ñòð. 41 |

(5.1)

?n + b1 ? ?bn = 0, ?i ? ?bi = 0 (i = 2, . . . , n ? 1).

The determinant of coefficients by b0 , b1 , and bn is equal to ?? 3 . Since ? = 0 then

the systems (5.1) has a solution. Therefore one can assume that X1 = G1 + ?D. Let

a = 1,

n

Xa = Ga + ?µ Pµ + ?D.

µ=0

Since

[X1 , Xa ] = ?(?0 ? ?n )P1 ? ?1 M + ? ?µ Pµ ,

[X1 , Xa ] ? ?Xa = ??Ga ? ??D ? (?0 ? ?n )P1 ? ?1 M,

we shall assume that

Xa = Ga + ?M + ?P1 + ?D.

Then

[X1 , Xa ] = (?? ? ?)M + ??P1 (2 ? a ? k).

If ?? ? ? = 0 then we shall consider that ? = 0, ? = 0. Since

[X1 , [X1 , Xa ]] = ?2??M + ? 2 ?P1 ,

? ?

then F containts M ??P1 , ?2M +?P1 and whence M, P1 ? F . That is why Ga +?D ?

?

F.

?

Let ?? ? ? = 0. If ? = 0 then P1 ? F . Since [X1 , P1 ] = [G1 + ?D, P1 ] = ?M + ?P1

? ?

then M ? F and therefore Ga + ?D ? F . If ? = 0 then ? = 0. It proves that F is a

splitting algebra. The proposition is proved.

?

On subalgebras of the Lie algebra of the extended Poincar? group P (1, n)

e 167

The record F : W1 , . . . , Ws means that we deal with the subalgebras W1 ? , . . .,

+F

Ws ? .

+F

In virtue of Propositions 4.4 and 5.1 we conclude that the subalgebras of the

algebra M ? possessing a nonzero projection onto D are exhausted with respect

+D

? (1, n) conjugation by the following algebras [see notations (4.2)]:

to P

D : 0, ?(i), V2 (s, t) (i = 0, 1, . . . , n ? 1, s = 0, 1, t = s, s + 1, . . . , n);

G1 + ?1 D, . . . , Gk + ?k D, ?D : 0, ?(i), ?(k), V2 (k + 1, t),

?(i) ? V2 (k + 1, t), ?(k) ? V2 (k + 1, t), ?(r) ? ?r+1,k+1 (j),

?(r) ? ?r+1,k+1 (j) ? V2 (k + j + 1, s) (k = 1, . . . , n ? 1, i = 0, 1, . . . , k,

t = k + 1, . . . , n ? 1, r = 0, 1, . . . , k ? 1, j = 1, . . . , k ? r,

s = k + j + 1, . . . , n ? 1).

These algebras must then be simplified using transformations contained in the nor-

malizer of each algebra in the group of O(1, n) automorphisms. If, for example, the

normalizer contains exp(?J12 ) then instead of G1 + ?1 D, G2 + ?2 D we can take

G1 + ?1 D, G2 .

Proposition 5.2. Let L be a subalgebra of AO(n), and F be the subdirect sum of L

?

and D . The algebra F possesses only the splitting extensions in AP (1, n).

Proposition 5.2 is proved by virtue of Propositions 2.1 and 3.2.

Proposition 5.3. Let L1 be a subalgebra of AO(n ? 1), L2 = D, J0n or L2 =

D + ?J0n , where ? = 0, ? 2 = 0, 2? + 1 = 0. If F is a subdirect sum of the

?

algebras L1 and L2 then every subalgebra F of the algebra K with the property

?

?(F ) = F is conjugated to the algebra (W1 + W2 ) ? , where W1 ? U , W2 ? V1 =

+F

G1 , . . . , Gn?1 .

?

Proof. Let L2 = D, J0n . On the basis of Propositions 2.2 and 5.1 algebra F contains

the elements

n n?1

X2 = D +

X1 = J0n + ?i Pi , ?j Gj .

i=0 j=1

?

?i Pi ? ?j Gj then D + ?i Pi ? F . Therefore one can suppose

Since [X1 , X2 ] =

? ? ?

that D ? F . Whence J0n ? F and F ? F .

Let L2 = D + ?J0n . Since [D + ?J0n , Pa ] = Pa , [D + ?J0n , Ga ] = ??Ga (a =

?

1, . . . , n ? 1), then by virtue of Proposition 5.2 one can admit that F contains the

subdirect sum of F and subalgebra of the algebra P0 , Pn . Evidently

exp(?0 P0 + ?n Pn ) · (D + ?J0n + ?0 P0 + ?n Pn ) · exp(??0 P0 ? ?n Pn ) =

= D + ?J0n + (?0 ? ?0 + ??n )P0 + (?n + ??0 ? ?n )Pn .

Since ? 2 = 1, then coefficients by P0 , Pn can be transformed into zero. On the basis

? ?

of the conditions ? 2 = 1, [D + ?J0n , F ? M] ? F ? M it is not difficult to get that

?

F ? F.

?

Let W = F ? M, Y = ?a Ga + ?i Pi ? W . Since

[D + ?J0n , Y ] = ?? ?a Ga ? ?(?0 Pn + ?n P0 ) + ?i Pi

168 L.F. Barannik, W.I. Fushchych

and ? 2 = 1 then one can assume that Y = ?a Ga + ?0 P0 + ?n Pn . By the direct

calculations we find that

[D + ?J0n , Y ] = ?? ?a Ga + (?0 ? ??n )P0 + (?n ? ??0 )Pn ,

[D + ?J0n , [D + ?J0n , Y ]] =

?a Ga + (? 2 ? ? 2??n + ?0 )P0 + (? 2 ?n ? 2??0 + ?n )Pn .

= ?2

The determinant ? constructed by the coefficients of ?a Ga , P0 , Pn in Y and the

vectors received is equal to ?(2? + 1)(?n ? ?0 ). If ? = 0 then ?a Ga , P0 , Pn ? W .

2 2

If ? = 0 then ?n = ±?0 . When ?n = ?0 we get that

[D + ?J0n , Y ] ? (1 ? ?)Y = ? ?a Ga .

If ?n = ??0 then

[D + ?J0n , Y ] ? (1 + ?)Y = (?2? ? 1) ?a Ga .

The proposition is proved.

Proposition 5.4. The subalgebras of the algebra M ? J0n , D containing J0n or

+

having the property that their projection F onto J0n , D coincides with D + ?J0n ,

?

where ? = 0, ? 2 = 1, 2? + 1 = 0, are exhausted with respect to P (1, n) conjugation

by the following algebras [see notation (4.2)]:

F : 0, ?(a), ?(a), V2 (1, d) (a = 0, 1, . . . , n ? 1, d = 1, . . . , n ? 1);

(G1 , . . . , Gk ) ? : 0, ?(i), ?(k), V2 (k + 1, t), ?(i) ? V2 (k + 1, t),

+F

?(k) ? V2 (k + 1, t), ?(r) ? ?r+1,k+1 (j), ?(r) ? ?r+1,k+1 (j) ? V2 (k + j + 1, s)

(i = 0, 1, . . . , k, t = k + 1, . . . , n ? 1, r = 0, 1, . . . , k ? 1, j = 1, . . . , k ? r,

s = k + j + 1, . . . , n ? 1, k = 1, . . . , n ? 1).

The proof of Proposition 5.4 is based on Proposition 5.3.

Proposition 5.5. Let L1 be a subalgebra of AO(n ? 1), L2 = 2D ? J0n , F a

? ?

subdirect sum of L1 and L2 , and F such subalgebra of K that ?(F ) = F . The

?

algebra F is conjugated to the algebra W ? , where W ? M and satisfies the

+F

following condition: if Y ? W and projection of Y onto V1 = G1 , . . . , Gn?1 is

?a Ga + ?(P0 ? Pn ).

equal to ?a Ga then W contains ?a Ga + ?P0 and ?M or

Proposition 5.6. Let L1 be a subalgebra of AO(n ? 1), L2 = D + J0n + ?M

(? ? {0, 1}), and F the subdirect sum of L1 and L2 . If a subspace W of the space

M is invariant under F then W = W1 + W2 , where W1 ? U , W2 ? V1 .

The proof of Propositions 5.5 and 5.6 is similar to that of Proposition 5.3.

Let ? = (?0 ? ?n )/2. Since

1

exp(?P0 ) · (D + J0n + ?0 P0 + ?n Pn ) · exp(??P0 ) = D + J0n + (?0 + ?n )M,

2

? ?

then further we shall suppose that the projection of the algebra F ? AP (1, n) onto

D + J0n , P0 , Pn contains D + J0n + ?M , where ? ? {0, 1}. Proposition 5.6 gives the

considerable information on the structure of such algebras.

?

On subalgebras of the Lie algebra of the extended Poincar? group P (1, n)

e 169

Proposition 5.7. Let L1 be a subalgebra of AO(n ? 1), L2 = D ? J0n + ?P0

(? ? {0, 1}), and F the subdirect sum of the algebras L1 and L2 . If a subspace W of

the space M is invariant under F , then W contains its own projection onto P0 , Pn

and [L1 , W ] ? W , [?P0 , W ] ? W .

?

Proof. On the basis of Proposition 4.3 [Li , W ] ? W (i = 1, 2). Let M = {Y ?

?

?

M|[L1 , Y ] = 0}, and W be a projection of W onto M. It is easy to see that the matrix

diag [2, 0] is the matrix of the operator D ? J0n in the basis P0 + Pn , P0 ? Pn of

the space P0 , Pn and in the basis of the space M| P0 , Pn the matrix of the same

?

operator is the unit one. Whence by Lemma 3.1 we conclude that W contains its own

projection onto P0 , Pn . It remains for us to note that for arbitrary

n?1

Y= (?j Pj + ?j Gj )

j=1

we have [D ? J0n + ?P0 , Y ] = Y + [?P0 , Y ]. The proposition is proved.

Proposition 5.8. Let F be a subalgebra of the algebra AO(1, n) generated by J0n

?

and Ga , where a runs through some subset I of the set {1, 2, . . . , n ? 1}. If F

?

is a subalgebra of AP (1, n) with ?(F ) = F , then within the conjugation with

?

respect to the group of translations the algebra F contains elements Ga (a ? I) and

J0n + ?i Pi (i = 1, . . . , n ? 1).

Proposition 5.9. Let L be a subalgebra of the algebra AP (1, n), X = Jab + ?J0n +

?Pc , Y = Gc + ?i Pi (i = 1, . . . , n), where ? = 0, ? = 0, and a, b, and c are

different numbers of {1, 2, . . . , n ? 1}. If X, Y ? L then L contains Gc .

Theorem 5.1. Let L be an Abelian subalgebra of the algebra K and ?(L) = 0. If

?

?(L) = J0n then L is P (1, n) conjugated to the subdirect sum of algebras L1 , L2 ,

J0n , where L1 ? AH(2d), L2 = 0, or L2 = P2d+1 , . . . , P2d+s . If ?(L) = D then

?

L is P (1, n) conjugated to the subdirect sum of L1 , L2 , D , where L1 ? AH(2d),

L2 = 0 or L2 = G2d+1 , . . . , G2d+s . If ?(L) = D, J0n or ?(L) = D + ?J0n , where

?

? = 0, ? 2 = 1 then L is P (1, n) conjugated to the subdirect sum of algebras ?(L)

and L1 ? AH(2d). If ?(L) = D + J0n , then L is conjugated to the subdirect sum of

the algebras L1 , L2 , L3 , where L1 ? AH(2d), L2 ? M , L3 = J0n + D .

Proof. If ?(L) = J0n then in view of Propositions 2.2 and 4.3 the algebra L

contains its own projection onto M, P0 ? Pn , G1 , . . . , Gn?1 . Since [J0n , Ga ] = ?Ga ,

[J0n , M ] = ?M , [J0n , P0 ? Pn ] = P0 ? Pn then this projection is equal to zero.

Therefore L is the subdirect sum of L1 ? AH(2d) and L2 ? P2d+1 , . . . , Pn?1 . If

L2 = 0 then by Witt’s theorem L2 is conjugated to P2d+1 , . . . , P2d+s .

If ?(L) = D then in virtue of Propositions 4.3 and 5.2 the projection of L onto

U is equal to 0.

If ?(L) = D, J0n or ?(L) = D + ?J0n , where ? = 0, ? 2 = 1, 2? + 1 = 0, then by

Proposition 5.3 the algebra L is conjugated to the subdirect sum of the algebras ?(L)

and L1 ? AH(2d). With ?(L) = 2D ? J0n Proposition 5.5 is applicable.

Let ?(L) = D ? J0n . On the basis of Propositions 2.2 and 4.3 the projection of L

onto G1 , . . . , Gn?1 is equal to 0. Applying the O(1, n) automorphism corresponding

to the matrix diag [1, . . . , 1, ?1] we get that ?(L) = D + J0n . According to Proposi-

tion 5.2 the projection of L onto P1 , . . . , Pn?1 is equal to 0. Since [J0n +D, P0 +Pn ] =

0, [J0n + D, P0 ? Pn ] = 2(P0 ? Pn ) then by Propositions 2.1 and 4.3 the projection of

L onto P0 , Pn belongs to P0 + Pn . The theorem is proved.

170 L.F. Barannik, W.I. Fushchych

Corollary 1. The maximal Abelian subalgebras of the algebra K with the condi-

?

tion ?(K) = 0 are exhausted with respect to P (1, n) conjugation by the following

algebras:

AH(n ? 1) ? J0n , D , AH(n ? 1) ? M, J0n , D ,

AH(2d) ? P2d+1 , . . . , Pn?1 , J0n ,

AH(2d) ? G2d+1 , . . . , Gn?1 , D (d = 0, 1, . . . , [(n ? 2)/2]).

The written algebras are not conjugated mutually.

Corollary 2. Let n ? 3, Xt = ?1 J12 + ?2 J34 + · · · + ?t J2t?1,2t ; ?1 = 1, 0 < ?2 ?

· · · ? ?t ? 1; t = 1, . . . , [(n?1)/2]; s = 1, . . . , [(n?2)/2]; ? > 0. The one-dimensional

subalgebras of the algebra K with the condition ?(K) = 0 are exhausted with respect

?

to P (1, n) conjugation by the following algebras:

J0n ; D ; D + ?J0n ; J0n + P1 ; D + G1 ; D + J0n + M ;

Xt + ?D + ?J0n (? ? 0); Xt + ?J0n ; Xt + ?(D + J0n + M ) ;

Xt + G2s+1 + ?D ; Xs + P2s+1 + ?J0n .

The written algebras are not conjugated mutually.

?

Proposition 5.10. The one-dimensional subalgebras of the algebra P (1, n) are ex-

?

hausted wtih respect to the P (1, n) conjugation by the one-dimensional subalgebras

of the algebra K and the following algebras:

J12 + ?1 J34 + · · · + ?n/2?1 Jn?1,n + ?D ,

J12 + ?1 J34 + · · · + ?n/2?1 Jn?1,n + P0 ,

where n ? 0 (mod 2), ? ? 0, 0 < ?1 ? · · · ? ?n/2?1 ? 1.

?

6. Subalgebras of the algebras AP (1, 4)

In this section we make use of the previous results to provide a classification of

? ?

all subalgebras of AP (1, 4) with respect to P (1, 4) conjugation.

? ? ? ?

Let F be an subalgebra of AP (1, 4) such that ?(F ) = F . An expression F + W

?

means that W is a subspace of U , [F, W ] ? W , and F ? U ? W . As concerns the

? ? ?

algebras F + W1 , . . . , F + Ws we will use the notation F : W1 , . . . , Ws .

ñòð. 41 |