ñòð. 49 |

P0 = ?x0 , P1 = x1 ?x0 , 1

And what is more, there is one new representation of the Lie algebra of the

extended Poincar? group AP (1, 1), where the basis operators P0 , P1 , J01 are of the

e

form (21) and the generator of dilations reads D = x0 ?x0 + ??u , ? = 0, 1.

In [10], we have studied realizations of the Poincar? algebras AP (n, m) with n +

e

m ? 2 by LVFs in the space with n + m independent and one dependent variables.

It was established, in particular, that, provided the generators of translations Pµ ,

218 R.Z. Zhdanov, W.I. Fushchych

µ = 0, 1, . . . , n+m?1 can be reduced to the form Pµ = ?xµ , each representation of the

algebra AP (n, m) with n + m > 2 is equivalent to the standard linear representation

Jµ? = gµ? x? ?x? ? g?? x? ?xµ ,

Pµ = ?xµ ,

where

?

? 1, µ = ? = 1, . . . , n,

?1, µ = ? = n + 1, . . . , m,

gµ? =

?

0, µ=?

and the summation over the repeated indices from 0 to n + m is understood. In view

of the results obtained in the present paper, it is not but natural to assume that if

there will be no additional constraints on basis elements Pµ , then new representations

will be obtained. Investigation of this problem is in progress now and will be reported

elsewhere.

3. Conclusions

Our search for new representations of the Galilei algebra and its extensions was moti-

vated not only by an aspiration to a completeness (which is very important) but also

by a necessity to have new Galilei-invariant equations. Since the representations of

the groups G(1, 1), G1 (1, 1), G2 (1, 1) obtained in the present paper are in most cases

nonlinear in the ?eld variables u, v, PDEs admitting these will be principally di?erent

from the standard Galilei-invariant models used in quantum theory. Nevertheless, bei-

ng invariant under the Galilei group and, consequently, obeying the Galilei relativistic

principle, they ?t into the general scheme of selecting admissible quantum mechanics

models.

Furthermore, (1+1)-dimensional PDEs having extensive symmetries are the most

probable candidates to the role of integrable models. A peculiar example is the seven-

parameter family of the nonlinear Schr?dinger equations suggested by Doebner and

o

Goldin [11]. As established in [12] in the case when the number of space variables

is equal to one, all subfamilies with exceptional symmetry are either linearizable

or integrable by quadratures. Another example is the Eckhaus equation which is

invariant under the generalized Galilei group (see, e.g., [8]) and is linearizable by a

contact transformation [13].

But even in the case where a Galilei-invariant equation can not be linearized or

integrated in some way, one can always utilize the symmetry reduction procedure

[1, 2, 4] to obtain its exact solutions. And the wider is a symmetry group admitted by

the PDE considered, the more e?cient is an application of the mentioned procedure

(for more details see [4]).

Thus, PDEs invariant under the Galilei group G(1, 1) and its extensions possess

a number of attractive properties and certainly deserve a detailed study. We intend

to devote one of our future publications to construction and investigation of PDEs

invariant under the groups G(1, 1), G1 (1, 1), G2 (1, 1) having the generators given in

Theorems 1–3.

Acknowledgments. One of the authors (R.Zh.) is supported by the Alexander

von Humboldt Foundation. The authors are thankful to Victor Lahno and Roman

Cherniha for critical reading the manuscript and valuable comments.

On new representations of Galilei groups 219

1. Ovsjannikov L.V., Group analysis of di?erential equations, Moscow, Nauka, 1978.

2. Olver P.J., Applications of Lie group to di?erential equations, New York, Springer, 1986.

3. Barut A., Raczka R., Theory of group representations and applications, Warszawa, Polish Sci-

enti?c Publishers, 1980.

4. Fushchych W.I., Shtelen W.M., Serov N.I., Symmetry analysis and exact solutions of nonlinear

equations of mathematical physics, Kyiv, Naukova Dumka, 1989 (translated into English by

Kluwer Academic Publishers, Dordrecht, 1993).

5. Fushchych W.I, Cherniha R.M., Galilei-invariant nonlinear equations of Schr?dinger-type and

o

their exact solutions I, Ukrain. Math. J., 1989, 41, 1349–1357.

6. Fushchych W.I., Cherniha R.M., Galilei-invariant nonlinear equations of Schr?dinger-type and

o

their exact solutions II, Ukrain. Math. J., 1989, 41, 1687–1694.

7. Fushchych W.I., Cherniha R.M., Galilei-invariant nonlinear systems of evolution equations,

J. Phys. A: Math. Gen., 1995, 28, ¹ 19, 5569–5579.

8. Rideau G., Winternitz P., Evolution equations invariant under two-dimensional space-time

Schr?dinger group, J. Math. Phys., 1993, 34, ¹ 2, 558–570.

o

9. Rideau G., Winternitz P., Nonlinear equations invariant under the Poincar?, similitude and

e

conformal groups in two-dimensional space-time, J. Math. Phys., 1990, 31, ¹ 5, 1095–1106.

10. Fushchych W.I., Zhdanov R.Z., Lahno V.I., On linear and nonlinear representations of the

generalized Poincar? groups in the class of Lie vector ?elds, J. Nonlinear Math. Phys., 1994, 1,

e

¹ 3, 295–308.

11. Doebner H.-D., Goldin G., Properties of nonlinear Schr?dinger equation associated with diffeo-

o

morphism group representation, J. Phys. A: Math. Gen., 1994, 27, 1771–1780.

12. Nattermann P., Zhdanov R.Z., On integrable Doebner–Goldin equations, J. Phys. A: Math.

Gen., 1996, 29, ¹ 11, 2869–2886.

13. Calogero F., Xiaoda J., C-integrable nonlinear partial di?erential equations, J. Math. Phys.,

1991, 32, ¹ 4, 875–888.

W.I. Fushchych, Scienti?c Works 2004, Vol. 6, 220–246.

On the classi?cation of subalgebras

of the conformal algebra with respect

to inner automorphisms

L.F. BARANNYK, P. BASARAB-HORWATH, W.I. FUSHCHYCH

We give a complete justi?cation of the classi?cation of inequivalent subalgebras of the

conformal algebra with respect to the inner automorphisms of the conformal group,

and we perform the classi?cation of the subalgebras of the conformal algebra AC(1, 3).

1 Introduction

The necessity of classifying the subalgebras of the conformal algebra is motivated by

many problems in mathematics and mathematical physics [1, 2]. The conformal algeb-

?

ra AC(1, n) of Minkowski space R1,n contains the extended Poincar? algebra AP (1, n)

e

and the full Galilei algebra AG4 (n ? 1) (also known as the optical algebra). The

classi?cation of the subalgebras of the conformal algebra AC(l, n) is almost reducible

?

to the classi?cation of the subalgebras of the algebras AP (1, n) and AG4 (n ? 1).

Patera, Winternitz and Zassenhaus [1] have given a general method for the classi-

?cation of the subalgebras of inhomogeneous transformations. Using this method, the

?

classi?cation of the subalgebras AP (1, n), AP (1, n), and AG4 (n ? 1) was carried out

in Refs. [1–9] for n = 2, 3, 4. In Refs. [7–11], this general method was supplemented by

many structural results which made possible the algorithmization of the classi?cation

of the subalgebras of the Euclidean, Galilean, and Poincar? algebras for spaces of arbi-

e

trary dimensions. Indeed, this was done in Refs. [9] and [10], where the subalgebras

of AC(1, n) were classi?ed up to conjugation under the conformal group C(1, n) for

n = 2, 3, 4.

In order to perform the symmetry reduction of di?erential equations, it is necessary

to identify the subalgebras of the symmetry algebra (of the equation) which give

the same systems of basic invariants. This observation has led to the introduction

in Ref. [12] of the concept of I-maximal subalgebras: a subalgebra F is said to be

I-maximal if it contains every subalgebra of the symmetry algebra with the same

invariants as F . In Ref. [13], all I-maximal subalgebras of AC(1, 4), classi?ed up to

C(1, 4)-conjugation, were found in the representation de?ned on the solutions of the

eikonal equation. Using these subalgebras, reductions of the eikonal and Hamilton–

Jacobi equations to di?erential equations of lower order were obtained in Refs. [9]

and [12]. We note that the list of I-maximal subalgebras for a given algebra can di?er

according to the equation being investigated.

In the above works, the question of the connection between conjugation of the

? ? ?

subalgebras of the algebra AP (1, n) under the group P (1, n) (or the group Ad AP (1, n)

?

of inner automorphisms of the algebra AP (1, n)) and the conjugacy of these subal-

gebras under the group C(1, n) was not dealt with. This, and the same problem for

J. Math. Phys., 1998, 39, ¹ 9, P. 4899–4922.

On the classi?cation of subalgebras of the conformal algebra 221

subalgebras of the Galilei algebra AG4 (n?1), is the problem we address in the present

article.

Since the group analysis of di?erential equations is of a local nature, we concentrate

on conjugacy of the subalgebras under the group of inner automorphisms of the

algebra AC(1, n). Going over to conjugacy under C(1, n) is not complicated, and

requires only a further identi?cation of the subalgebras under the action of at most

three discrete symmetries. The results of this paper allow us to obtain a full classi?-

cation of the subalgebras of AC(1, n) for low values of n. On the basis of these results,

we give at the end of this paper a classi?cation of the algebra AC(1, 3) with respect

to its group of inner automorphisms. The list of subalgebras obtained in this way can

be used for the symmetry reduction of any system of di?erential equations which are

invariant under AC(1, 3).

2 Maximal subalgebras of the conformal algebra

We denote by Ad L the group of inner automorphisms of the Lie algebra L. Unless

otherwise stated, conjugacy of subalgebras of L means conjugacy with respect to the

group Ad L. We consider Ad L1 as a subgroup of Ad L2 whenever L1 is a subalgebra

of L2 . The connected identity component of a Lie group H is denoted by H1 .

Let R1,n (n ? 2), be Minkowski space with metric g?? , where (g?? ) = diag [1, ?1,

. . . , ?1] and ?, ? = 0, 1, . . . , n. The transformation de?ned by the equations

x? = x? (y0 , y1 , . . . , yn ), ? = 0, 1, . . . , n

of a domain U ? R1,n into R1,n , is said to be conformal if

?xµ ?x? µ?

g = ?(x)g?? ,

?y ? ?y ?

where ?(x) = 0 and x = (x0 , x1 , . . . , xn ). The conformal transformations of R1,n

form a Lie group, the conformal group C(1, n). The Lie algebra AC(1, n) of the group

C(1, n) has as its basis the generators of pseudorotations J?? , the translations P? , the

nonlinear conformal translations K? , and the dilatations D, where ?, ? = 0, 1, . . . , n.

These generators satisfy the following commutation relations:

[J?? , J?? ] = g?? J?? + g?? J?? ? g?? J?? ? g?? J?? ,

[P? , J?? ] = g?? P? ? g?? P? , [P? , P? ] = 0, [K? , J?? ] = g?? K? ? g?? K? ,

(1)

[K? , K? ] = 0, [D, P? ] = P? , [D, K? ] = ?K? , [D, J?? ] = 0,

[K? , P? ] = 2(g?? D ? J?? ).

The pseudo-orthogonal group O(2, n+1) is the multiplicative group of all (n+3)?

(n + 3) real matrices C satisfying C t E2,n+1 C = E2,n+1 , where E2,n+1 = diag [1, 1, ?1,

. . . , ?1]. We denote by Iab the (n + 3) ? (n + 3) matrix whose entries are zero except

for 1 in the (a, b) position, with a, b = 1, 2, . . . , n + 3. The Lie algebra AO(2, n + 1) of

O(2, n + 1) has as its basis the following operators:

?12 = I12 ? I21 , ?ab = ?Iab + Iba (a < b; a, b = 3, . . . , n + 3),

?ia = ?Iia ? Iai (i = 1, 2; a = 3, . . . , n + 3),

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

which satisfy the commutation relations

[?ab , ?cd ] = ?ad ?bc + ?bc ?ad ? ?ac ?bd ? ?bd ?ac (a, b, c, d = 1, 2, . . . , n + 3),

where (?ab ) = E2,n+1 . Let us denote by R2,n+1 the pseudo-Euclidean space of n + 3

dimensions with metric ?ab . The matrices of the group O(2, n + 1) and the algebra

AO(2, n + 1) will be identi?ed with operators acting on the left on R2,n+1 . Then, with

this convention, O(2, n + 1) is the group of isometries of R2,n+1 .

It is known (see for instance Ref. [9]) that there is a homomorphism ? : O(2, n +

1) > C(1, n) with kernel {±En+3 }, where {En+3 } is the unit (n + 3) ? (n + 3) matrix.

Thus we are able to identify O(2, n + 1) with C(1, n). This homomorphism of groups

induces an isomorphism f of the corresponding Lie algebras, f : AO((2, n + 1) >

AC(1, n), which is given by

f (??+2,?+2 ) = J?? , f (?1,?+2 ? ??+2,n+3 ) = P? ,

f (?1,?+2 + ??+2,n+3 ) = K? , f (?1,n+3 ) = ?D (?, ? = 0, 1, . . . , n).

We shall in this article identify the two algebras, using this isomorphism, so that we

can write the previous equations as

??+2,?+2 = J?? , ?1,?+2 ? ??+2,n+3 = P? ,

?1,?+2 + ??+2,n+3 = K? , ?1,n+3 = ?D (? < ?; ?, ? = 0, 1, . . . , n).

We shall use the matrix realization of the conformal algebra.

Each matrix C which belongs to the identity component O1 (2, n + 1) of the group

O(2, n + 1) is a product of matrices which are rotations in the x1 x2 and xa xb planes

(a < b; a, b = 3, . . . , n + 3) and hyperbolic rotations in the xi xa planes (i = 1, 2;

a = 3, . . . , n + 3). Thus each such matrix C can be given as a ?nite product of

matrices of the form exp X, where X ? AO(2, n + 1). From this, it follows that each

inner automorphism of the algebra AO(2, n + 1) is a mapping

?C : Y > CY C ?1 , (2)

where Y ? AO(2, n + 1) and C ? O1 (2, n + 1), and conversely each mapping of this

type is an inner automorphism of the algebra AO(2, n + 1).

In the process of our investigation mappings of the above type (2) will occur for

certain matrices C ? O(2, n + 1), so we call these types of mappings O(2, n + 1)-

automorphisms of the algebra AO(2, n + 1) corresponding to the matrix C.

If G is the group of O(2, n + 1)-automorphisms of the algebra AO(2, n + 1), and

H is the subgroup of G consisting of its inner automorphisms, then H is normal in

G and [G : H] ? 4. Representatives of the cosets of G/H di?erent from the identity

will be

C1 = diag [?1, 1, . . . , 1, ?1], C2 = diag [1, 1, ?1, 1 . . . , 1],

(3)

C3 = diag [?1, 1, ?1, 1, . . . , 1, ?1],

or

C1 = diag [?1, 1, . . . , 1, ?1, 1], C2 = diag [1, 1, ?1, 1 . . . , 1],

(4)

C3 = diag [1, ?1, ?1, 1, . . . , 1, ?1, 1].

On the classi?cation of subalgebras of the conformal algebra 223

Given a subspace V of R2,n+1 , there is a maximal subalgebra of AO(2, n + 1)

which leaves V invariant. We call this algebra the normalizer in AO(2, n + 1) of the

subspace V .

Let Q1 , . . . , Qn+3 be a system of unit vectors in R2,n+1 . Then the normalizer in

AO(2, n + 1) of the isotropic subspace Q1 + Qn+3 is the extended Poincar? algebra

e

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

AP (1, n) = P0 , P1 , . . . , Pn

where denotes semidirect sum, and ? denotes direct sum of algebras; AO(1, n) =

J?,? : ?, ? = 0, 1, . . . , n . The normalizer in AO(2, n + 1) of the completely isotropic

subspace Q1 + Qn+3 , Q2 + Qn+2 is the ful1 Galilei algebra

AG4 (n ? 1) = M, P1 , . . . , Pn?1 , G1 , . . . , Gn?1 (AO(n ? 1)? R, S, T ? Z ),

where

M = P0 + Pn , Ga = J0a ? Jan (a = 1, . . . , n ? 1), R = ?(J0n + D),

1 1

S = (K0 + Kn ), T = (P0 ? Pn ), Z = J0n ? D.

2 2

ñòð. 49 |