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. 39
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Theorem 3.1. Let K1 , K2 , . . . , Kq be primary parts of a subalgebra F of the algebra
?
AO(1, n), and V a subspace of the space U invariant under F . Then V = V1 ?
?
· · · ? Vq ? V , where Vi = [Ki , V ] = [Ki , Vi ], [Kj , Vi ] = 0 when j = i (i, j = 1, . . . , q),
?
V = {X ? V |[F, X] = 0}. If the primary algebra K is the subdirect sum of the
? ? ?
irreducible subalgebras of the algebras AO(W1 ), AO(W2 ), . . . , AO(Wr ), respectively,
then nonzero subspaces W of the space U with the condition [K, W ] = W are
exhausted with respect to O(1, n) conjugation by the spaces W1 , W1 ? W2 , . . . , W1 ?
W2 ? · · · ? Wr .
Proof. From the complete reducibility of algebra F it follows that V = V ? V ,
where V is the maximal subspace of the space V , annulled by F . Further we shall
?
suppose that V = V . From Proposition 3.1 one can suppose that F ? AO(m), m ? n.
Let Ki be a subdirect sum of irreducible parts Ki1 , . . . , Kisi , Vij = [Kij , V ], ?a be a
sa
?Vaj .
projection of V onto
j=1
In view of Lemma 3.1 ?a (V ) ? V and that is why
q
??a (V ).
V=
a=1
Since Ka annuls in ?a (V ) only the zero subspace, then [Ka , V ] = [Ka , ?a (V )] =
?a (V ).
158 L.F. Barannik, W.I. Fushchych

Let primary algebra K be a subdirect sum of irreducible subalgebras of algebras
? 1 ), AO(W2 ), . . . , AO(Wr ), respectively. If W is a nonzero subspace of the space
? ?
AO(W
r
?Wj
?=
j=1

and [K, W ] = W then in view of Witt’s mapping theorem there exists such isometry
B ? O(?) that B(W ) = W1 ?· · ·?Ws (1 ? s ? r) and the space Wi is invariant under
BKB ?1 (i = 1, . . . , s). Whence BKB ?1 is a subdirect sum of irreducible subalgebras
? ? ?
of algebras AO(W1 ), AO(W2 ), . . . , AO(Wr ), respectively. Since irreducible parts of
?
the algebra L ? AO(n) are defined uniquely up to conjugation then one may consider
that BKB ?1 = K. The theorem is proved.
? ?
On the basis of Theorem 3.1 the description of splitting subalgebras F ? AP (1, n),
?
for which ?(F ) is a completely reducible algebra and has no isotropic invariant
subspaces in the space U , reduces to the description of irreducible subalgebras of
the algebras AO(1, k) and AO(k) (k = 2, 3, . . . , n). The rest of the cases can be
?
reduced to the case of the algebra AG(n ? 1) ? J0n , D .
+

4. On the subalgebras of the extended Galilei algebra
?
The aim of this section is to study subalgebras of the algebra AG(n ? 1) with
?
respect to P (1, n) conjugation. The main result concerning this problem is contained
in Theorem 4.1. Theorem 4.2 gives a description of all Abelian subalgebras of the
?
algebra AG(n ? 1). As a corollary, we obtain the list of maximal Abelian subalgebras
?
and one-dimensional subalgebras of the algebra AG(n ? 1).
?
The basis elements of the extended Galilei algebra AG(n ? 1) satisfy the following
commutation relations:
[Jab , Jcd ] = gad Jbc + gbc Jad ? gac Jbd ? gbd Jac , [Pa , Jbc ] = gab Pc ? gac Pb ,
[Pa , Pb ] = 0, [Ga , Jbc ] = gab Gc ? gac Gb , [Ga , Gb ] = 0, [Pa , Gb ] = ?ab M,
[Pa , M ] = [Ga , M ] = [Jab , M ] = 0, [P0 , Jab ] = [P0 , M ] = [P0 , Pa ] = 0,
[P0 , Ga ] = Pa (a, b, c, d = 1, . . . , n ? 1).

Let V1 = G1 , . . . , Gn?1 be a Euclidean space with orthonormal basis G1 , . . .,
Gn?1 , V2 = [P0 , V1 ] (n ? 3), M = V1 + V2 + P0 , M . We settle on identifying the
group O(n ? 1) with the isometry group O(V1 ), O(V2 ). If W is a subspace of V1 and
dim W = k then according to Witt’s theorem for every a, 0 ? a ? n ? k ? 1, there
exists an isometry Ba ? O(V1 ) such that

Ba (W ) = V1 (a + 1, a + k) = Ga+1 , Ga+2 , . . . , Ga+b .

Further, in spaces V1 , V2 we shall consider only subspaces V1 (a, b), V2 (a, b) = [P0 ,
V1 (a, b)]. We call them elementary spaces. The basis Ga , Ga+1 , . . . , Gb of the space
V1 (a, b) and the basis Pa , Pa+1 , . . . , Pb of the space V2 (a, b) we shall call canonical.
Let W1 , W2 be subspaces of some vector space W over the field R and W1 ? W2 =
0. If ? : W1 > W2 is an isomorphism then we denote as (W1 , W2 , ?) the space
{Y + ?(Y )|Y ? W1 }. As I(W1 , W2 ) we denote the isomorphism of elementary spaces
W1 and W2 , by which the canonical basis of W1 is mapped to the canonical basis of
W2 with numeration of the basis of elements maintained.
?
On subalgebras of the Lie algebra of the extended Poincar? group P (1, n)
e 159

?
Let AG(n?1) = AG(n?1)/ M . For the generators of the AG(n?1) we preserve
?
the notation of the generators of the algebra AG(n?1). By ? , ?0 , ?1 , and ?2 we denote
?
the projection of AG(n ? 1) and AG(n ? 1) onto AO(n ? 1) ? P0 , P0 , V1 , and V2 ,
respectively.
?
Let F be a subalgebra of the AO(n ? 1) ? P0 , F an subalgebra of the AG(n ? 1)
? ?
such that ? (F ) = F . If algebra F is conjugated to the algebra W ? , where W is
+F
?
the F -invariant subspace of space V1 + V2 , then F is called splitting in the algebra
?
AG(n ? 1). The notion of a splitting subalgebra of the algebra AG(n ? 1) is defined
analogously.
Proposition 4.1. Let L1 be a subalgebra of the AO(n ? 1), L2 be a subalgebra of the
P0 , and F be the subdirect sum of L1 and L2 . If P0 ? F then the algebra F only
/
has splitting extensions in the algebra AG(n ? 1) if and only if L1 is a semisimple
algebra or L1 is not conjugated to any subalgebra of the algebra AO(n ? 2). When
P0 ? F , the algebra F only has splitting extensions in the AG(n ? 1) if and only if
L1 is not conjugated to any subalgebra of the algebra AO(n ? 2).
Proof. If L1 is a semisimple algebra and L2 = P0 then by Whitehead’s theorem [23]
?
P0 ? F . Let us assume that L2 = P0 and P0 ? F . Let F be an subalgebra of the
/
?
AG(n ? 1) such that ? (F ) = F . If L1 is not conjugated to any subalgebra of the
?
AO(n ? 2) then by Proposition 2.2 the algebra F is splitting. If L1 is conjugated to
some subalgebra of AO(n ? 2) then F = X ? F1 where X = 0, X , and F1 are
subalgebras of the algebra AO(n ? 2) ? P0 . The algebra
?
F = P1 , . . . , Pn?1 , G1 , . . . , Gn?2 , X + Gn?1 ? 1
+F
is not splitting by Lemma 2.1. The case L2 = 0 can be treated similarly.
Let P0 ? F . If L1 ? AO(n ? 2) then algebra P0 + Gn?1 ? 1 is nonsplitting.
+L
If L1 is not conjugated to any subalgebra of the algebra AO(n ? 2) then by way of
? ?
complete reducibility of the algebra L1 we get that P0 ? F and whence algebra F is
splitting. The proposition is proved.
Proposition 4.2. The subalgebra F of the algebra AO(n?1)? P0 has only splitting
?
extensions in the AG(n ? 1) if and only if F is a semisimple algebra.
Lemma 4.1. Let W1 = Y1 , . . . , Ym , W2 = Z1 , . . . , Zm be Euclidean spaces over
the field R, O(Wi ) the isometry group of Wi (i = 1, 2), 0 < ?1 ? ?2 ? · · · ? ?t ,
S0 = 0, Sj = Zt+1 , . . . , Zt+j (j = 1, . . . , m?t). The subspaces of the space W1 ?W2
t t

are exhausted with respect to O(W1 ) ? O(W2 ) conjugation by the following spaces:
O, Y1 , . . . , Yr , Z1 , . . . , Zs , Y1 , . . . , Yr , Z1 , . . . , Zs (r, s = 1, . . . , m),
Y1 , . . . , Yk , Yk+1 + ?1 Z1 , . . . , Yk+t + ?t Zt ? Sj t

(k = 1, . . . , m ? 1, t = 1, . . . , m ? k, j = 0, 1, . . . , m ? t),
Y1 + ?1 Z1 , . . . , Yt + ?t Zt ? Sj (t = 1, . . . , m, j = 0, 1, . . . , m ? t).
t


Proof. Let N be a subspace of W1 ? W2 and N = W1 ? W2 , where Wi is a subspace
of Wi (i = 1, 2). If Bi = N ? Wi , Ni is a projection of N onto Wi (i = 1, 2)
and then N1 /B1 ? N1 /B2 . Let dim B1 = k. By Witt’s theorem the space B1 is
=
conjugated to the space Y1 , . . . , Yk . If dim (N1 |B1 ) = t then N contains elements
Yk+j + ?1j Z1 + · · · + ?tj Zt (j = 1, . . . , t), and moreover the matrix A = (?ij ) is
nonsingular. The matrix A can be represented uniquely in the form CT , where C is
an orthogonal matrix and T is a positively definite symmetric matrix.
160 L.F. Barannik, W.I. Fushchych

The isometry diag [Em , C ?1 , Em?t ] maps N onto the space to which the matrix
?1
C ?1 (CT ) = T corresponds. There exists such orthogonal matrix C1 that C1 T C1 =
diag [?1 , . . . , ?t ]. The isometry diag [Ek , C1 , Em?k?t , C1 , Em?t ] maps N onto the space
?1
to which the matrix C1 T C1 corresponds. Therefore N is conjugated to the space

B1 ? Yk+1 + ?1 Z1 , . . . , Yk+t + ?t Zt ? B2 ,

where 0 < ?1 ? ?2 ? · · · ? ?t . The lemma is proved.
Let K be the primary subalgebra of the algebra AO(n ? 1) which is a subdirect
sum of irreducible subalgebras of the algebras AO(V1 (1, q)), AO(V1 (q + 1, 2q)), . . .,
AO(V1 ((r ? 1)q + 1, rq)), respectively, and W nonzero subspace of the space M with
the property [K, W ] = W . If ?1 (W ) = 0 then by way of Theorem 3.1 W is conjugated
to the space V2 (1, iq) (1 ? i ? r). If ?2 (W ) = 0 then W is conjugated to V1 (1, iq)
(1 ? i ? r). Let us suppose that ?1 (W ) = 0, ?2 (W ) = 0. Then W is a subdirect
sum of ?1 (W ), ?2 (W ), where ?1 (W ) = V1 (1, m) and ?2 (W ) coincides with V2 (1, k) or
V2 (m + 1, m + l) or a subdirect sum of V2 (1, k) and V2 (m + 1, m + l) (k ? m). Every
number of k, m, and l is divisible by q. Let us consider the case when ?2 (W ) is a
subdirect sum of V2 (1, k) and V2 (m + 1, m + l). In the space W we choose the basis
in the following form:
i i
Ga + ?a Pi , ?c Pi
(4.1)
(a = 1, . . . , m, c = m + 1, . . . , m + t, i = 1, . . . , k, m + 1, . . . , m + l).

The coefficients of the decomposition we write down as the corresponding columns of
the matrix
A1 B1
?= ,
A2 B2

having m + t columns and k + l lines. We call the matrix ? a coupling matrix of
elementary spaces in the space W . With the coupling matrix we shall carry out the
transformations corresponding to definite O(n?1) automorphisms and transformations
to new bases of the form (4.1). Let C1 ? O(k), C2 ? O(m ? k), C3 ? O(l), S =
diag [C1 , C2 ], T be a t ? m matrix, and T2 a nonsingular matrix of degree t. The most
general admissible transformations of the coupling matrix have the form

C1 A1 S ?1 + C1 B1 T1
A1 B1 C1 B1 T2
> .
C3 A2 S ?1 + C3 B2 T1
A2 B2 C3 B2 T2

If B2 = 0 then according to Theorem 3.1 for some matrices C3 , T2 , the following
equality is correct:

0 0
C3 B2 T2 = ,
0 ?1

where ?1 = diag (µ1 Eq , . . . , µa Eq ], µ1 = · · · = µa = 1. By this transformation algebra
K is left invariant. Applying Theorem 3.1 again we get that with k = m the matrix
A2 can be transformed into matrix
?2 0
,
0 0
?
On subalgebras of the Lie algebra of the extended Poincar? group P (1, n)
e 161

where ?2 is a square matrix of degree bq. For simplicity we shall assume that ?2 is
a coupling matrix of elementary spaces in the subdirect sum of the spaces V1 (1, bq)
and V2 (bq + 1, 2bq). One can admit that

K = diag [Ab , Ab ] = {diag [X, . . . , X |X ? A},
2b

where A is an irreducible subalgebra of the algebra AO(q). Since for every matrix
Y ? Ab the equality ?2 Y = Y ?2 takes place then ?2 = QS, where S is a symmetric
matrix, Q is an orthogonal matrix, and Y · Q = Q · Y . Applying the automorphism
diag [E, Q?1 ] we transform the coupling matrix ?2 into S. There exists such matrix
C ? O(bq) that

CSC ?1 = diag [?1 E(1) , ?2 E(2) , . . . , ?t E(t) ],

where ?i = ?j when i = j, and E(i) is the unit matrix (i, j = 1, . . . , t). The
automorphism diag [C, C] transforms K into diag [CAb C ?1 , CAb C ?1 ] and the coup-
ling matrix S into CSC ?1 . If Y ? CAb C ?1 then Y (CSC ?1 ) = (CSC ?1 )Y . Whence
Y = diag [Y1 , Y2 , . . . , Yt ], where deg Yi = deg E(i) . The further decomposition of
?? ?
the blocks Yi by O(2bq) automorphisms diag [C, C], where C = diag [C1 , . . . , Ct ],
deg Ci = deg E(i) does not change the coupling matrix. Since irreducible parts of an
algebra are defined uniquely then by the considered transformations of the coupling
matrix the algebra K is left invariant. That is why one can suppose that with k = m
?2 0
C3 A2 S ?1 + C3 B2 T1 = ,
0 0

where ?2 = diag [?1 Eq , . . . , ?b Eq ], 0 < ?1 ? · · · ? ?b , and (a + b)q = l or ?1 = · · · =
?b = 0 and aq = l. If B1 = 0 then for some C1 , T2 we have
0 0
C1 B1 T2 = ,
0 ?3

where ?3 = diag [Eq , . . . , Eq ].
The complete classification of coupling matrices one can get for large n.
Further we shall use the following notation:
M = P0 , M, P1 , . . . , Pn?1 , G1 , . . . , Gn?1 , m = [(n ? 1)/2],
m
?(n ? 1) = ?i J2i?1,2i |?i = 0, 1 ,
i=1

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( 145 .)



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