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. 109
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cY ? [X, Y ] = (? ? c?4 )P0 + (c?1 ? ?2 )P1 + (c?2 + ?1 )P2 + c?3 P3 + (c?4 + ?)P4 .

According to lemma 3 (? ? c?4 )P0 + (c?4 + ?)P4 , (c?1 ? ?2 )P1 + (c?2 + ?1 )P2 ? A. If
?4 = 0 then lemma 4 yields P0 , P4 ? A. If c?1 ??2 = 0, c?2 +?1 = 0 then ?1 = ?2 = 0.
Thereafter using lemma 1 we can put ?1 = ?2 = 0. Since c?3 P3 ? A one can admit
that ?3 = 0. Thus the lemma is proved.
Lemma 8. Let A be a subalgebra of LP (1, 4), ? = exp(??Kb ) (? ? R, ? = 0). If
P0 + P4 , Pb + ? ?1 P4 ? A (1 ? b ? 3) then the algebra ?(A) contains P0 and P4 .
Proof. According to the Campbell–Hausdorff formula we have
1
?(P0 + ? ?1 P4 ) = ? ?1 P4 + ?(P0 + P4 ).
?(P0 + P4 ) = P0 + P4 ,
2
This gives that P0 + P4 , P4 ? ?(A), therefore P0 , P4 ? ?(A). Thus this lemma is
proved.

3. The non-splitting subalgebras of the LP (1, 4) algebra
Let F be an subalgebra of LP (1, 4) such that ?(F ) = F . An expression F + W
means that [F, W ] ? W and F ? V ? W . As concerns the non-splitting algebras
F + W1 , . . . , F + Ws we will use the notation F : W1 , . . . , Ws .
Theorem. Let ?, ?, ?, µ, ? ? R, ? > 0, ? > 0, µ ? 0 and this takes place for
all labelling variables. The non-splitting subalgebras of the LP (1, 4) algebra are
exhausted by the non-splitting subalgebras of the LP (1, 3) algebra and the following
subalgebras:
J12 + ?P0 : P3 , P4 , P1 , P2 , P3 , P4 ;
J12 + P0 + P3 : P4 , P1 , P2 , P4 ;
J12 +?P3 : P4 , P0 +P4 , P0 , P4 , P1 , P2 , P4 , P0 +P4 , P1 , P2 , P0 , P1 , P2 , P4 ;
J12 + P0 : P0 + P4 , P3 , P0 + P4 , P1 , P2 , P3 ;
J12 + J34 + ?P0 : 0, P1 , P2 , P1 , P2 , P3 , P4 ;
J12 + cJ34 + ?P0 : 0, P1 , P2 , P3 , P4 , P1 , P2 , P3 , P4 (0 < c < 1);
J04 + ?P3 : P1 , P2 , P0 + P4 , P1 , P2 , P0 , P1 , P2 , P4 ;
J12 +cJ04 +?P3 : 0, P0 +P2 , P0 , P4 , P1 , P2 , P0 +P4 , P1 , P4 , P0 , P1 , P2 , P4
(c > 0);
482 W.I. Fushchych, A.F. Barannik, L.F. Barannik, V.M. Fedorchuk

K3 +P2 : P1 , P0 +P4 , P1 , P0 +P4 , P1 +?P3 , P0 +P4 , P1 , P3 , P0 , P1 , P3 , P4 ;
K3 + P4 : P1 , P2 , P0 + P4 , P1 + ?P3 , P2 , P0 + P4 , P1 , P2 , P0 + P4 , P1 , P2 , P3 ;
K3 ? J12 + ?P4 : 0, P0 + P4 , P0 + P4 , P3 , P1 , P2 , P0 + P4 , P1 , P2 , P0 +
P 4 , P1 , P2 , P3 ;
J12 + ?P0 , J34 + µP0 : 0, P1 , P2 , P1 , P2 , P3 , P4 ; J12 , J34 + ?P0 , P1 , P2 ;
J04 + ?P3 , J12 + µP3 : 0, P0 + P4 , P0 , P4 , P1 , P2 , P0 + P4 , P1 , P2 , P0 , P1 ,
P 2 , P4 ;
J04 , J12 + ?P3 : 0, P0 + P4 , P0 , P4 , P1 , P2 , P0 + P4 , P1 , P2 , P0 , P1 , P2 , P4 ;
J12 + P0 + P4 , K3 + µP4 ; J12 , K3 + P4 ;
J12 + µP3 , K3 + P4 , P0 + P4 ; J12 + ?P3 , K3 , P0 + P4 ;
J12 + P0 + P4 , K3 + µP4 , P1 , P2 ; J12 , K3 + P4 , P1 , P2 ;
J12 + µP4 , K3 + P4 , P0 + P4 , P3 ; J12 + P4 , K3 , P0 + P4 , P3 ;
J12 + µP3 , K3 + P4 , P0 + P4 , P1 , P2 ; J12 + ?P3 , K3 , P0 + P4 , P1 , P2 ;
J12 + µP4 , K3 + P4 , P0 + P4 , P1 , P2 , P3 ; J12 + P4 , K3 , P0 + P4 , P1 , P2 , P3 ;
K1 + µP2 + P3 , K2 + µP1 + ?P2 ; K1 , K1 ± P2 , P3 ;
K1 + P2 , K2 + P1 + ?P2 , P3 ; K1 + ?P2 + P3 , K2 + ?1 P1 + ?2 P2 , P0 + P4 ;
K1 + P3 , K2 + µP1 + ?P2 , P0 + P4 ; K1 + µ2 P2 + µ3 P3 , K2 + P4 , P0 + P4 , P1 ;
K1 + P2 + ?P3 , K2 + ?P3 , P0 + P4 , P1 ; K1 + P2 , K2 + ?P3 , P0 + P4 , P1 ;
K1 + P3 , K2 + µP3 , P0 + P4 , P1 ; K1 , K2 + P3 , P0 + P4 , P1 ;
K1 + P2 , K2 + ?1 P1 + ?2 P2 , P0 + P4 , P3 ; K1 , K2 ± P2 , P0 + P4 , P3 ;
K1 + P2 + ?P3 , K2 + ?P3 , P0 + P4 , P1 + ?P3 ;
K1 + P3 , K2 + µP3 , P0 + P4 , P1 + ?P3 ; K1 , K2 + P3 , P0 + P4 , P1 + ?P3 ;
K1 + P3 , K2 , P0 + P4 , P1 , P2 ; K1 + P4 , K2 + ?P3 , P0 + P4 , P1 , P2 ;
K1 + P2 , K2 , P0 + P4 , P1 , P3 ; K1 + P2 , K2 + ?P4 , P0 + P4 , P1 , P3 ;
K1 , K2 + P4 , P0 + P4 , P1 , P3 ; K1 , K2 + P3 , P0 + P4 , P1 + ?P3 , P2 ;
K1 + P4 , K2 + µP3 , P0 + P4 , P1 + ?P3 , P2 ; K1 + P3 , K2 , P0 , P1 , P2 , P4 ;
K1 + P4 , K2 , P0 + P4 , P1 , P2 , P3 ; K3 , J04 + ?P1 , P0 + P4 , P1 + ?P3 , P2 ;
K3 , J04 + ?P2 : P1 , P0 + P4 , P1 , P0 + P4 , P1 + ?P3 , P0 + P4 , P1 , P3 , P0 , P1 ,
P 3 , P4 ;
K3 , J04 + ?P3 , P0 + P4 , P1 , P2 ; K3 , J04 + ?1 P1 + ?2 P2 , P0 + P4 , P1 + ?P3 ;
K3 , J04 + ?2 P2 + ?3 P3 , P0 + P4 , P1 ;
K3 , J12 + cJ04 + ?P3 : P0 + P4 , P0 + P4 , P1 , P2 (c > 0);
K3 , J04 + µ1 P3 , J12 + µ2 P3 : P0 + P4 , P0 + P4 , P1 , P2 (µ2 + µ2 > 0);
1 2
K1 , K2 , J12 + ?P3 ; K1 , K2 , J12 + P0 + P4 , P3 ; K1 , K2 , J12 + ?P3 , P0 + P4 ;
K1 + P2 , K2 ? P1 , J12 + ?P3 , P0 + P4 ; K1 + P2 , K2 ? P1 , J12 , P0 + P4 , P3 ;
K1 , K2 , J12 + ?P3 , P0 + P4 , P1 , P2 ; K1 , K2 , J12 + ?P3 , P0 , P1 , P2 , P4 ;
K1 , K2 , J12 + P4 , P0 + P4 , P1 , P2 , P3 ;
K1 , K2 , J04 + ?P1 : P0 + P4 , P3 , P0 + P4 , P1 + ?P3 , P0 + P4 , P1 + ?P3 , P2 ;
K1 , K2 , J04 + ?P2 : P0 + P4 , P1 + ?P3 , P0 + P4 , P1 , P3 ;
K1 , K2 , J04 + ?P3 : 0, P0 + P4 , P0 + P4 , P1 , P0 + P4 , P1 , P2 , P0 , P1 , P2 , P4 ;
K1 , K2 , J04 + ?1 P1 + ?2 P2 , P0 + P4 , P1 + ?P3 ;
K1 , K2 , J04 + ?1 P1 + ?3 P3 , P0 + P4 ; K1 , K2 , J04 + ?2 P2 + ?3 P3 , P0 + P4 , P1 ;
K1 , K2 , J12 + cJ04 + ?P3 : 0, P0 + P4 , P0 + P4 , P1 , P2 , P0 , P1 , P2 , P4 (c > 0);
K1 + P2 , K2 + P1 + ?P2 + µP3 , K3 + µP2 + ?P3 ; K1 , K2 ± P2 , K3 + ?P3 ;
K1 + P2 , K2 + ?1 P1 + ?2 P2 + ?P3 , K3 + ?1 P1 + ?2 P2 + ?3 P3 , P0 + P4 ;
K1 + P2 , K2 + ?1 P1 + ?2 P2 , K3 + ?P1 + ?2 P2 + ?3 P3 , P0 + P4 ;
K1 + P2 , K2 + ?1 P1 + ?2 P2 , K2 + µP2 + ?P2 , P0 + P4 ;
Continuous subgroups of the Poincar? group P (1, 4)
e 483

K1 , K2 ± P2 , K3 + ?P3 , P0 + P4 ; K1 + P2 , K2 + ?P3 , K3 + ?P2 + ?P3 , P0 + P4 , P1 ;
K1 + P2 , K2 , K3 + µP2 + ?P3 , P0 + P4 , P1 ;
K1 , K2 + P3 , K3 + ?P2 + ?P3 , P0 + P4 , P1 ; K1 , K2 , K3 ± P3 , P0 + P4 , P1 ;
K1 + P3 , K2 , K3 , P0 + P4 , P1 , P2 ; K1 , K2 , K3 + P4 , P0 + P4 , P1 , P2 ;
K1 + ?P3 , K2 , K3 + P4 , P0 + P4 , P1 , P2 ; K1 + P4 , K2 , K3 , P0 + P4 , P1 , P2 , P3 ;
K1 ± ?P1 , K2 ± ?P2 , J12 ? K3 ;
K1 + ?P1 + µP2 , K2 ? µP1 + ?P2 , J12 ? K3 , P0 + P4 (? 2 + µ2 > 0);
K1 + ?P2 , K2 ? ?P1 , J12 ? K3 , P0 + P4 , P3 ;
K1 , K2 , J12 ? K3 + ?P4 , P0 + P4 , P1 , P2 , sP3 (s = 0, 1);
J12 + J34 , J12 ? J24 , J23 + J14 , J34 + ?P0 : 0, P1 , P2 , P3 , P4 ;
K1 , K2 , J04 + ?P3 , J12 + µP3 : 0, P0 + P4 , P0 + P4 , P1 , P2 , P0 , P1 , P2 , P4 ;
K1 , K2 , J04 , J12 + ?P3 : 0, P0 + P4 , P0 + P4 , P1 , P2 , P0 , P1 , P2 , P4 ;
K1 , K2 , K3 ± P3 , J12 ; K1 , K2 , K3 + ?P3 , J12 + P0 + P4 ;
K1 + P2 , K2 ? P1 , K3 + ?P3 , J12 + µP3 , P0 + P4 ;
K1 , K2 , K3 ± P3 , J12 + µP3 , P0 + P4 ; K1 , K2 , K3 , J12 + ?P3 , P0 + P4 ;
K1 +P2 , K2 ?P1 , K3 , J12 , P0 +P4 , P3 ; K1 , K2 , K3 +P4 , J12 +µP3 , P0 +P4 , P1 , P2 ;
K1 , K2 , K3 , J12 + ?P3 , P0 + P4 , P1 , P2 ;
K1 , K2 , K3 + P4 , J12 + µP4 , P0 + P4 , P1 , P2 , P3 ;
K1 , K2 , K3 , J12 + P4 , P0 + P4 , P1 , P2 , P3 ; K1 , K2 , K3 , J04 + ?P1 , P0 + P4 ;
K1 , K2 , K3 , J04 + ?P2 , P0 + P4 , P1 ; K1 , K2 , K3 , J04 + ?P3 , P0 + P4 , P1 , P2 ;
K1 , K2 , K3 , J12 + cJ01 + ?P3 : P0 + P4 , P0 + P4 , P1 , P2 (c > 0);
K1 , K2 , K3 , J04 + µ1 P3 , J12 + µ2 P3 : P0 + P4 , P0 + P4 , P1 , P2 (µ2 + µ2 > 0).
1 2
Proof. The subalgebras of LO(1, 4) are classified by Patera et at [19]. For every
algebra Fedorchuk [6, 7] has found invariant subspaces of the space V . Using these
results together with lemmas 1–8, we will find the non-splitting subalgebras of the
LP (1, 4) algebra. Below we consider some examples in detail.
Let A be a subalgebra LP (1, 4), W = A ? V .
Suppose that ?(A) = J12 . Within the automorphism exp(t1 P1 +t2 P2 ) the algebra
A contains the element X = J12 + ?P0 + ?P3 + ?P4 (?, ?, ? ? R). Since

exp(tJ04 )(?P0 ?P4 ) = (? cosh t ? ? sinh t)P0 + (? cosh t ? ? sinh t)P4

then if P0 +P4 ? W one can write X = J12 +et (???)P0 +?P3 . Since exp(?J13 )(X) =
?J12 + et (? ? ?)P0 ? ?P3 , we consider ? ? ? ? 0. If ? ? ? > 0 then putting t =
? ln(???), we obtain the algebra W + J12 +P0 +?P3 . Applying the automorphism
?
exp(tK3 ), one can put ? = 0. If ? ? ? = 0 then A = W + J12 + ?P3 , ? = 0.
?
Let P0 + P4 ? W . If P3 , P4 ? W then ? > 0, ? = ? = 0. If W = P4 or
W = P1 , P2 , P4 then ? = 0. Applying the automorphism exp(tJ03 ) we reduce this
case to the following ones ? = ? = 1 or ? = 0, ? > 0.
Suppose that ?(A) = K1 , K2 , J12 + cJ04 (c > 0) one can suppose that A contains
the elements
4 4
X 1 = K1 + ? i Pi , X2 = K2 + ?i Pi , X3 = J12 + cJ04 + ?P3 .
0 0

Obviously, [X1 , X2 ] = (?2 ? ?1 )(P0 + P4 ) + (?0 ? ?4 )P2 ? (?0 ? ?4 )P1 . If ?0 ? ?4 = 0
or ?0 ? ?4 = 0 then using lemma 1, we obtain P1 , P2 , ? A. Therefore P0 + P4 ? A
and one can put ?i = ?i = 0 for i = 0, 1, 2. Later, [X3 , X1 ] = K2 ? cK1 ? c?4 P0 ,
484 W.I. Fushchych, A.F. Barannik, L.F. Barannik, V.M. Fedorchuk

[X3 , X2 ] = ?K1 ? cK2 ? c?4 P0 . Therefore ?3 = ?3 = 0, ?4 P4 + c?4 (P4 ? P0 ), ??4 P4 +
c?4 (P4 ? P0 ) ? A. The determinant constructed by the coefficients of P4 , P4 ? P0 is
equal to c(?2 + ?2 ). If ?2 + ?2 = 0 then P4 , P4 ? P0 ? A. So we have the algebra
4 4 4 4
K1 , K2 , J12 + cJ04 + ?P3 , P0 + P4 , P1 , P2 , sP0 (s = 0, 1).
Let ?0 ? ?4 = 0, ?0 ? ?4 = 0, ?3 = ?3 = 0. Obviously,
[X3 , X1 ] = K2 ? cK1 + ?1 P2 + ?2 P1 ? c?0 (P0 + P4 ),
[X3 , X2 ] = ?K1 ? cK2 + ?1 P2 ? ?2 P1 ? c?0 (P0 + P4 ),
[X3 , X1 ] + cX1 ? X2 = (c?1 ? ?2 ? ?1 )P1 + (c?2 + ?1 ? ?2 )P2 ? ?0 (P0 + P4 ),
[X3 , X2 ] + X1 + cX2 = (?1 + c?1 ? ?2 )P1 + (?2 + c?2 + ?1 )P2 + ?0 (P0 + P4 ).
If on the right-hand side of one of the last two equalities some coefficients of P1 ,
P2 are non-zero, so by lemmas 1 and 3 P1 , P2 , P0 + P4 ? A. Let c?1 ? ?2 ? ?1 = 0,
c?2 + ?1 ? ?2 = 0, ?1 + c?1 ? ?2 = 0, ?2 + c?2 + ?1 = 0. The determinant formed by the
coefficients of ?1 , ?2 , ?1 , ?2 is equal c2 (4 + c2 ). We obtain ?1 = ?2 = 0, ?1 = ?2 = 0,
?0 (P0 + P4 ), ?0 (P0 + P4 ) ? A and therefore
W ? V.
A = W + K1 , K2 , J12 + cJ04 + ?P3 ,
Let ?(A) = J12 , J13 , J23 , J04 . Because of the simplicity of the algebra J12 , J13 ,
J23 one can assume that A contains the elements J12 , J13 , J23 , X = J04 + ?i Pi
?i Pi ? A,
(i = 1, 2, 3). Applying lemma 1 to [J12 , X], [J13 , X], we conclude that
i.e. A is a splitting algebra.
When the algebra ?(A) coincides with one of the following algebras: K3 , J04 ,
K1 , K2 , J04 , K1 , K2 , K3 , J04 , one has to apply lemma 6. If ?(A) contains J12 +cJ04 ,
Ka , where a ? I ? {1, 2, 3}, then we apply lemma 7. Thus, this theorem is proved.

1. Aghassi J.J., Roman P., Santilli R.M., Phys. Rev. D, 1970, 1, 2753–2765.
2. Aghassi J.J., Roman P., Santilli R.M., J. Math. Phys., 1970, 11, 2297–2301.
3. Bacry H., Combe Ph., Sorba P., Connected subgroups of the Poincar? group, I and II, Preprints,
e
Marseille, 1972.
4. Bacry H., Combe Ph., Sorba P., Rep. Math. Phys., 1974, 5, 145–186.
5. Bacry H., Combe Ph., Sorba P., Rep. Math. Phys., 1974, 5, 361–392.
6. Fedorchuk V.M., Preprint 78-18, Mathematical Institute, Kyiv, 1978.
7. Fedorchuk V.M., Ukrainian Math. J., 1979, 31, 717–722.
8. Fedorchuk V.M., Ukrainian Math. J., 1981, 33, 696–700.
9. Fedorchuk V.M., Fushchych W.I., in Group-Theoretical Methods in Physics, Proc. Int. Seminar,
Svenigorod, 1979, Moscow, Nauka, 1980, 61–66.
10. Fushchych W.I., Teor. Math. Fiz., 1970, 4, 360–367.
11. Fushchych W.I., Krivsky I.Yu., Nucl. Phys. B, 1968, 7, 79–87.
12. Fushchych W.I., Krivsky I.Yu., Nucl. Phys. B, 1969, 14, 537–544.
13. Fushchych W.I., Nikitin A.G., J. Phys. A: Math. Gen., 1980, 13, 2319–2330.
14. Fushchych W.I., Serov N.I., J. Phys. A: Math. Gen., 1983, 16, 3645–3656.
15. Fushchych W.I., Serov N.I., Dokl. Akad. Nauk USSR, 1983, 273, 543–546.
16. Kadyshevsky V.G., Fiz. Elem. Chastiz At. Yad., 1980, 11, 5–36.
17. Lassner W., Die Zussamenhangenden Untergruppen der Poincare Gruppe, Preprint, Leipzig, 1970.
18. Patera J., Winternitz P., Zassenhaus H., J. Math. Phys., 1975, 16, 1597–1614.
19. Patera J., Winternitz P., Zassenhaus H., J. Math. Phys., 1976, 17, 717–728.
W.I. Fushchych, Scientific Works 2000, Vol. 2, 485–497.

The Galilean relativistic principle
and nonlinear partial differential equation
W.I. FUSHCHYCH, R.M. CHERNIHA
The second-order partial differential equations invariant under transformations of Galilei,
rotation, scale and projection are described.

1. Introduction
From the mathematical point of view the Galilean relativistic principle (in a restri-
cted sense) is nothing other than the requirement of the equations of motion to be
invariant under the linear transformations

t > t = t, xa > xa = xa + va t, a = 1, 2, 3,

va being transformation parameters (the inertial reference system velocity v compo-
nent). These transformations form a three-parameter Lie group. In order to construct
linear and nonlinear partial differential equations (PDE)

LU (t, x) = 0, x = (x1 , . . . , xn )

(where L is a linear or nonlinear operator, which is invariant under the Galilean
transformations) it is also necessary to give the law of transformation for the depen-
dent variable of U (t, x). Under different transformation laws of the function U (t, x)
we obtain different classes of PDE.
As is well known, the linear heat equation in the (n + 1)-dimensional space

? = ? 2 /?x2 + · · · + ? 2 /?x2 , U = U (t, x),
?U = ?U0 , n
1
(1.1)
? = const
U0 = Ut = ?U/?t,

is invariant under the following transformations:

t > t = t, xa > xa = xa + va t, (1.2)
a = 1, n,

1 1
U > U = exp ? va xa + va t (1.3)
,
2 2

va being the transformation parameters.
(1.3) defines the transformation law for the dependent function U (t, x) under the
Galilean transformations (1.2).
The 1 (n2 + 3n + 6)-dimensional algebra with basic elements
2

1
Ga = t?a ? ?xa U ?U , (1.4a)
?a = ?/?xa , ?U = ?/?U, a = 1, n,
2
Jab = xa ?b ? xb ?a , (1.4b)
a = b, a, b = 1, n,
J. Phys. A: Math. Gen., 1985, 18, P. 3491–3503.
486 W.I. Fushchych, R.M. Cherniha

1 1
? = t2 ?t + txa ?a ? |x|2 = xa xa ,
?|x|2 + nt U ?U , (1.4c)
4 2

k = constant, (1.4d)
D = 2t?t + xa ?a + kU ?U ,

(1.4e)
P 0 = ?t , Pa = ? a

(where the repeated indices imply summation) is maximal in the Lie restriction invari-
ance algebra (IA) of (1.1).
The set of operators (1.4) forms a Lie algebra, which will be noted by the symbol
SLi(1, n), i.e. the special Lie algebra. This name is natural because in the previ-
ous century Lie [10] (see also Ovsyannikov [13]) was the first to calculate the
maximal IA of the two-dimensional U (t, xa ) heat equation. The maximal IA of the
(3 + 1)-dimensional Schr?dinger equation, which coincides with (1.1) (differing only
o
by constant coefficients), was calculated by Niederer [11]. For some more details on
this, see, for example, Fushchych and Nikitin [6, 7].
From the group-theoretical point of view (1.3) defines the projective representation
of the group (1.2). Apart from the projective representation (1.3) the group (1.2) has

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