued functions v1 , v2 , w1 , w2 , then on the set of its solutions realization
(37) of the algebra AE(3) is realized, where [3, 7]
1
J1 = (w2 ?v1 ? v1 ?w2 + w1 ?v2 ? v2 ?w1 ),
2
1
J2 = (v2 ?v1 ? v1 ?v2 + w2 ?w1 ? w1 ?w2 ), (39)
2
1
J3 = (w1 ?v1 ? v1 ?w1 + v2 ?w2 ? w2 ?v2 ).
2
Making the change of variables
u1 u2 u3 u1 u2 u3
v1 = u4 sin sin cos + cos cos sin ,
2 2 2 2 2 2
u1 u2 u3 u1 u2 u3
? sin
v2 = u4 cos cos cos sin sin ,
2 2 2 2 2 2
u1 u2 u3 u1 u2 u3
? sin
w1 = u4 cos sin cos cos sin ,
2 2 2 2 2 2
u1 u2 u3 u1 u2 u3
w2 = u4 sin cos cos + cos sin sin
2 2 2 2 2 2
reduces the above LVFs to the form (26) with g = 0.
On the solution set of the Maxwell equations the realization of the Euclid algebra
(37), where

Ja = ??abc (Eb ?Ec + Hb ?Hc ) , a = 1, 2, 3,

is realized [19].
This realization is reduced to the form (26) under g = 0 with the help of the
change of variables

E1 = u6 sin u1 cos u2 ,
E2 = u6 cos u1 cos u2 ,
E3 = u6 sin u2 ,
H1 = u4 (cos u1 sin u3 + sin u1 sin u2 cos u3 ) + u5 sin u1 cos u2 ,
H2 = u4 (cos u1 sin u2 cos u3 ? sin u1 sin u3 ) + u5 cos u1 cos u2 ,
H3 = ?u4 cos u2 cos u3 + u5 sin u2 .

Taking the Dirac matrices ?µ in the Majorana representation we can represent the
Dirac equation as the system of eight real equations for eight real-valued functions
On covariant realizations of the Euclid group 309

0 3 0 3
?1 , . . . , ?1 , ?2 , . . . , ?2 (for details, see e.g. [7]). With this choice of ?-matrices, on the
set of solutions of the Dirac equation realization of the Euclid algebra (37) with
13
J1 = ? ? 1 ??1 + ? 1 ??1 ? ? 1 ??1 ? ? 1 ??1 + ? 2 ??2 + ? 2 ??2 ? ? 2 ??2 ? ? 2 ??2 ,
2 1 0 3 2 1 0
0 1 2 3 0 1 2 3
2
1
J2 = ? ? 1 ??1 + ? 1 ??1 + ? 1 ??1 ? ? 1 ??1 ? ? 2 ??2 + ? 2 ??2 + ? 2 ??2 ? ? 2 ??2 ,
2 3 0 1 2 3 0 1
0 1 2 3 0 1 2 3
2
11
J3 = ? ?1 ??1 ? ?1 ??1 + ?1 ??1 ? ?1 ??1 + ?2 ??2 ? ?2 ??2 + ?2 ??2 ? ?2 ??2
0 3 2 1 0 3 2
0 1 2 3 0 1 2 3
2
is realized on the set of solutions of the Dirac equation.
Making the change of variables
u1 u2 u3 u1 u2 u3
0
?1 = u4 cos cos sin + sin sin cos ,
2 2 2 2 2 2
u1 u2 u3 u1 u2 u3
? cos
1
?1 = u4 sin cos sin sin cos ,
2 2 2 2 2 2
u1 u2 u3 u1 u2 u3
= ?u4 cos ? sin
2
?1 cos cos sin sin ,
2 2 2 2 2 2
u1 u2 u3 u1 u2 u3
= ?u4 sin
3
?1 cos cos + cos sin sin ,
2 2 2 2 2 2
u1 u2 u3 + u6 u1 u2 u3 + u6
? cos
0
?2 = u5 sin sin sin cos cos +
2 2 2 2 2 2
u1 u2 u3 + u8 u1 u2 u3 + u8
? cos
+ u7 sin cos sin sin cos ,
2 2 2 2 2 2
u1 u2 u3 + u6 u1 u2 u3 + u6
= ?u5 sin ?
1
?2 cos cos + cos sin sin
2 2 2 2 2 2
u1 u2 u3 + u8 u1 u2 u3 + u8
? u7 sin ? cos
sin cos cos sin ,
2 2 2 2 2 2
= ?u5 cos u1 cos u2 sin u3 +u6 + sin u1 sin u2 cos u3 +u6
2
?2 2 2 2 2 2 2
u1 u2 u3 + u8 u1 u2 u3 + u8
+ u7 cos sin sin + sin cos cos ,
2 2 2 2 2 2
u1 u2 u3 + u6 u1 u2 u3 + u6
? sin ?
3
?2 = u5 cos sin cos cos sin
2 2 2 2 2 2
u1 u2 u3 + u8 u1 u2 u3 + u8
? u7 cos ? sin
cos cos sin sin
2 2 2 2 2 2
reduces the above realization to the form (26) with g = 0.

5 Covariant realizations of the Lie algebra
of the group E(4)
We recall that the basis elements of the Lie algebra of the Euclid group E(4) ful?ll
the following commutation relations:

(40)
[P? , P? ] = 0,

[Jµ? , P? ] = ?µ? P? ? ??? Pµ , (41)
310 R.Z. Zhdanov, V.I. Lahno, W.I. Fushchych

[J?? , Jµ? ] = ??µ J?? + ??? J?µ ? ??? J?µ ? ??µ J?? , (42)

where ?, ?, µ, ? = 1, 2, 3, 4.
Using the results of the previous sections and the fact that the Lie algebra of
the rotation group O(4) is the direct sum of two algebras AO(3) we will obtain
a description of covariant realizations of the Lie algebra (40)–(42) within the class of
LVFs
Pµ = ?µ? (x, u)?x? + ?µi (x, u)?ui ,
Jµ? = ?µ?? (x, u)?x? + ?µ?i (x, u)?ui
with Jµ? = ?J?µ . Here the indices µ, ?, ? take the values 1, 2, 3, 4 and the index i
takes the values 1, . . . , n.
As we consider covariant realizations, mutually commuting operators Pµ satisfy
(20) with N = 4. Hence due to Lemma 1 it follows that they can be reduced to the
form Pµ = ?xµ , µ = 1, 2, 3, 4. Next, using the commutation relations (41) we establish
that the operators Jµ? have the following structure:
Jµ? = x? ?xµ ? xµ ?x? + fµ?? (u)?x? + gµ?i (u)?ui (43)
with arbitrary su?ciently smooth fµ?? , gµ?i .
In what follows we will restrict our considerations to the case when in (43) fµ?? ?
0. This means geometrically that the transformation groups generated by the opera-
tors Jµ? in the space of independent variables are standard rotations in the planes
(xµ , x? ). With this restriction LVFs Jµ? take the form
Jµ? = x? ?xµ ? xµ ?x? + Jµ? , (44)
where
Jµ? = gµ?i (u)?ui (45)
and, furthermore, gµ?i (u) = ?g?µi (u).
Inserting LVFs (44) into (42) we come to conclusion that the operators Jµ? satisfy
the commutation relations of the Lie algebra of the rotation group O(4)
[J?? , Jµ? ] = ??µ J?? + ??? J?µ ? ??? J?µ ? ??µ J?? . (46)
An exhaustive description of inequivalent realizations of the above Lie algebra
within the class of LVFs (45) is given below. It is based on results of Section 2 and
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