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AH(n) ? D ; AH(n) ? S + T ; AH(n) ? T [n ? 0 (mod 2)];
AH(2d) ? T ? W [2d + 1, n] (d = 0, 1, . . . , [(n ? 1)/2]);
AH(2d) ? M[2d + 1, n] (d = 0, 1, . . . , [(n ? 1)/2]);
AH(2d) ? G2d+1 + T + W [2d + 1, n] (d = 0, 1, . . . [(n ? 1)/2]);
r
AH(2d) ? J(d + 1, r) + S + T ? La (d = 0, 1, . . . [(n ? 2)/2]; r = d +
a=d+1
1, . . . , [n/2]).
6. Classification of subalgebras of the algebras ASCh(3) and ASch(3)
In this section we make use of the previous results to provide a classification of
all subalgebras of the algebras ASch(3) and ASch(3).
? ?
Let AG(3) = (AO(3) ? T ) ? M[1, 3] and AG(3) = AG(3)/ M . Subalgebras of
+
? ?
the algebras AG(3) and AG(3) were classified up to conjugacy under G(3) and G(3),
respectively, in [5]. Further simplification of these subalgebras is being realized by
SL(2, R) automorphisms.
Theorem 6.1. Let ?, ?, ?, ?, µ ? R, and ? > 0, ? > 0, ? = 0. The splitting
subalgebras of the algebra AG(3) are exhausted with respect to Sch(3) conjugation
by the splitting subalgebras of the algebra AG(2) (see [2]) and by the following
algebras (the subalgebras preceded by the sign ? are subalgebras of ASch(3)):
? G1 + P2 , P3 ; G1 + P2 , P1 + ?P3 ; ? G1 + ?P1 + P3 , G2 + ?P3 ;
G1 + ?P1 + P3 , G2 + ?P1 + ?P3 ; G1 + ?P1 + P3 , G2 + ?P1 ;
G1 + P2 + ?P3 , G2 ? P1 + ?P2 + ?P3 ; G1 + P2 + ?P3 , G2 ? P1 ;
G1 + P2 , G2 ? P1 + ?P2 + ?P3 ; ? P1 , P2 , P3 ; ? G1 , P2 , P3 ; G1 + P2 , P1 , P3 ;
G1 , P1 + ?P2 , P3 ; G1 + P3 , G2 + ?P3 , P1 ; G1 + P3 , G2 , P1 ; G1 , G2 + P3 , P1 ;
G1 + ?P1 , G2 + P1 , P3 ; ? G1 + ?P1 , G2 , P3 ; G1 + ?P1 , G2 + P1 , P1 + ?P3 ;
G1 + ?P1 , G2 , P1 + ?P3 ; G1 + P2 + ?P3 , G2 + ?P3 , P1 ; G1 + P2 , G2 + ?P3 , P1 ;
G1 + P2 , G2 ? P1 , P3 ; G1 + P2 , G2 ? P1 + ?P2 , P3 ;
G1 + P2 + ?P3 , G2 + µP3 , P1 + ?P3 ;
G1 ? P2 + ?P3 , G2 + P1 + ?P2 + ?P3 , G3 + ?P1 + ?P2 + µP3 (µ ? ?2 ? = 0);
G1 ? P2 , G2 + P1 + ?P2 + ?P3 , G3 + ?P2 + ?P3 ;
G1 ? P2 + ?P3 , G2 + P1 , G3 + ?P1 + ?P3 ; G1 , P1 , P2 , P3 ; G1 , G2 , P1 , P3 ;
G1 + P2 , G2 , P1 , P3 ; G1 , G2 + P3 , P1 + ?P3 , P2 ; G1 , G2 + P3 , P1 , P2 ;
G1 , G2 , P1 + ?P3 , P2 ; G1 + P2 , G2 ? P1 + ?P2 , G3 + ?P1 + ?P2 , P3 ;
G1 + P2 , G2 ? P1 + ?P2 , G3 + ?P2 , P3 ; G1 + P2 , G2 ? P1 + ?P2 , G3 , P3 ;
G1 , G2 + P2 , G3 + ?P1 + ?P2 , P3 ; G1 + P2 , G2 ? P1 , G3 + ?P1 , P3 ;
G1 + P2 , G2 ? P1 , G3 , P3 ; G1 , G2 , P1 , P2 , P3 ;
G1 , G2 + P1 , G3 , P2 , P3 ; G1 , G2 , G3 , P1 , P2 , P3 ;
T : ? W [1, 3], G1 + P2 , P1 , P3 , G1 , P1 , P2 , P3 , G1 + P3 , G2 , P1 , P2 ,
M[1, 2] + W [3], M[1, 3];
J12 : ? W [3], M[3], ? W [1, 3], ? W [1, 2] + V [3], L1 + W [3], V [3] + W [1, 3],
L1 + M[3], M[1, 2] + W [3], M[1, 3];
Continuous subgroups of the generalized Schr?dinger groups
o 465

J12 + T : ? W [3], M[3], ? W [1, 3], V [3] + W [1, 3], M[1, 2] + W [3], M[1, 3];
J12 , T : ? W [3], M[3], ? W [1, 3], V [3] + W [1, 3], M[1, 2] + W [3], M[1, 3];
AO(3): ? 0, ? W [1, 3], M[1, 3]; AO(3) ? T : ? 0, ? W [1, 3], M[1, 3].
Theorem 6.2. The nonsplitting subalgebras of the algebra AG(3) are exhausted
with respect to Sch(3) conjugation by the nonsplitting subalgebras of the algebra
AG(2) [2] and by the following algebras:
T + G1 : ? W [2, 3], P1 + ?P2 , P3 , G2 + ?P3 , P2 , G2 + ?P1 + ?P3 , P2 ;
W [1, 3]; G2 , P2 , P3 , G2 + ?P1 , P2 , P3 , M[2] + P1 + ?P3 ,
G2 + ?P3 , P1 + ?P3 , P2 , G2 + ?P3 , P1 , P2 , V [2] + W [1, 3], M[2, 3],
G2 + ?P1 , G3 , P2 , P3 , M[2, 3] + W [1] (? > 0, ? > 0);
J12 + G3 : ? 0, W [3], ? W [1, 2], ? V [1, 2], W [1, 3], V [1, 2] + W [3], M[1, 2],
M[1, 2] + W [3];
J12 + T + ?G3 : ? 0, W [3], ? W [1, 2], W [1, 3], M[1, 2], M[1, 2] + W [3];
J12 + ?G3 : L1 , L1 + W [3] (? > 0);
J12 , T + G3 : ? 0, W [3], ? W [1, 2], W [1, 3], M[1, 2], M[1, 2] + W [3];
J12 + ?G3 , T + G3 : W [3], W [1, 3], M[1, 2] + W [3] (? > 0);
J12 + G3 , T : W [3], W [1, 3], M[1, 2] + W [3];
J12 + P3 , T : ? 0, ? W [1, 2], M[1, 2];
J12 + ?P3 , T + G3 : 0, W [1, 2], M[1, 2].
The written algebras are not mutually conjugated.
?
Theorem 6.3. The subalgebras of the algebra AG(3) are exhausted with respect
to Sch(3) conjugation by the subalgebras of the algebra AG(2) (see [2]), by the
algebras preceded by the sign ? in Theorems 6.1 and 6.2, by algebras obtained from
algebras written in Theorems 6.1 and 6.2 by adding the generator M , and by the
following algebras:
T ± M, P1 , P2 , P3 ;
J12 + ?M : W [3], W [1, 3], W [1, 2] + V [3] (? > 0);
J12 + T ± ?M : W [3], W [1, 3] (? > 0);
J12 + ?M, T : W [3], W [1, 3] (? > 0);
J12 + ?M, T + G3 : 0, W [1, 2] (? > 0);
J12 + ?P3 + ?M, T ± M : 0, W [1, 2] (? > 0, ? > 0);
J12 + ?P3 , T ± M : 0, W [1, 2] (? > 0);
J12 + P3 + ?M, T : 0, W [1, 2] (? > 0);
A0(3) ? T ± M : 0, W [1, 3].
The written algebras are not mutually conjugated.
Theorem 6.4. Let ? ? R, ? > 0. The subalgebras of the algebra ASch(3) which are
nonconjugated to subalgebras of the algebras AG(3) and ASch(2) are exhausted
with respect to Sch(3) conjugation by the following algebras:
D : ? W [1, 3], ? G1 , P2 , P3 , G1 , P1 + ?P2 , P3 , G1 , G2 , P1 + ?P3 , P2 ,
G1 , G2 , P1 , P3 , V [1] + W [1, 3], M[1, 2] + W [3], M[1, 3];
S + T, G1 ? ??1 P2 , G2 + ?P1 , G3 , P3 (0 < ? ? 1); S + T ? M[1, 3];
+
J12 +?D : ? W [3], M[3], ? W [1, 3], W [1, 2]+V [3], W [1, 2]+M[3], M[1, 2]+W [3],
M[1, 3];
S + T + ?J12 : M[3], L1 + M[3], N1 + M[3], M[1, 3];
S + T + 2J12 , G1 + P2 + ?P3 , G2 ? P1 ? ?G3 ;
466 L.F. Barannik, W.I. Fushchych

S + T + J12 : G1 + P2 + M[3], G1 + P2 + N1 + M[3];
D, T : ? W [1, 3], V [1, j] + W [1, 3] (j = 1, 2, 3);
J12 + ?D, T : ? W [3], M[3], ? W [1, 3], W [1, 2] + M[3], M[1, 2] + W [3], M[1, 3];
J12 , D : ? W [3], M[3], ? W [1, 3], ? W [1, 2]+V [3], W [1, 2]+M[3], M[1, 2]+W [3],
M[1, 3];
J12 , S + T : M[3], L1 + M[3], M[1, 3];
J12 , D, T : ? W [3], M[3], ? W [1, 3], W [1, 2] + M[3], M[1, 2] + W [3], M[1, 3];
ASL(2, R) ? M[1, 3]; J12 ? ASL(2, R): M[3], M[1, 3];
+
AO(3) ? D : ? 0, ? W [1, 3], M[1, 3]; AO(3) ? S + T : ? 0, M[1, 3];
A)(3) ? D, T : ? 0, ? W [1, 3], M[1, 3]; AO(3) ? ASL(2, R): ? 0, M[1, 3];
S+T +J12 +?(G1 +P2 ) : M[3], G2 ?P1 +M[3], N1 +M[3], G2 ?P1 +N1 +M[3].
The written algebras are not mutually conjugated.
Theorem 6.5. Let ?, ?, ? ? R, and ? > 0, ? = 0. The subalgebras of the algebra
ASch(3) are exhausted with respect to Sch(3) conjugation by subalgebras of the
?
algebra AG(3), by subalgebras of the algebra ASch(2) (see [2]), by algebras prece-
ded by the sign ? in Theorem 6.4, by algebras obtained from algebras written in
Theorem 6.4 by adding the generator M , and by the following algebras:
D +?M : W [1, 3], V [1]+W [2, 3]; J12 +?D +?M : W [3], W [1, 3], W [1, 2]+V [3];
v v
S + T + 2J12 + ?M, G1 + P2 + 2P3 , G2 ? P1 ? 2G3 ;
D + ?M, T ? W [1, 3]; J12 + ?M, D : W [3], W [1, 3], W [1, 2] + V [3];
+
J12 + ?M, D + ?M : W [3], W [1, 3], W [1, 2] + V [3];
J12 , D + ?M : W [3], W [1, 3], W [1, 2] + V [3];
J12 + ?D + ?M, T : W [3], W [1, 3]; J12 + ?M, D + ?M, T : W [3], W [1, 3];
J12 , D + ?M, T : W [3], W [1, 3]; AO(3) ? D + ?M : 0, W [1, 3];
AO(3) ? S + T + ?M ; AO(3) ? D + ?M, T : 0, W [1, 3].
The written algebras are not mutually conjugated.

7. Conclusions
The results of the present paper may be summarized in the following way.
(1) The completely reducible subalgebras of the algebra AO(n) ? ASL(2, R) have
been identified (Theorem 3.1).
(2) The subalgebras of AO(n) ? ASL(2, R) which possess only splitting extensions
in the algebra ASch(n) have been described (Theorem 4.1).
(3) We have established that the description of the splitting subalgebras of the
algebra ASch(n) whose projections onto ASL(2, R) are not equal to S + T is
reduced to the description of the splitting subalgebras of ASch(n) whose projections
onto AO(n) are equal to zero or to primary algebras (Theorem 4.2).
(4) The maximal Abelian subalgebras and the one-dimensional subalgebras of
the algebras ASch(n) and ASch(n) have been explicitly found (the corollaries to
Theorems 5.1 and 5.2).
(5) The classification of the subalgebras of ASch(3) and ASch(3) with respect
to Sch(3) conjugation and Sch(3) conjugation, respectively, has been carried out
(Theorems 6.1–6.5). This classification gives the possibility to construct the wide
Continuous subgroups of the generalized Schr?dinger groups
o 467

?
classes of exact solutions of the nonlinear, Schrodinger-type equations in [15–18],
??
? ?? + ?|?|3/4 ? = 0,
i
?t
?(?? ?) ?(?? ?) ? ?2
??
? ?? + ? (? ?) · ? = 0,
i
?t ?Xa ?Xa
which are invariant under Sch(3).
Acknowledgment
We are grateful to the referee for his valuable remarks.

1. Fushchych W.I., Cherniha R.M., J. Phys. A: Math. Gen., 1985, 18, 3491.
2. Burdet G., Patera J., Perrin M., Winternitz P., Ann. Sci. Math. Quebec, 1978, 2, 81.
3. Fushchych W.I., Nikitin A.G., Symmetry of Maxwell’s equations, Dordrecht, Reidel, 1987.
4. Boyer C., Sharp R.T., Winternitz P., J. Math. Phys., 1976, 17, 1439.
5. Fushchych W.I., Barannik A.F., Barannik L.F., The continuous subgroups of the generalized Galilei
group. I, Preprint 85.19, Kyiv, Institute of Mathematics, 1985.
6. Fushchych W.I., Barannik A.F., Barannik L.F., Fedorchuk V.M., J. Phys. A: Math. Gen., 1985, 18,
2893.
7. Barannik L.F., Barannik A.F., Subalgebras of the generalized Galilei algebra, in Group-Theoretical
Studies of Equations of Mathematical Physics, Kyiv, Institute of Mathematis, 1985.
8. Fushchych W.I., Barannik A.F., Barannik L.F., Ukr. Math. J., 1986, 38, 67.
9. Barannik L.F., Fushchych W.I., J. Math. Phys., 1987, 28, 1445.
10. Patera J., Winternitz P., Zassenhaus H., J. Math. Phys., 1975, 16, 1957.
?
11. Goursat E., Ann. Sci. Ecole Norm. Sup., 1889, 6, 9.
12. Jacobson N., Lie algebras, New York, Dover, 1962.
13. Barannik L.F., Fushchych W.I., On continuous subgroups of the generalized Schr?dinger groups,
o
Preprint 87.16, Kyiv, Institute of Mathematics, 1987.
14. Lang S., Algebra, MA, Addison-Wesley, 1965.
15. Fushchych W.I., Symmetry in the problems of the mathematical physics, in Algebraic Studies in
Mathematical Physics, Kyiv, Institute of Mathematics, 1982.
16. Fushchych W.I., Serov N.I., J. Phys. A: Math. Gen., 1987, 20, L929.
17. Fushchych W.I., Cherniha R.M., Exact solutions of multidimensional nonlinear Schr?dinger-type
o
equations, Preprint 86.85, Kyiv, Institute of Mathematics, 1986.
18. Fushchych W.I., Shtelen W.M., Theoret. Mat. Fiz., 1983, 56, 387.
W.I. Fushchych, Scientific Works 2001, Vol. 3, 468–472.

Подалгебры алгебры Пуанкаре AP (2, 3)
и симметрийная редукция нелинейного
ультрагиперболического уравнения
Даламбера. II
Л.Ф. БАРАННИК, В.И. ЛАГНО, В.И. ФУЩИЧ

Настоящая работа является продолжением статьи [1].
1. Классификация подалгебр алгебры AP (2, 2). В качестве базиса алгебры
AO(2, 2) возьмем систему таких генераторов:
1 1 1
B1 = ? (J14 + J23 ), B2 = (J24 ? J13 ), (J12 ? J34 ),
B3 =
2 2 2
1 1 1
C1 = (J14 ? J23 ), C2 = ? (J13 + J24 ), C3 = (J12 + J34 ),
2 2 2
Пусть AP (1, 2) = P1 , P3 , P4 + Jab | a, b = 1, 3, 4 , AP (2, 1) = P1 , P2 , P3 +
? ?
Jab | a, b = 1, 2, 3 . Проведем на основании [2] классификацию подалгебр алгебры
AP (2, 2) относительно эквивалентности.
Теорема 1. Расщепляемые подалгебры алгебры AP (2, 2) с точностью до экви-
валентности исчерпываются расщепляемыми подалгебрами K ? AP (1, 2), L ?
AP (2, 1), алгебрами K ? P2 , L ? P4 и такими алгебрами:
F1 = P1 + P3 , P2 + P4 , F2 = B1 ? B3 ,
Fj = B2 : 0, P1 + P3 , P1 + P3 , P2 + P4 , P1 + P3 , P2 ? P4 , j = 3, 4, 5, 6,
Fj = B3 : 0, P1 + P3 , P2 + P4 , j = 7, 8,
Fj = ?B1 + B3 + C2 : 0, P1 ? P3 , P1 + P3 , P2 + P4 , j = 9, 10, 11,
F12 = B1 ? B3 + C3 , F13 = B1 ? B3 ? C3 ,
Fj = B2 + eC2 : 0, P1 + P3 , P2 + P4 , P1 + P3 , P2 + P4 , P1 + P3 , P2 ? P4 ,
0 < e < 1, j = 14, . . . , 18,
F19 = B2 + C2 , P2 + P4 , Fj = B2 ? eC3 : 0, P1 + P3 , P2 ? P4 , e > 0, j = 20, 21,
F22 = B3 + eC3 , 0 < |e| < 1, F23 = B1 ? B3 , B2 ,
F24 = B1 ? B3 , C2 , F25 = B1 ? B3 , C3 , F26 = B2 , C2 , F27 = B2 , C3 ,
F28 = B3 , C3 , F29 = B2 + dC2 , B1 ? B3 , d > 0, d = 1,
F30 = B2 ? dC3 , B1 ? B3 , d > 0, F31 = B1 ? B3 ? C2 , C1 ? C3 ,
F32 = B1 ? B3 , B2 ? C2 , C1 ? C3 , F33 = AO(2, 2).
Доказательство. Среди эквивалентных расщепляемых подалгебр алгебры AP (2, 2)
из перечня, приведенного в [2], выбираем одну подалгебру, а остальные исклю-
чаем. Поскольку все случаи в чем-то аналогичны, ограничимся рассмотрением
только нескольких, наиболее характерных, случаев.
Так как
1 1
B1 ? B3 = ? (x1 + x3 )(?2 + ?4 ) + (x2 ? x4 )(?1 ? ?3 ),
2 2
Укр. мат. журн., 1989, 41, № 5, 579–584.
Подалгебры алгебры Пуанкаре AP (2, 3) 469

то
B1 ? B3 , P1 ? P3 ? P2 + P4 , P1 ? P3 ,
B1 ? B3 , P1 ? P3 , P2 + P4 ? P1 ? P3 , P2 + P4 .

Далее ранги алгебр B1 , B2 , B3 , B1 ? B3 , B2 , C2 , B1 ? B3 , B2 + dC2 , C1 ? C3
равны 3. Поскольку эти алгебры суть подалгебры AO(2, 2), то функция x2 + x2 ?
1 2
x2 ? x2 является их инвариантом. Следовательно, рассматриваемые подалгебры
3 4
эквивалентны алгебре AO(2, 2). Теорема доказана.
Теорема 2. Нерасщепляемые подалгебры алгебры AP (2, 2) исчерпываются с то-
чностью до эквивалентности нерасщепляемыми подалгебрами K ? AP (1, 2),
L ? AP (2, 1), алгебрами K ? P2 , L ? P4 и такими алгебрами:
Kj = B1 ? B3 + P1 : 0, P1 ? P3 , P1 ? P3 , P2 ? P4 , j = 1, 2, 3,
K4 = B1 ? B3 + P2 , P1 ? P3 ,
K5 = B1 ? B3 + C1 ? C3 + P3 , P1 ? P3 , P2 + ?P4 , ? > 0,
Kj = B1 ? B3 + C1 ? C3 + P4 : 0, P1 ? P3 , j = 6, 7,
Kj = B1 ? B3 ? C1 + C3 + P2 : 0, P1 ? P3 , j = 8, 9,
K10 = B1 ? B3 ? C1 + C3 + P1 , P1 ? P3 , P2 + ?P4 , ? > 0,
Kj = B2 + C2 + P2 + P4 : 0, P1 + P3 , j = 11, 12,
Kj = B2 + C2 + P2 : P2 + P4 , P1 + P3 , P2 + P4 , j = 13, 14,
K15 = B1 ? B3 + P2 ? P4 , C1 ? C3 , K16 = B1 ? B3 + P4 , C1 ? C3 ? P4 ,
K17 = B1 ? B3 + P2 , C1 ? C3 ? P4 ,
K18 = B1 ? B3 + C1 ? C3 , B2 + C2 + ?P4 , ? > 0,
K19 = B1 ? B3 ? C1 + C3 , B2 + C2 + ?P2 , ? > 0,
K20 = B1 ? B3 , B2 + C2 + P2 + P4 ,
Kj = B2 + 3C2 , B1 ? B3 + P2 ? P4 : 0, P1 ? P3 , j = 21, 22,
K23 = B2 + 3C2 , B1 ? B3 + P4 , P2 + P4 ,
K24 = B1 ? B3 , 3B2 + C2 , C1 ? C3 + P2 + P4 .
Доказательство. Если ранг алгебры L ? AP (2, 2) равен r, а генераторы L имеют
r
fi (x1 , . . . , xs )?i , где s ? r, то L ? P1 , . . . , Pr .
вид
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