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1. Subalgebras with completely reducible projection onto AO(1, 3)
0, P0 , P1 , M , P0 , P3 , M, P1 , P1 , P2 , M, P1 , P2 , P0 , P1 , P2 ,
P 1 , P2 , P3 , P 0 , P1 , P2 , P3 ;
J12 : 0, P0 , P3 , M , P0 , P3 , P1 , P2 , P0 , P1 , P2 , M, P1 , P2 ,
P1 , P2 , P3 , P 0 , P1 , P2 , P3 ;
J12 + P0 : 0, P3 , P1 , P2 , P1 , P2 , P3 ;
J12 ± P3 : 0, P0 , P1 , P2 , P0 , P1 , P2 ;
J12 ± 2T : 0, M , P1 , P2 , M, P1 , P2 .
On the classi?cation of subalgebras of the conformal algebra 241

2. Subalgebras whose projection onto AO(1, 3) is not completely redu-
cible

G1 : P2 , M, P1 , M, P2 , M, P1 + ?P 2 , M, P1 , P2 ,
P0 , P1 , P3 , P0 , P1 , P2 , P3 (? = 0);
G1 ± P2 : 0, M , M, P1 , P0 , P1 , P3 ;
G1 + 2T : 0, P2 , M , M, P1 , M, P2 , M, P1 + ?P2 ,
M, P1 , P2 (? = 0);
G1 , G2 : M, P1 , P2 , P0 , P1 , P2 , P3 ;
G1 + ?P2 , G2 ? ?P1 , M , G1 + ?P2 , G2 ? ?P1 + ?P2 , M (? = ±1, ? = 0);
G1 + ?P2 , G2 + 2T, M, P1 (? ? R);
G1 ± P2 , G2 , M, P1 , G1 , G2 + 2T, M, P1 , P2 ;
G1 , G2 , J12 : M, P1 , P2 , P0 , P1 , P2 , P3 ;
G1 , G2 , J12 ± 2T, M, P1 , P2 , G1 + ?P2 , G2 ? ?P1 , J12 , M (? = ±1).

C. Subalgebras of AG1 (2) J03 with nonzero projection
onto J03
We divide also this class into two subclasses which are distinguished by whether or
not they have a completely reducible projection onto AO(1, 3).
1. Subalgebras with completely reducible projection onto AO(1, 3)

J03 : 0, P1 , M , P0 , P3 , M, P1 , P1 , P2 , P0 , P1 , P3 , M, P1 , P2 ,
P 0 , P1 , P2 , P3 ;
J03 + P1 : 0, P2 , M , P0 , P3 , M, P2 , P1 , P2 , P3 ;
J12 + ?J03 : 0, M , P0 , P3 , P1 , P2 , M, P1 , P2 ,
P0 , P1 , P2 , P3 , (? = 0);
J12 , J03 : 0, M , P0 , P3 , P1 , P2 , M, P1 , P2 , P0 , P1 , P2 , P3 .

2. Subalgebras with projections onto AO(1, 3) which are not completely
reducible

G1 , J03 : 0, M , P2 , M, P1 , M, P2 , M, P1 + ?P2 , M, P1 , P2 ,
P0 , P1 , P3 , P0 , P1 , P2 , P3 (? = 0);
G1 , J03 + P2 : 0, M , M, P1 , M, P1 + ?P2 , P0 , P1 , P3 , (? = 0);
G1 , J03 + P1 : M , M, P2 ;
G1 , J03 + P1 + ?P2 , M (? = 0);
G1 , G2 , J03 : 0, M , M, P1 , M, P1 , P2 , P0 , P1 , P2 , P3 ;
G1 , G2 , J03 + P1 , M , G1 , G2 , J03 + P2 , M, P1 ;
G1 , G2 , J12 + ?J03 : 0, M , M, P1 , P2 , P0 , P1 , P2 , P3 (? = 0);
G1 , G2 , J12 , J03 : 0, M , M, P1 , P2 , P0 , P1 , P2 , P3 .
242 L.F. Barannyk, P. Basarab-Horwath, W.I. Fushchych

D. Subalgebras of AP (1, 3) which are not conjugate
to subalgebras of AG1 (2) J03
This class consists of those subalgebras of the Poincar? algebra AP (1, 3) whose projec-
e
tion onto AO(1, 3) do not possess isotropic invariant subspaces in R1,3 . Since the
projections are simple algebras, then each subalgebra of the fourth class splits. The
full list of such algebras is
AO(1, 2) : 0, P3 , P0 , P1 , P2 , P0 , P1 , P2 , P3 ;
AO(3) : 0, P0 , P1 , P2 , P3 , P0 , P1 , P2 , P3 ;
AO(1, 3) : 0, P0 , P1 , P2 , P3 .

E. Subalgebras of AG1 (2) J03 , D which are not conjugate
to subalgebras of AG1 (2) J03
Let K be a subalgebra of AG1 (2) J03 , D with nonzero projection onto D , and
? be the projection of K onto J03 , D . By Propositions IV.2.3 and IV.2.5 in
let ?
?
?
Ref. [9], the algebra K, as a subalgebra of AP (1, 3), is split whenever ?(K) is one of
the subalgebras 1) D ; 2) ?D ? J03 (? = ±1, 0, 2); 3) D, J03 . This leads us to
dividing this class of subalgebras into two subclasses of nonsplittable subalgebras K of
?
AP (1, 3), denoted by D and E, for which the projection onto G1 , G2 is non-zero, and
?
for which ?(K) is J03 ± D and J03 ? 2D respectively. It is also useful to distinguish
the subclass A of subalgebras having zero projection onto G1 , G2 . The subalgebras in
this subclass di?er from the other subalgebras in that their projections onto AO(1, 3)
are completely reducible algebras of linear transformations of Minkowski space R1,3 .
All the other subalgebras are split, and we divide them formally into subclasses B
and C, depending on the dimension of their projection onto D, J03 .
1. Subalgebras with zero projection on G1 , G2
D : P 0 , P 0 , P3 , P 0 , P1 , P2 , P 1 , P2 , P3 , P 0 , P1 , P2 , P3 ;
J12 + ?D : P0 , P3 : P0 , P3 , P0 , P1 , P2 , P1 , P2 , P3 ,
P0 , P1 , P2 , P3 (? > 0);
J12 , D : P0 , P3 : P0 , P3 , P0 , P1 , P2 , P1 , P2 , P3 , P0 , P1 , P2 , P3 ;
J03 + ?D (0 < ? ? 1);
J03 + ?D, M (0 < |?| ? 1);
J03 + ?D : P1 , P0 , P3 , P1 , P2 , P0 , P1 , P3 , P0 , P1 , P2 , P3 (? > 0);
J03 + ?D : M, P1 , M, P1 , P2 , (? = 0);
J03 ? D ± 2T : 0, P1 , M , P1 , P2 , M, P1 , M, P1 , P2 ;
J03 , D : 0, P1 , M , P0 , P3 , P1 , P2 , M, P1 , M, P1 , P2 ,
P 0 , P1 , P3 , P 0 , P1 , P2 , P3 ;
?J12 + ?J03 + ?D (0 < ? ? ?, ? = ±1);
J12 + ?J03 + ?D, M (0 < |?| ? |?|);
?J12 + ?J03 + ?D : P0 , P3 , P1 , P2 , P0 , P1 , P2 , P3 (? = ±1, ?, ? > 0);
J12 + ?J03 + ?D, M, P1 , P2 (? = 0, ? = 0);
J12 + ?(J03 ? D ± 2T ) : 0, M , P1 , P2 , M, P1 , P2 (? = 0);
On the classi?cation of subalgebras of the conformal algebra 243

J12 + ?J03 , D : 0, M , P1 , P2 , P0 , P3 , M, P1 , P2 ,
P0 , P1 , P2 , P3 (? = 0);
J03 + ?D, J12 + ?D : P0 , P3 , P1 , P2 , M, P1 , P2 ,
P0 , P1 , P2 , P3 (?2 + ? 2 = 0);
J03 + ?D, J12 + ?D : (|?| ? 1, ? ? 0, |?| + ? = 0);
J03 + ?D, J12 + ?D, M : (|?| ? 1, ? ? 0, |?| + ? = 0);
J03 + ?D, J12 + ?D, M, P1 , P2 : (?, ? ? R, ?2 + ? 2 = 0);
J03 ? D ± 2T, J12 + 2?T : 0, M , P1 , P2 , M, P1 , P2 ;
J03 ? D, J12 ± T : 0, M , P1 , P2 , M, P1 , P2 ;
J03 , J12 , D : 0, M , P0 , P3 , P1 , P2 , M, P1 , P2 , P0 , P1 , P2 , P3 .
2. Subalgebras with two-dimensional projection onto J03 , D and non-
zero projection onto G1 , G2
G1 , J03 , D : P2 , M, P1 , M, P2 , M, P1 + ?P2 , M, P1 , P2 ,
P 0 , P1 , P3 , P 0 , P1 , P2 , P3 ;
G1 , G2 , J03 , D : M, P1 , P2 , P0 , P1 , P2 , P3 ;
G1 , G2 , J12 + ?J03 , D : M, P1 , P2 , P0 , P1 , P2 , P3 (? = 0);
G1 , G2 , J03 + ?D, J12 + ?D, P1 , P2 (|?| ? 1, ? ? 0, |?| + ? = 0);
G1 , G2 , J03 + ?D, J12 + ?D, P0 , P1 , P2 , P3 (?2 + ? 2 = 0);
G1 , G2 , J03 , J12 , D : M, P1 , P2 , P0 , P1 , P2 , P3 .
3. Split subalgebras with one-dimensional projection onto J03 , D and
nonzero projection onto G1 , G2
G1 + D : P0 , P1 , P3 , P0 , P1 , P2 , P3 ;
G1 , D : P0 , P1 , P3 , P0 , P1 , P2 , P3 ;
G1 + D, G2 , P0 , P1 , P2 , P3 , G1 , G2 , D, P0 , P1 , P2 , P3 ;
G1 , J03 + ?D : P2 , M, P1 , M, P2 , M, P1 + ?P2
(|?| ? 1, ? = 0, ? = 0);
G1 , J03 + ?D : M, P1 , P2 , P0 , P1 , P3 , P0 , P1 , P2 , P3 (? = 0);
G1 , G2 , J03 + ?D, M, P1 , P2 (0 < |?| ? 1);
G1 , G2 , J03 + ?D, P0 , P1 , P2 , P3 (? = 0);
G1 , G2 , J12 + ?D, P0 , P1 , P2 , P3 (? = 0);
G1 , G2 , J12 , D, P0 , P1 , P2 , P3 ;
G1 , G2 , J12 + ?J03 + ?D, M, P1 , P2 (0 < |?| ? |?|);
G1 , G2 , J12 + ?J03 + ?D, P0 , P1 , P2 , P3 (? = 0).
4. Nonsplit subalgebras of AG1 (2) J03 ? D with nonzero projection
onto G1 , G2 and J03 ? D
J03 ? D, G1 ± P2 : 0, M , M, P1 , P0 , P1 , P3 ;
J03 ? D ± 2T, G1 + ?P2 , M, P1 ;
J03 ? D ± 2T, G1 , M, P1 , P2 , J03 ? D + M, G1 , P2 ;
244 L.F. Barannyk, P. Basarab-Horwath, W.I. Fushchych

J03 ? D, G1 + ?P2 , G2 ? ?P1 + ?P2 , M (? = ±1, ? ? R);
J03 ? D, G1 ± P2 , G2 , M, P1 , J03 ? D ± 2T, G1 , G2 , P1 , P2 , M ;
J12 + ?(J03 ? D), G1 + ?P2 , G2 ? ?P1 , M (? = ±1, ? = 0);
J12 + ?(J03 ? D ± 2T ), G1 , G2 , M, P1 , P2 (? = 0);
J12 ± 2T, J03 ? D, G1 , G2 , M, P1 , P2 ;
J12 + 2?T, J03 ? D ± 2T, G1 , G2 , M, P1 , P2 (? ? R);
J12 , J03 ? D, G1 + ?P2 , G2 ? ?P1 , M (? = ±1).
5. Nonsplit subalgebras of AG1 (2) J03 ? 2D with nonzero projection
onto G1 , G2 and J03 ? 2D
J03 ? 2D, G1 + 2T : 0, M , P2 , M, P1 , M, P2 , M, P1 + ?P2 ,
M, P1 , P2 (? = 0);
J03 ? 2D, G1 , G2 + 2T : M, P1 , M, P1 , P2 .

F. Subalgebras of AP (1, 3) not conjugate to subalgebras
?
of AP (1, 3) and of AG1 (2) J03 , D
This class consists of those subalgebras of AP (1, 3) whose projection onto AO(1, 3)
do not have invariant isotropic subspaces in R1,3 and with a nonzero projection onto
D . We have
AO(1, 2) ? D : 0, P3 , P0 , P1 , P2 , P0 , P1 , P2 , P3 ;
AO(3) ? D : 0, P0 , P1 , P2 , P3 , P0 , P1 , P2 , P3 ;
AO(1, 3) ? D : 0, P0 , P1 , P2 , P3 .

G. Subalgebras of AG4 (2) which are not conjugate
to subalgebras of AP (1, 3)
?
Let K be a subalgebra of AG4 (2) and ? (K) its projection onto AGL(2, R). By Proposi-
tions V.2.1 and V.2.2 of Ref. [9], the algebra K belongs to this class if and only if
? (K) is conjugate to one of the following algebras: S + T , S + T + Z (subdirect
sum), ASL(2, R) = R, S, T , AGL(2, R) = R, S, T, Z . Because of this, we divide
this seventh class into three subclasses, each of which consists of subalgebras having
a corresponding projection onto AGL(2, R); those sub-algebras whose projections are
either ASL(2, R) or AGL(2, R) are put into the same subclass.
1. Subalgebras whose projection onto AGL(2, R) is S + T
S + T : 0, M , G1 , P1 , M , G1 ? ??1 P2 , G2 + ?P1 , M ,
G1 , G2 , P1 , P2 , M (0 < |?| ? 1);
S + T ± M , S + T + ?J12 ± M (? = 0);
S + T + ?J12 : 0, M , G1 + ?P2 , G2 ? ?P1 , M , G1 , G2 , P1 , P2 , M
(? = ±1, ? = 0);
S + T + ?J12 : G1 + ?P2 , G1 + ?P2 , M , G1 + ?P2 , G1 ? ?P2 ,
G2 + ?P1 , M (? = ±1);
S + T + ?J12 ± M, G1 + ?P2 (? = ±1);
On the classi?cation of subalgebras of the conformal algebra 245

S + T + ?J12 + ?G1 + P2 : 0, M , G2 ? ?P1 , M ,
G1 ? ?P2 , G2 + ?P1 , M , G2 ? ?P1 , G1 ? ?P2 , G2 + ?P1 , M (? = ±1);
J12 , S + T : 0, M , G1 + ?P2 , G2 ? ?P1 , M ,
G1 , G2 , P1 , P2 , M (? = ±1);
J12 ± M, S + T + ?M (? ? R);
J12 , S + T ± M .
2. Subalgebras whose projection onto AGL(2, R) is the subdirect sum
S+T + Z
S + T + ?Z : 0, M , G1 , P1 , M , G1 ? ? ?1 P2 , G2 + ?P1 , M ,
G1 , G2 , P1 , P2 , M (0 < |?| ? 1, ? = 0);
S + T, Z : 0, M , G1 , P1 , M , G1 ? ??1 P2 , G2 + ?P1 , M ,
G1 , G2 , P1 , P2 , M (0 < |?| ? 1);
S + T + ?J12 + ?Z : 0, M , G1 + ?P2 , G2 ? ?P1 , M ,
G1 , G2 , P1 , P2 , M (? = ±1, ? ?0, ? > 0);
S + T + ?J12 , Z : 0, M , G1 + ?P2 , G2 ? ?P1 , M ,
G1 , G2 , P1 , P2 , M (? = ±1, ? = 0);
S + T + ?J12 + ?Z : G1 + ?P2 , G1 + ?P2 , M ,
G1 + ?P2 , G1 ? ?P2 , G2 + ?P1 , M (? = ±1, ? = 0);
S + T + ?J12 , Z : G1 + ?P2 , G1 + ?P2 , M ,
G1 + ?P2 , G1 ? ?P2 , G2 + ?P1 , M (? = ±1);
J12 + ?Z, S + T + ?Z : 0, M , G1 + ?P2 , G2 ? ?P1 , M ,
G1 , G2 , P1 , P2 , M (? = ±1, |?| + |?| = 0);
J12 , S + T, Z : 0, M , G1 + ?P2 , G2 ? ?P1 , M ,
G1 , G2 , P1 , P2 , M (? = ±1).
3. Subalgebras whose projection onto AGL(2, R) contains ASL(2, R)
R, S, T : 0, M , G1 , P1 , M , G1 , G2 , P1 , P2 , M ;
J12 ? R, S, T : 0, M , G1 , G2 , P1 , P2 , M ;
J12 ± M ? R, S, T ;
R, S, T, Z : 0, M , G1 , P1 , M , G1 , G2 , P1 , P2 , M ;
R, S, T ? J12 + ?Z : 0, M , G1 , G2 , P1 , P2 , M (? = 0);
R, S, T ? J12 , Z : 0, M , G1 , G2 , P1 , P2 , M .

Acknowledgments
This work was supported in part by INTAS and DKNT of Ukraine. P. Basarab-
Horwath thanks the Wallenberg fund of Link?ping University and the Magnusson
o
Fund of the Swedish Academy of Sciences for travel grants, and the Mathematics
Institute of the Ukrainian Academy of Sciences in Kyiv for its hospitality.
Professor Wilhelm Fuschych died on April 7, 1997, after a short illness. This is
a great loss for his family, his many students, and for the scienti?c community. His
246 L.F. Barannyk, P. Basarab-Horwath, W.I. Fushchych

many and deep contributions to the ?eld of symmetry analysis of di?erential equations
have made the Kyiv school of symmetries known throughout the world. We take this
opportunity to express our deep sense of loss as well as our gratitude for all the
encouragement in research that Wilhelm Fushchych gave during the years we knew
him.

1. Patera J., Winternitz P., Zassenhaus H., Continuous subgroups of the fundamental groups of
physics. I. General method and the Poincar? group, J. Math. Phys., 1975, 16, 1597–1614.
e
2. Burdet G., Patera J., Perrin M., Winternitz P., The optical group and its subgroups, J. Math.
Phys., 1978, 19, 1758–1780.
3. Patera J., Winternitz P., Zassenhaus H., Continuous subgroups of the fundamental groups of
physics. II. The similitude group, J. Math. Phys., 1975, 16, 1615–1624.
4. Patera J., Winternitz P., Sharp R.T., Zassenhaus H., Subgroups of the similitude group of
three-dimensional Minkowski space, Can. J. Phys., 1976, 54, 950–961.
5. Burdet G., Patera J., Perrin M., Winternitz P., Sous-alg`bres de Lie de l’alg`bre de Schr?din-
e e o
ger, Ann. Sc. Math Quebec, 1978, 2, 81–108.
6. Fushchych W.I., Barannik A.F., Barannik L.F., Fedorchuk V.M., Continuous subgroups of the
Poincar? group P (1, 4), J. Phys. A, 1985, 18, 2893–2899.
e
7. Barannik L.F., Fushchych W.I., On subalgebras of the Lie algebra of the extended Poincar?
e
group P (1, n), J. Math. Phys., 1987, 28, 1445–1458.
8. Barannik L.F., Fushchych W.I., On continuous subgroups of the generalized Schr?dinger
o
groups, J. Math. Phys., 1989, 30, 280–290.
9. Fushchych W.I., Barannyk L.F., Barannyk A.F., Subgroup analysis of the Galilei and Poincar?
e
groups and reduction of nonlinear equations, Kyiv, Naukova Dumka, 1991 (in Russian).
10. Barannik L.F., Fushchych W.I., On continuous subgroups of the conformal group of Minkowski
space R1,n , Preprint 88.34, Kyiv, Inst. of Math., 1988 (in Russian).
11. Fushchych W.I., Barannyk L.F., Barannyk A.F., Continuous subgroups of the generalized
Euclidean group, Ukr. Math. J., 1986, 38, 67–72 (in Russian).
12. Barannyk A.F., Barannyk L.F., Fushchych W.I., Reduction and exact solutions of the eikonal
equation, Ukr. Math. J., 1991, 43, 461–474 (in Russian).
13. Barannyk A.F., Barannyk L.F., Fushchych W.I., Connected subgroups of the conformal group
C(1, 4), Ukr. Math. J., 1991, 43, 870–884 (in Russian).