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i?µ pµ + ?(??)?1/3 Sµ? F µ? ? = 0
? (2.7)

совместно с (0.1), (2.5), описывающих взаимодействие спинорного и электрома-
гнитного полей.
Решение системы (0.1), (2.5), (2.7) ищем в виде
(fµk x? + fk? xµ )xk
fµ?
= 2 2 ?2 (2.8)
Fµ? ,
(x2 )3
(x )

? ? ? k
? µ? = hµ? ? 2 (hµk x? + hk? xµ )x , (2.9)
H
(x2 )2 (x2 )3
??
(2.10)
?= ?(?),
(x2 )2

? ?
где fµ? = ?f?µ , hµ? = ?h?µ — зависящие от переменной w, ?µ — произвольные
действительные константы. С помощью формул (2.8)–(2.10) получаем частные то-
чные решения системы (0.1), (2.5), (2.7):
(bµk x? + bk? xµ )xk
bµ?
?2
Fµ? = ,
(x2 )2 (x2 )3
k
? µ? = cµ? ? 2 (?µk x? + ck? xµ )x ,
? c ?
(2.11)
H
(x2 )2 (x2 )3
?i(??)(Sµ? bµ? )?
??
? = 2 2 exp ?,
? 2 (??)?1/3
(x ) ?
зависящих от постоянных тензорных величин bµ? , cµ? .
?

1. Овсянников Л.В., Групповой анализ дифференциальных уравнений, М., Наука, 1978, 400 с.
2. Фущич В.И., Никитин А.Г., Симметрия уравнений Максвелла, Киев, Наук. думка, 1983, 197 с.
3. Born М., Infeld L., Foundations of the new field theory, Proc. Roy. Soc. A, 1934, 144, № 852,
4225.
W.I. Fushchych, Scientific Works 2001, Vol. 3, 150–174.

On subalgebras of the Lie algebra
?
of the extended Poincare group P (1, n)
?
L.F. BARANNIK, W.I. FUSHCHYCH
?
Some general results on the subalgebras of the Lie algebra AP (1, n) of the extended
? ?
Poincar? group P (1, n) (n ? 2) with respect to P (1, n) conjugation have been obtai-
e
?
ned. All subalgebras of AP (1, 4) that are nonconjugate to the subalgebras of AP (1, 4)
?
are classified with respect to P (1, 4) conjugation. The list of representatives of each
conjugacy class is presented.

1. Introduction
The systematic study of subalgebras of quantum mechanics transformation al-
gebras was begun in the fundamental paper by Patera, Winternitz and Zassenhaus
(PWZ) [1] in which the general method for classifying the subalgebras of a finite-
dimensional Lie algebra with a nontrivial solvable ideal with respect to some group
of automorphisms was suggested. This method is applied to classify all subalgebras
of Lie algebras of the following groups: the Poincar? group P (1, 3) [1], the extended
e
? (1, 2) [2], P (1, 3) [3], the de Sitter groups O(1, 4) [4], O(2, 3) [5],
?
Poincar? groups P
e
the optical groups Opt(1, 2) [5], Opt(1, 3) [6], the Euclidean group E(3) [7], the
Schr?dinger group Sch(2) [8], and the extended Schr?dinger group Sch(2) [8], the
o o
Poincar? group P (1, 4) [9–11], the Euclidean group E(5) [12, 13], the Galilei group
e
?
G(3) [12], and the extended Galilei group G(3) [12]. The application of the general
method had allowed us to study the subalgebras structure of the Lie algebra of the
generalized Euclidean group E(n) (n ? 2) [13]. The subalgebras of the algebras
?
AP (1, 3), AG(3), and AG(3) were described by another method [14–17].
The PWZ method needs the development for particular classes of algebras of its
generality. In the present paper we give the further development of the PWZ method
?
for extended Poincar? algebras AP (1, n) (n ? 2), denoted also by ASim(1, n). The
e
?
necessity in the description of subalgebras of AP (1, n) follows from certain problems
of theoretical and mathematical physics [1]. In particular, knowledge of the algebra
?
AP (1, n) subalgebras gives us the possibility to study the symmetry reduction for the
relativistically invariant scalar differential equation

? 2u, (?u)2 , u = 0,

where
2u = ux0 x0 ? ux1 x1 ? · · · ? uxn xn ,
(?u)2 = (ux0 )2 ? (ux1 )2 ? · · · ? (uxn )2 ,

and ? is a sufficiently smooth function [18–20]. The description of the algebra
? ?
AP (1, n) subalgebras allows us to solve the problem of the reduction of AP (1, n)
algebra representations on its subalgebras [21, 22].
J. Math. Phys., 1987, 28, № 7, P. 1445–1458.
?
On subalgebras of the Lie algebra of the extended Poincar? group P (1, n)
e 151

?
In Sec. 2 we describe the maximal reducible subalgebras of the algebra AO(1, n),
?
and in Sec. 3 we describe the completely reducible subalgebras of the algebra AO(1, n)
= AO(1, n) ? D , where D is the dilatation generator. Section 4 is devoted to study
?
of the subalgebras of the extended Galilei algebra AG(n ? 1), which is one of the
important subalgebras of the AP (1, n) algebra. In Sec. 5 which is the logical sequel to
Sec. 4, a number of assertions on subalgebras of the normalizer of isotropic subspace
?
of the Minkowski space M (1, n) in algebra AP (1, n) are conceived. Classification of
? ?
the AP (1, n) algebra subalgebras with respect to the P (1, 4) conjugation is carried
out in Sec. 6. The conclusions are summarized in Sec. 7.
?
2. Maximal reducible subalgebras of the algebra AO(1, n)
In this section we describe the maximal reducible subalgebras and the maximal
?
Abelian subalgebras of the algebra AO(1, n).
Let R be the real number field; Y1 , . . . , Ys is a vector space or Lie algebra
over R with the generators Y1 , . . . , Ys ; Rm is the m-dimensional arithmetical vector
space over R; U = M (1, n) is (1 + n)-dimensional pseudo-Euclidean space with the
scalar product
(X, Y ) = x0 y0 ? x1 y1 ? · · · ? xn yn ; (2.1)
O(1, n) is the group of the linear transformations of M (1, n) which conserve (X, X)
for every X ? M (1, n); Eq is the unit matrix of degree q. We suppose that O(1, n) is
realized as the group of the real matrices of degree n + 1.
?
We call the extended Poincar? group P (1, n) the multiplicative group of the matri-
e
ces
?? Y
,
0 1
where ? ? O(1, n), ? ? R, ? > 0, Y ? Rn+1 .
We denote by AG the Lie algebra of the Lie group G. Using the definition of Lie
algebra, we find that AO(1, n) consists of matrices
? ?
· · · ?0,n?1
0 ?01 ?02 ?0n
? ?01 ?1n ?
· · · ?1,n?1
0 ?12
? ?
? ?02 ?2n ?
??12 · · · ?2,n?1
0
? ?
X=? ?. (2.2)
. . . . .
..
? ?
. . . . .
.
. . . . .
? ?
? ?0,n?1 ??1,n?1 ??2,n?1 · · · ?n?1,n ?
0
??1n ??2n · · · ??n?1,n
?0n 0
Let Eik be the matrix of degree n + 2 which has the unity on the cross of ith line
and kth column and zeros on the other places (i, k = 0, 1, . . . , n + 1). It is easy to get
?
that the basis of the algebra AP (1, n) is formed by the matrices
D = E00 + E11 + · · · + Enn , J0a = ?E0a ? Ea0 , Jab = ?Eab + Eba ,
P0 = E0,n+1 , Pa = Ea,n+1 (a < b, a, b = 1, . . . , n).
The basis elements satisfy the following commutation relations:
[J?? , J?? ] = g?? J?? + g?? J?? ? g?? J?? ? g?? J?? , J?? = ?J?? ,
(2.3)
[P? , J?? ] = g?? P? ? g?? P? , [P? , P? ] = 0, [D, J?? ] = 0, [D, P? ] = P? ,
where g00 = ?g11 = · · · = ?gnn = 1, g?? = 0, when ? = ? (?, ? = 0, 1, . . . , n).
152 L.F. Barannik, W.I. Fushchych

The generators of turning J?? generate the algebra AO(1, n) and the translation
?
P? the commutative ideal N , and moreover AP (1, n) = N ? +(AO(1, n) ? D ). Let
? n) = {?En+1 |? ? R, ? > 0} ? O(1, n). Evidently, AO(1, n) = AO(1, n) ? D . It
?
O(1,
?
is easy to see that [X, Y ] = X · Y for all X ? AO(1, n), Y ? N . Let us identify N and
M (1, n) establishing correspondence between Pi and the (n + 1)-dimensional column
with unity on the ith place and zeros on the others (i = 0, 1, . . . , n).
Let C be such matrix of degree n + 2 over R that mapping ?C : X > CXC ?1
?
is an automorphism of the algebra AP (1, n). If C ? G, where G is a subgroup of
?
P (1, n), then ?C is called G automorphism. The subalgebras L and L of algebra
? ? ?
AP (1, n) are called P (1, n) conjugated if ?C (L) = L for some P (1, n) automorphism
?
?C of algebra AP (1, n). Let us identify ?C and C.
Let W a nondegenerate subspace of the space U . This subspace we also consider
to be pseudo-Euclidean relative to scalar product defined in U . Let O(W ) be the
?
group of isometries of the space W , O(W ) = O(W ) ? {?En+1 |? ? R, ? > 0}.
?
A subalgebra F ? AO(W ) is called irreducible if in W there does not exist any
F -invariant subspace different from O and W . Otherwise F is called reducible. If for
every F -invariant subspace W in W there exists an F -invariant subspace W in W
such that W = W ? W then it is called completely reducible.
?
Theorem 2.1. The maximal reducible subalgebras of algebra AO(1, n) are exhausted
?
with respect to O(1, n) conjugation by the following algebras: (1) AO(1, n ? 1) ? D ;
(2) AO(n) ? D ; (3) AO(1, k) ? AO (n ? k) ? D , where AO (n ? k) = Jab |a, b =
k + 1, . . . , n (k = 2, . . . , n ? 2); (4) G1 , . . . , Gn?1 ?
+(AO(n ? 1) ? J0n , D ), where
Ga = J0a ? Jan (a = 1, . . . , n ? 1).
?
Proof. If L is a maximal subalgebra of the algebra AO(1, n) then L = AO(1, n) or
L = L1 ? D , where L1 is a maximal subalgebra of the algebra AO(1, n). Let F be
a maximal reducible subalgebra of the algebra AO(1, n), U a subspace of the space
U invariant under F . If U is a degenerate space then it contains one-dimensional
F -invariant isotropic space W conjugated under O(1, n) to the space P0 + Pn . In
this case
F = {X ? AO(1, n)|X(P0 + Pn ) ? P0 + Pn }.
It is not difficult to show that
F = G1 , . . . , Gn?1 ?
+(AO(n ? 1) ? J0n ).
If U is a nondegenerate space of dimension r then it possesses an orthogonal
basis consisting of r vectors with nonzero length. Let r+ , r? be numbers of positive
and negative length vectors, in the given basis of the space U , respectively. These
numbers are independent of the choice of basis. In accordance with Witt’s mapping
1 1
theorem any two spaces U and U1 , for which r+ = r+ , r? = r? are mutually
conjugate under the group O(1, n). Obviously, r+ ? {0, 1}. Since U = U ? U ? and
U ? is invariant under F therefore F is O(1, n) conjugated to one of the algebras,
AO(1, n ? 1), AO(n), AO(1, k) ? AO (n ? k).
The theorem is proved.
Let
?
AE(n) = P1 , . . . Pn ?
+(AO(n) ? D ),
AE (n ? k) = Pk+1 , . . . , Pn ?
+AO (n ? k),
?
On subalgebras of the Lie algebra of the extended Poincar? group P (1, n)
e 153

?
and AG(n ? 1) is the extended Galilei algebra with the basis
(a, b = 1, . . . , n ? 1).
M = P0 + Pn , P0 , P1 , . . . , Pn?1 , G1 , . . . , Gn?1 , Jab
?
According to Theorem 2.1, the description of subalgebras of the algebra AP (1, n)
?
is reduced to the description with respect to the P (1, n) conjugation of irreducible
subalgebras of the algebra AO(1, n) and subalgebras of the following algebras:
+A ?
P0 ? E(n), (AP (1, k) ? AE (n ? k) ? D , +
?
AG(n ? 1) ? J0n , D (k = 2, . . . , n ? 1).

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