ñòð. 50 |

tions:

[Jab , Jcd ] = gad Jbc + gbc Jad ? gac Jbd ? gbd Jac , [Ga , Jbc ] = gab Gc ? gac Gb ,

[Pa , Jbc ] = gab Pc ? gac Pb , [Ga , Gb ] = 0, [Pa , Gb ] = ?ab M, [Ga , M ] = 0,

[Pa , M ] = 0, [Jab , M ] = 0, [R, S] = 2S, [R, T ] = ?2T, [T, S] = R,

[Z, R] = [Z, S] = [Z, T ] = [Z, Jab ] = 0, [R, Ga ] = Ga , [R, Pa ] = ?Pa ,

[R, M ] = 0, [R, Jab ] = 0, [S, Ga ] = 0, [S, Pa ] = ?Ga , [S, M ] = 0,

[S, Jab ] = 0, [T, Ga ] = Pa , [T, Pa ] = 0, [T, M ] = 0, [T, Jab ] = 0,

[Z, Ga ] = ?Ga , [Z, Pa ] = ?Pa , [Z, M ] = ?2M,

with a, b, c, d = 1, . . . , n ? 1.

From these commutation relations we ?nd that

R, S, T = ASL(2, R), R, S, T ? Z = AGL(2, R),

where R denotes the ?eld of real numbers.

Let F be a reducible subalgebra of AO(2, n + 1). That is, there exists in R2,n+1

a nontrivial subspace W which is invariant under F . If W is isotropic, then there

exists a totally isotropic subspace W0 ? W which is invariant under F . Since dim W0

is 1 or 2, then, by Witt’s theorem [14] there exists an isometry C ? O(2, n + 1) such

that CW0 is either Q1 + Qn+3 or Q1 + Qn+3 , Q2 + Qn+2 . Taking into account

that the matrices (3) do not change these subspaces and represent all the components

of the group O(2, n + 1) di?erent from the identity component O1 (2, n + 1), then

we may assume that the above C lies in O1 (2, n + 1), the identity component. Thus

there exists an inner automorphism ? of the algebra AO(2, n + 1) such that either

?

?(F ) ? AP (1, n) or ?(F ) ? AG4 (n ? 1).

If W is a nondegenerate subspace, then, by Witt’s theorem, it is isometric with

one of the following subspaces: R1,k (k ? 2), R2,k (k ? 1), Rk (k ? 1). Each of the

isometrics (3) leaves invariant each of these subspaces, so that we may assume that the

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

isometry which maps W onto one of these subspaces belongs to O1 (2, n+1). From this,

it follows that a subalgebra F is conjugate under the group of inner automorphisms

of the algebra AO(2, n + 1) to a subalgebra of one of the following algebras:

AO (1, k) ? AO (1, n ? k + 1),

(1)

where AO (1, k) = ?ab : a, b = 1, 3, . . . , k + 2 and

AO (1, n ? k + 1) = ?ab : a, b = 2, k + 3, . . . , n + 3 with n ? 3

and k = 2, . . . , [(n + 1)/2];

AO(2, k) ? AO(n ? k + 1), where

(2)

AO(n ? k + 1) = ?ab : a, b = k + 3, . . . , n + 3 with k = 0, 1, . . . , n.

In order to classify the subalgebras of these direct sums it is necessary to know

the irreducible subalgebras of algebras of the type AO(1, m) (m ? 2) and AO(2, m)

(m ? 3). It has been shown in Ref. [15] that AO(1, m) has no irreducible subalgebras

di?erent from AO(1, m). In Refs. [16] and [17] it has been shown that every semisimple

irreducible subalgebra of AO(2, m) (m ? 3) can be mapped by an automorphism of

this algebra onto one of the following algebras:

(1) AO(2, m);

ASU (1, (m/2)] when m is even;

(2)

v v

?12 + 3?13 + ?25 , ??15 + ?24 ? 3?23 , ?12 ? 2?45 when m = 3.

(3)

It follows then that when m > 3 is odd, the algebra AO(2, m) has no irreducible

subalgebras other than AO(2, m). If m = 2k and k ? 2, then, up to inner automor-

phisms, AO(2, m) has two nontrivial maximal irreducible subalgebras: ASU (l, k) ?

Y , and ASU (l, k) ? Y , where

Y = diag [J, ?J, J . . . , J]

Y = diag [J, . . . , J],

with

0 ?1

J= .

10

We note that a subalgebra L of AG4 (n?1) is conjugate under Ad AO(2, n+1) with

?

a subalgebra the algebra AP (1, n) if and only if the projection of L onto AGL(2, R) =

R, S, T ? Z is conjugate under Ad AGL(2, R) with a subalgebra of the algebra

R, T, Z .

Conjugacy under Ad AP (1, n) of subalgebras

3

of the Poincar? algebra AP (1, n)

e

The Poincar? group P (1, n) is the multiplicative group of matrices

e

? Y

,

0 1

On the classi?cation of subalgebras of the conformal algebra 225

where ? ? O(1, n) and Y ? Rn+1 . Let Iab , a, b = 0, 1, . . . , n + 1 be the (n + 2) ? (n + 2)

matrix whose entries are all zero except for the ab-entry, which is unity. Then a basis

for AP (1, n) is given by the matrices

J0a = ?I0a ? I0a , Jab = ?Iab + Iba , P0 = I0,n+1 , Pa = Ia,n+1 ,

with a < b; a, b = 1, . . . , n. These basis elements obey the commutation relations (1).

It is sometimes useful in calculations to identify elements of AO(1, n) with matrices

of the form

? ?

?02 · · · ?0n

0 ?01

? ?01 ?12 · · · ?1n ?

0

? ?

X = ? ?02 ??12 · · · ?2n ?

0

? ?

?· ·?

· · ·

?0n ??1n ??2n · · · 0

and elements of the space U = P0 , . . . , Pn are represented by n + 1-dimensional

columns Y . In this case, we take

?? ? ? ? ?

1 0 0

?0? ?1 ? ? ?

0

?? ? ? ? ?

P0 = ? . ? , P1 = ? . ?, . . . , Pn = ? ?

.

?.? ?. ? ? ?

.

. . .

0 0 1

and with this notation it is easy to see that [X, Y ] = XY . We endow the space U

with the metric of the pseudo-Euclidean space R1,n , so that the inner product of two

vectors

? ? ? ?

x0 y0

? x1 ? ? y1 ?

? ? ? ?

? . ?, ? . ?

?.? ?.?

. .

xn yn

is x0 y0 ? x1 y1 ? · · · ? xn yn . The projection of AP (1, n) onto AO(1, n) is denoted

by ?. We also note that AO(n), contained in AO(1, n), is generated by Jab (a < b;

?

a, b = 1, . . . , n).

Let B be a Lie subalgebra of the algebra AO(1, n) which has no invariant isotropic

subspaces in R1,n . Then B is conjugate under Ad AO(1, n) to a subalgebra of AO(n)

or to AO(1, k) ? C, where k ? 2 and C is a subalgebra of the orthogonal algebra

AO (n ? k) generated by the matrices Jab (a, b = k + 1, . . . , n). In the ?rst case, B is

not conjugate to any subalgebra of AO(n ? 1).

Proposition 1. Let B be a subalgebra of AO(n) which is not conjugate to a subalgebra

of AO(n ? 1). If L is a subalgebra of AP (1, n) and ?(L) = B, then L is conjugate to

?

an algebra W C, where W is a subalgebra of P1 , . . . , Pn , and C is a subalgebra of

B ? P0 . Two subalgebras W1 C1 and W2 C2 of this type are conjugate to each

other under Ad AP (1, n) if and only if they are conjugate under Ad AO(n).

Proof. The algebra B is a completely reducible algebra of linear transformations of

the space U and annuls only the subspace P0 (other than the null subspace itself).

Thus, by Theorem 1.5.3 [9], the algebra L is conjugate to an algebra of the form

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

W C where W ? P1 , . . . Pn and C ? B ? P0 . Now let W1 C1 , and W2 C2 be of

this form, conjugate under Ad AP (1, n). Then there exists a matrix ? ? P1 (1, n) such

that ?? (W1 C1 ) = W2 C2 , and from this it follows that ?? (B1 ) = B2 for some

? ? O1 (1, n). Let V = P1 , . . . , Pn . Since [B1 , V ] = V , then [B2 , ?? (V )] = ?? (V )

and ?? (V ) = V . Thus we can assume that ? = diag [1, ?1 ] where ?1 ? SO(n), so

that the given algebras are conjugate under Ad AO(n). The converse is obvious.

Proposition 2. Let B = AO(1, k) ? C, where k ? 2 and C ? AO (n ? k). If L

is a subalgebra of AP (1, n) and ?(L) = B then L is conjugate to L1 ? L2 where

?

L1 = AO(1, k) or L1 = AP (1, k), and L2 is a subalgebra of the Euclidean algebra

AE (n ? k) with basis Pa , Jab (a, b = k + 1, . . . , n). Two subalgebras of this form,

L1 ? L2 and L1 ? L2 are conjugate under Ad AP (1, n) if and only if L1 = L1 and L2

is conjugate to L2 under the group of E (n ? k)-automorphisms.

Proof. The proof is as in the proof of Proposition 1.

Lemma 1. If C ? O(1, n) and C(P0 + Pn ) = ?(P0 + Pn ) then ? = 0 and

? ?

1 + ?2 (1 + v 2 ) ?1 + ?2 (1 ? v 2 )

t

?v B

? ?

2? 2?

? ?

? ?,

v ?v

B (5)

C=? ?

? ?1 + ?2 (1 + v 2 ) ?

1 + ?2 (1 ? v 2 )

?v t B

2? 2?

where B ? B(n?1), v is an (n?1)-dimensional column vector, v 2 is the scalar square

of v and v t is the transpose of v. Conversely, every matrix C of this form satis?es

C(P0 + Pn ) = ?(P0 + Pn ).

Proof. Proof is by direct calculation.

Lemma 2. Let C ? O(1, n) have the form (5), with ? > 0. Then

C = diag [1, B, 1] exp[(? ln ?)J0n ] exp(??1 G1 ? · · · ? ?n?1 Gn?1 ),

where Ga = J0a ? Jan and

? ?

?1

?.? ?1

? . ? = B v.

.

?n?1

Proof. Direct calculation gives us

? ?

cosh ? 0 sinh ?

exp(??J0n ) = ? ?

0 En?1 0

sinh ? 0 cosh ?

and

? ?

b2 b2

b t

1+

? ?

2 2

? ?

exp(??1 G1 ? · · · ? ?n?1 Gn?1 ) = ? ?,

b ?b

En?1

? ?

? ?

b b2

2

b 1?

t

2 2

On the classi?cation of subalgebras of the conformal algebra 227

where b = (?1 , . . . , ?n?1 )t . On putting ? exp ? we have

?2 ? 1

?2 + 1

cosh ? = , sinh ? = .

2? 2?

Since we have

?2 ?? ?

?2 ? 1 b2 b2

? +1

b t

0 1+

? 2? 2? ? ? ?

2 2

? ?? ?

? ?? ?=

b ?b

0 En?1 0 En?1

? ?? ?

? ?2 ? 1 ?2 + 1 ? ? b2 ?

b2

b 1?

t

0

2? 2? 2 ?2

?

1 + ? (1 + b ) ?1 + ? (1 ? b )

2 2

2 2

?bt

? ?

2? 2?

? ?

=? ?,

b ?b

En?1

? ?

? ?

ñòð. 50 |