ñòð. 53 |

C4 = exp (S + T ) , C5 = exp(?(S + T ))

2

(see (3) and (10) for the notation).

Theorem 3. Let L1 and L2 be subalgebras of AG4 (n ? 1) which are not conjugate

under Ad AG4 (n ? 1) with subalgebras of

(AO(n ? 1) ? D, J0n

M, T, P1 , . . . , Pn?1

and

AO(n ? 1) ? S + T, Z .

Then the subalgebras L1 and L2 are conjugate under Ad AC(1, n) if and only if they

are conjugate under Ad AG4 (n ? 1).

236 L.F. Barannyk, P. Basarab-Horwath, W.I. Fushchych

Table 1. Action of automorphisms on elements of AG4 (n ? 1) for n ? 2.

Element of

Restrictions

? ?1 ? ?2 ?C1 ?C4 ?C5

AG4 (n ? 1)

?P1 ?P1 ?G1 ?P1

P1 K1

?Pa ?Ga ?Pa a = 2, . . . , n ? 1

Pa Ka Pa

K0 ? Kn 2T ?M

M M M

J01 + J1n ?(J01 + J1n ) ?G1

G1 G1 P1

J0a + Jan J0a + Jan ?Ga a = 2, . . . , n ? 1

Ga Ga Pa

?J1a a = 2, . . . , n ? 1

J1a J1a J1a J1a J1a

a, b = 2, . . . , n ? 1

Jab Jab Jab Jab Jab Jab

?R ?R

R Z R R

1

(K0 ? Kn ) ?S

S T T S

2

1

?T

T S M S T

2

?Z

Z R Z Z Z

Proof. If the subalgebras L1 and L2 are conjugate under Ad AG4 (n?1) then they are

conjugate under AdAC(1, n). Now suppose that they are conjugate under AdAC(1, n).

In order to prove their conjugacy under Ad AG4 (n ? 1) it is su?cient (by Lemma 14)

to show that for an arbitrary ? ? Ad AG4 (n ? 1) and for each matrix ? of the form

(10), the subalgebra ??(L1 )??1 either equals ?(L1 ) or is not contained in AG4 (n?1),

for then the only possibility is that they are conjugate under Ad AG4 (n ? 1).

If the projection of ?(L1 ) onto G1 , . . . , Gn?1 is nonzero, then, using Table 1, the

subalgebra ??(L1 )??1 contains an element Y whose projection for some a, 1 ? a ?

n?1 onto J0a , Jan is of the form ?(J0a +Jan ) with ? = 0. If ??(L1 )??1 ? AG4 (n?1),

then the projection of Y onto J0a , Jan would have the form µ(J0a ?Jan ) which would

imply ? = µ = ?µ = 0, an obvious contradiction.

Now let the projection of ?(L1 ) onto G1 , . . . , Gn?1 be zero. Denote by ? ?(L1 )

the projection of ?(L1 ) onto R, S, T . If ? ?(L1 ) = R, S, T , then R, S, T ? ?(L1 ).

From this it follows that ?2 ?(L1 )??1 is not a subset of AG4 (n ? 1). If we assume that

2

?1 ?(L1 )??1 ? AG4 (n ? 1), we obtain, from Table 1, that the projection of ?(L1 )

1

onto P1 , . . . , Pn , M is zero, and consequently we have either ?(L1 ) = R, S, T or

?(L1 ) = R, S, T ? Z . In this case, ?1 ?(L1 )??1 = ?(L1 ). If ? ?(L1 ) = R +

1

?1

?S, T + ?S , with ? = 0, then ?2 ?(L1 )?2 is not contained in AG4 (n ? 1). If we had

?1 ?(L1 )??1 ? AG4 (n ? 1), then the projection of ?(L1 ) onto P1 , . . . , Pn , M would

1

be zero. But then ?(L1 ) would be conjugate under Ad AG4 (n ? 1) with a subalgebra

of AO(n ? 1) ? R, T, Z , which contradicts the assumptions of the theorem. The

theorem is proved.

Theorem 4. Let L1 and L2 be subalgebras of the algebra

(AO(n ? 1) ? D, J0n )

L = M, T, P1 , . . . , Pn?1

having nonzero projection on J0n and D and are not conjugate under Ad L with

(AO(n ? 1) ? D, J0n ). Then L1 and L2 are

subalgebras of the algebra M, T

conjugate under Ad AC(1, n) if and only if they are conjugate under Ad L or if there

exists an automorphism ? ? Ad L such that ?(L1 ) = ?L2 ??1 where ? is one of the

matrices ?2 , C5 , ?2 C5 (see Table 1).

Proof. If ? ? Ad AG4 (n ? 1), then ? = ?C where C is a matrix of the form (9). By

theorem IV.3.4 of Ref. [9], the subalgebra L1 is, up to an automorphism of Ad AG4 (n?

On the classi?cation of subalgebras of the conformal algebra 237

1), one of the following algebras:

(U1 + U2 + U3 ) F, where U1 ? M , U2 ? T , U3 ? P1 , . . . , Pn?1

(1)

and F ? AO(n ? 1) ? D, J0n ;

(U1 + U2 ) F, where U1 ? T , U2 ? P1 , . . . , Pn?1

(2)

and F is a subalgebra of AO(n ? 1) ? R, M ;

(U1 + U2 ) F, where U1 ? M , U2 ? P1 , . . . , Pn?1

(3)

and F is a subalgebra of AO(n ? 1) ? Z, T .

By assumption, the projection of L1 onto P1 , . . . , Pn?1 is nonzero.

If ?(L1 ) = L2 , then in formula (9) ? = 0 or ? = ? because for other values

of ? the projection of ?(L1 ) onto G1 , . . . , Gn?1 is nonzero. For this reason, ?1 =

· · · = ?n?1 = 0 and so ? ? Ad L or ?C5 ? ? Ad L. Let there be automorphisms

?1 , ?2 ? Ad AG4 (n ? 1) with ??1 (L1 )? = ?2 (L2 ) where ? is one of the matrices (10).

If Ad L did not contain ?1 and ?C5 ?1 , then the projection of ?1 (L1 ) on G1 , . . . , Gn?1

would be nonzero, and so, by Table 1, ?2 (L2 ) would not be in AG4 (n ? 1). Thus ?j or

?C5 ?j belongs to Ad L for each j = 1, 2. For ? = ?1 the projection of ??1 (L1 )? onto

K1 , . . . , Kn?1 is nonzero, so we have ? = ?2 . In this case ??2 (L2 )? = ?2 (?L2 ?).

Using Lemma 14, the theorem is proved.

In a similar way, one proves the following results.

Theorem 5. Let B be a subalgebra of the algebra

(AO(n ? 1) ? D, T )

N = M, P1 , . . . , Pn?1

and let B have nonzero projection onto D . Then B is conjugate under Ad AC(1, n)

to the algebra

F = (W1 ? W2 ) (12)

E,

where E is a subalgebra of the algebra AO(n ? 1) ? D , W1 ? P1 , . . . , Pn?1 and W2

is one of the algebras 0, P0 , Pn , Pn , P0 , Pn . If W2 = Pn , or W2 = P0 , Pn

then the subalgebra W1 E is not conjugate under Ad AO(n ? 1) with any subalgebra

(AO(n ? 2) ? D ). Subalgebras F1 , F2 of the type (12) of the

of P1 , . . . , Pn?2

algebra N with nonzero projection onto D , which are not conjugate under Ad N to

(AO(n ? 1) ? D ), will be conjugate under AC(1, n) if and

subalgebras of M, T

only if they are conjugate under Ad L or when there exists an automorphism ? ? Ad L

with ?(F1 ) = ?2 F2 ??1 (see (10)), where L = AO(n ? 1) (we consider Ad AO(n ? 1)

2

to be a subgroup of Ad AC(1, n)).

Theorem 6. Let B be a subalgebra of the algebra

(AO(n ? 1) ? J0n , T )

N = M, P1 , . . . , Pn?1

and let B have nonzero projection onto J0n . Then B is conjugate under Ad AC(1, n)

with the algebra

(13)

F =W E,

(AO(n ? 1) ? J0n ) and W

where E is a subalgebra of the algebra P1 , . . . , Pn?1

is one of the algebras 0, M , P0 , Pn . Let L = N D . Subalgebras F1 , F2 of the

238 L.F. Barannyk, P. Basarab-Horwath, W.I. Fushchych

type (13) of the algebra N which are not conjugate under Ad N with subalgebras of

(AO(n ? 1) ? J0n , T ), will be conjugate under Ad AC(1, n) if and

the algebra M

only if they are conjugate under Ad L or if there exists an automorphism ? ? Ad L

with ?(F1 ) = ?F2 ??1 where ? is one of the matrices ?2 , C5 , ?2 C5 (see Table 1).

Theorem 7. Let L1 , L2 be subalgebras of the algebra L = M, S + T, Z ? AO(n ? 1)

which have nonzero projection onto S + T . The algebras L1 and L2 are conjugate

under Ad AC(1, n) if and only if they are conjugate under Ad L or if there exists an

automorphism ? ? Ad L such that ?(L1 ) = ?1 L2 ??1 (see Table 1).

1

Subalgebras of AC(1, 3)

6

We recall that in this article the conformal algebra AC(1, 3) is realized as the pseudo-

orthogonal algebra AO(2, 4). It turns out that it is convenient to divide the subalgeb-

ras of AO(2, 4) into seven classes:

(1) subalgebras not having invariant isotropic subspaces in R2,4 ;

(2) subalgebras conjugate to subalgebras of AG1 (2);

(3) subalgebras conjugate to subalgebras of AG1 (2) J03 and having nonzero

projection onto J03 ;

(4) subalgebras conjugate to subalgebras of AP (1, 3) but not conjugate to subalgeb-

ras of AG1 (2) J03 ;

(5) subalgebras conjugate to subalgebras of AG1 (2) J03 , D but not conjugate to

subalgebras of AG1 (2) J03 ;

(6) subalgebras conjugate to subalgebras of AP (1, 3) but not conjugate to subalgeb-

ras of AG1 (2) J03 , D ;

(7) subalgebras conjugate to subalgebras of AG4 (2) but not conjugate to subalgeb-

ras of AP (1, 3).

Since subalgebras conjugate under Ad AC(1, 3) are identi?ed, we omit mentioning

conjugacy when referring to classes. So, for instance, we shall consider the second class

as consisting of subalgebras of AG1 (2). In order to have a better survey of subalgebras

it is convenient to split the classes into subclasses corresponding to certain properties

of the projections of the subalgebras of a class onto the homogeneous part of the

algebra.

The division of the set of subalgebras of AC(1, 3) into the classes (1)–(7) allows

us easily to construct the set of subalgebras of each of the algebras AG1 (2), AP (1, 3),

?

AP (1, 3), AG4 (2). Up to conjugacy under Ad AC(1, 3) we have

(a) the set of subalgebras of AG1 (2) coincides with class (2);

(b) the set of subalgebras of AP (1, 3) is the union of classes (2), (3) and (4);

?

(c) the set of subalgebras of AP (1, 3) coincides with the union of classes (2)–(6);

(d) the set of subalgebras of AG4 (2) is the union of classes (2), (3), (5), and (7).

We use the notation F : U1 , . . . , Um for U1 F.

F, . . . , Um

On the classi?cation of subalgebras of the conformal algebra 239

A. Subalgebras not possessing invariant isotropic subspaces

in R2,4

This class is divided into subclasses by the existence for the subalgebras of invariant

irreducible subspaces of a particular kind in the space R2,4 .

1. Irreducible subalgebras of AO(2, 4)

AC(1, 3);

ASU (1, 2) = P0 + K0 + 2J12 , P0 + K0 + K3 ? P3 , P1 + K1 + 2J02 ,

P3 + K3 + K0 ? P0 , K2 ? P2 + 2J13 , P2 + K2 ? 2J01 ,

D + J03 , K1 ? P1 ? 2J23 ;

ASU (1, 2) = P0 + K0 ? 2J12 , P0 + K0 + K3 ? P3 , P1 + K1 ? 2J02 ,

P3 + K3 + K0 ? P0 , K2 ? P2 ? 2J13 , P2 + K2 + 2J01 ,

D + J03 , K1 ? P1 + 2J23 ;

ASU (1, 2) ? P0 + K0 ? 2J12 ? K3 + P3 ;

ASU (1, 2) ? P0 + K0 + 2J12 ? K3 + P3 ;

P0 + K0 ? 2J12 ? K3 + P3 ? P1 + K1 + 2J02 , P3 + K3 + K0 ? P0 ,

K2 ? P2 + 2J13 ;

P0 + K0 + 2J12 ? K3 + P3 ? P1 + K1 ? 2J02 , P3 + K3 + K0 ? P0 ,

K2 ? P2 ? 2J13 .

2. Irreducible subalgebras AO(1, 4)

AC(3).

3. Irreducible subalgebras of AO(2, 3)

AC(1, 2);

v v

P2 + K2 + 3(P1 + K1 ) + K0 ? P0 , D + J02 ? 3J01 , P0 + K0 ? 2(K2 ? P2 ) ;

v v

P2 + K2 ? 3(P1 + K1 ) + K0 ? P0 , D + J02 + 3J01 , P0 + K0 ? 2(K2 ? P2 ) .

4. Subalgebras of AO(2, 2) ? AO(2) with irreducible projection onto

AO(2, 2)

J01 ? D, K0 ? P0 ? P1 ? K1 , P0 + K0 ? K1 + P1 ?

? P0 + K0 + K1 ? P1 ? F, where F = 0 or F = J23 ;

J01 + D, K0 ? P0 + P1 + K1 , P0 + K0 + K1 ? P1 ?

? P0 + K0 ? K1 + P1 ? F, where F = 0 or F = J23 ;

AC(1, 1), AC(1, 1) ? J23 , where AC(1, 1) = P0 , P1 , K0 , K1 , J01 , D ;

J01 ? D, K0 ? P0 ? P1 ? K1 , P0 + K0 ? K1 + P1 ?

? P0 + K0 + K1 ? P1 + ?J23 (? = 0);

J01 + D, K0 ? P0 + P1 + K1 , P0 + K0 + K1 ? P1 ?

? P0 + K0 ? K1 + P1 + ?J23 (? = 0).

5. Subalgebras of the type AO(2, 1) ? F with F ? AO(3)

AC(1) ? L, where AC(1) = D, P0 , K0 ,

and L is one of the algebras: 0, J12 , J12 , J13 , J23 .

240 L.F. Barannyk, P. Basarab-Horwath, W.I. Fushchych

6. Subalgebras of AO(2) ? AO(4) having an irreducible projection

P0 + K0 ; P0 + K0 ? 2J12 + ?(K3 ? P3 ) (|?| ? 1);

P0 + K0 ? J12 , K3 ? P3 ; P0 + K0 ? J12 + J13 , J23 ;

P0 + K0 ? 2J12 + ?(K3 ? P3 ), 2J13 ? ?(K2 ? P2 ),

2J23 + ?(K1 ? P1 ) (? = ±1);

P0 + K0 ? 2J12 + ?(K3 ? P3 ), 2J13 ? ?(K2 ? P2 ), 2J23 + ?(K1 ? P1 ) ?

? 2J12 ? ?(K3 ? P3 ) (? = ±1);

P0 + K0 ? K1 ? P1 , K2 ? P2 , K3 ? P3 , J12 , J13 , J23 ;

P0 + K0 + 2?J12 (? = 0, |?| = 1);

P0 + K0 + 2?J12 + ?(K3 ? P3 ) (? = 0, |?| = 1, ? ? ?, ? = 1);

2J12 + ?(P0 + K0 ), K3 ? P3 + ?(P0 + K0 )

(? = 0, ? ? 0, with |?| = 1 when ? = 0);

?(P0 + K0 ) + 2?J12 ? K3 + P3 ? 2?J12 + K3 ? P3 , 2?J13 ? K2 + P2 ,

2?J23 + K1 ? P1 (? ? 0);

2?J12 + K3 ? P3 , 2?J13 ? K2 + P2 , 2?J23 + K1 ? P1 (? = ±1);

2?J12 + K3 ? P3 , 2?J13 ? K2 + P2 , 2?J23 + K1 ? P1 ?

? 2?J12 ? K3 + P3 (? = ±1);

K1 ? P1 , K2 ? P2 , K3 ? P3 , J12 , J13 , J23 .

7. Subalgebras of AO(1, 2) ? AO(1, 2)

P1 + K1 , P2 + K2 , J12 ? K0 ? P0 , K3 ? P3 , J03 ;

P1 + K1 + 2?J03 , P2 + K2 + K0 ? P0 , 2?J12 + K3 ? P3 (? = ±1);

P1 + K1 , P2 + K2 , J12 ? K3 ? P3 .

B. Subalgebras of AG1 (2)

The classical Galilei algebra AG1 (2) is the semidirect sum of a solvable ideal, generated

by P1 , P2 , M, T , and the Euclidean algebra AE(2) = G1 , G2 , J12 . The projection

of AG1 (2) onto AO(1, 3) coincides with AE(2), which has, up to inner automorphi-

sms, the subalgebras 0, J12 , G1 , G1 , G2 , G1 , G2 , J12 . The ?rst two subalgebras

are completely reducible algebras of linear transformations of Minkowski space R1,3 ,

whereas the others are not of this type. Thus we divide this class into two subclasses A

and B.

ñòð. 53 |