ñòð. 65 |

Integrating these we have

??µ = 2g?? x? xµ ? g?? x? x? ??µ + F?µ (u), ?? = 2?x? u + G? (u),

where F?µ , g? are arbitrary smooth functions, ?, ?, µ, ? = 1, n + m.

Next, we make use of commutational relations [D, K? ] = K? . Direct computation

shows that the following equalities hold

[D, K? ] = [xµ ?µ + ?u?u , 2g?? x? (xµ ?µ + ?u?u ) ? gµ? xµ x? ?? +

+ F?? (u)?? + G? (u)?u ] = 2g?? x? (xµ ?µ + ?u?u ) ?

? (gµ? xµ x? )?? + (?uF??u ? F?? )?? + ?(uG?u ? G? )?u .

On linear and non-linear representations of the generalized Poincar? groups

e 261

Comparison of the right-hand sides of the above equalities yields the system of

PDE

G? = ?(uG?u ? G? ), (16)

2F?? = ?uF??u , ?, ? = 1, n + m.

In the following, we will consider the cases ? = 0 and ? = 1 separately.

Case 1, ? = 0. Then it follows from (16) that F?? = 0, G? = 0, ?, ? = 1, n + m,

i.e. operators Kµ are given by (14) with ? = 0. It is not difficult to verify that the

rest of commutational relations (6) also holds.

Case 2, ? = 1. Integrating the equations (16) we get

F?? = C?? u2 , G? = C? u2 ,

where C?? , C? are arbitrary real constants.

Next, from the commutational relations for K? , Jµ? it follows that

C?? = C??? , C? = 0,

where C is an arbitrary constant, ?, ? = 1, n + m.

Thus, operators Kµ have the form

Kµ = 2gµ? x? D ? (g?? x? x? )?µ + Cu2 ?µ . (17)

Easy check shows that the operators (17) commute, whence it follows that all

commutational relations of the conformal algebra hold.

If in (17) C = 0, then we have the case (14) with ? = 1. If C = 0, then after

rescaling the dependent variable u = u|c|1/2 we obtain the operators (15). Theorem

is proved.

Note 2. Nonlinear representations of the conformal algebra given by (13) with ? = 1

and (15) are realized on the set of solutions of the eikonal equations [3, 14]

gµ? uxµ ux? ± 1 = 0

and on the set of solutions of d’Alembert–eikonal system [15]

gµ? uxµ x? ± (n + m ? 1)u?1 = 0.

gµ? uxµ ux? ± 1 = 0,

Thus, the Poincar? group P (n, m) with max{n, m} ? 3 has no truly nonlinear

e

representations. The only hope to obtain nonlinear representations of the Poincar?

e

group is to study the case when max{n, m} < 3.

4 Representations of the algebras AP (n, m),

AP (n, m), AC(n, m) with max{n, m} < 3

Representations of algebras AP (1, 1), AP (1, 1), AC(1, 1) in the class of Lie vector

fields were completely described by Rideau and Winternitz [9]. They have established,

in particular, that the Poincar? algebra AP (1, 1) has no nonequivalent representations

e

distinct from the standard one (9), while algebras AP (1, 1), AC(1, 1) admit nonlinear

262 W.I. Fushchych, R.Z. Zhdanov, V.I. Lahno

representations. In the paper [10] nonlinear representations of the Poincar? algebra

e

AP (1, 2)

Pµ = ?µ , J12 = x1 ?2 + x2 ?1 + ?u ,

(18)

J13 = x1 ?3 + x3 ?1 + cos u?u , J23 = x2 ?3 ? x3 ?2 ? sin u?u ,

were constructed and besides that, it was proved that an arbitrary representation

of the algebra AP (1, 2) in the class of Lie vector fields is equivalent either to the

standard representation or to (18).

In the paper [11] we have constructed nonlinear representations of the algebras

AP (2, 2) and AC(2, 2).

Theorem 4. Arbitrary representation of the Poincar? algebra AP (2, 2) in the class

e

of Lie vector fields is equivalent to the following representation:

Pµ = ?µ , µ = 1, 4,

J12 = x1 ?2 ? x2 ?1 + ??u , J13 = x3 ?1 + x1 ?3 + ? cos u?u ,

(19)

J14 = x4 ?1 + x1 ?4 ? ? sin u?u , J23 = x3 ?2 + x2 ?3 + ? sin u?u ,

J24 = x4 ?2 + x2 ?4 ± ? cos u?u , J34 = x4 ?3 ? x3 ?4 ± ??u ,

where ? = 0, 1.

Proof. When, proving the theorem 1, we have established that the operators Pµ , J??

can be reduced to the form

Jµ? = gµ? x? ?? ? g?? x? ?µ + Fµ?? (u)?? + Gµ? (u)?u , (19a)

P µ = ?µ ,

where Fµ?? = ?F?µ? , Gµ? = ?G?µ are arbitrary smooth functions; µ, ?, ? = 1, 4.

Consider the triplet of operators J12 , J13 , J23 . From commutational relations (4)

we obtain the following system of nonlinear ordinary differential equations for functi-

ons G12 , G13 , G23 :

? ? ? ?

G23 = G13 G12 ? G12 G13 , G13 = G12 G23 ? G23 G12 ,

(20)

? ?

G12 = G13 G23 ? G23 G13 ,

(a dot means differentiations with respect to u).

Multiplying the first equation of the system (20) by G23 , the second — by G13

and the third — by G12 and summing we get an equality

G2 = G2 + G2 . (21)

12 13 23

In the following one has to consider cases G12 = 0 and G12 = 0 separately.

Case 1, G12 = 0. General solution of the algebraic equation (21) reads

(22)

G12 = f (u), G13 = f (u) cos g(u), G23 = f (u) sin g(u),

where f (u), g(u) are arbitrary smooth functions.

Substitution of (22) into (20) yields gf 2 = f . Since f (u) = g12 = 0, the equality

?

?1

holds. Consequently, the general solution of the system (20) is of the form

g=f

?

G12 = g ?1 , G13 = g ?1 cos g, G23 = g ?1 sin g,

? ? ?

where g = g(u) is an arbitrary smooth function.

On linear and non-linear representations of the generalized Poincar? groups

e 263

On making the change of variables

x? = x? , ? = 1, 4, u = g(u),

which does not alter the structure of operators Pµ , Jµ? (19a), we reduce operators

J12 , J23 , J13 to the form

J12 = x1 ?2 ? x2 ?1 + ?u + F12? (u)?? ,

(23)

J23 = x3 ?2 + x2 ?3 + (sin u)?u + F23? (u)?? ,

J13 = x3 ?1 + x1 ?3 + (cos u)?u + F13? (u)?? ,

where F12? , F23? , F13? , ? = 1, 4 are arbitrary smooth functions.

Substitution of (23) into (4) yields the system of linear ordinary differentional

equations, which for general solution reads

? ? ? ?

F121 = V + W, F122 = W ? V, F123 = Q, F131 = V cos u ? Q,

? ? ? ?

F132 = W cos u, F133 = Q cos u ? V, F231 = V sin u, F232 = W sin u ? Q,

?

F233 = Q sin u ? W, F124 = R, F134 = R cos u ? C1 sin u,

F234 = R sin u + C1 cos u.

Here V , W , Q, R are arbitrary smooth functions on u, C1 is an arbitrary constant.

The change of variables

x1 = x1 ? V (u), x2 = x2 ? W (u),

x3 = x3 ? Q(u), x4 = x4 ? R(u)du, u =u

reduce operators J12 , J23 , J13 to the form

J12 = x1 ?2 ? x2 ?1 + ?u ,

J13 = x3 ?1 + x1 ?3 ? C1 sin u?u + cos u?u , (24)

J23 = x3 ?2 + x2 ?3 + C1 sin u?u + sin u?u ,

the rest of basis elements of the algebra AP (2, 2) having the form (19a).

Computing commutational relations (4) for operators Jab ; ?, ? = 1, 4 given by

formulae (19a) with µ = 1, 3, ? = 4 and (24) we obtain system of equations for

unknown functions Fµ4 ? , Gµ4 ; ? = 1, 4; µ = 1, 3. General solution of the system

reads

G14 = ? sin u, G24 = ± cos u, G34 = ±1, C1 = 0,

F141 = F242 = F343 = C2 , F?4? = 0, ? = ?,

where C2 is an arbitrary constant.

Substituting the result obtain into the formulae (19a) and making the change of

variables

x? = x? , ? = 1, 3; x4 = x4 + C2 ; u =u

we conclude that operators J?4 , ? = 1, 3 are given by (19) with ? = 1.

Case 2, G12 = 0. In this case from (21) it follows that G12 = G13 = G23 = 0.

Computing commutators of operators J12 , J14 and J12 , J24 we get G14 = G24 . Next,

computing commutator of operators J13 , J23 we came to conclusion that G34 = 0.

264 W.I. Fushchych, R.Z. Zhdanov, V.I. Lahno

Substitution of operators Jµ? from (19a) with Gµ? = 0, µ, ? = 1, 4 into commu-

tational relations (4) yields a homogenerous system of linear algebraic equations for

functions Fµ?? . Its general solution can be represented in the form

Fµ?? = Fµ (u)g?? ? F? (u)gµ? , µ, ?, ? = 1, 4,

where Fµ (u) are arbitrary smooth functions.

Consequently, operators (19a) take the form

J?? = g?? (x? + F? (u))?? ? g?? (x? + F? (u))?? .

P µ = ?µ ,

Making in the above operators the change of variables xµ = xµ + Fµ (u), µ = 1, 4,

u = u we arrive at formulae (19) with ? = 0. Theorem is proved.

Theorem 5. Arbitrary representations of the extended Poincar? algebra AP (2, 2) in

e

the class of Lie vector fields is equalent to one of the following representations:

1) Pµ , J?? are of the form (19) with ? = 1, D = xµ ?µ ;

2) Pµ , J?? are of the form (19) with ? = 0, D = xµ ?µ + ?1 u?u , ?1 = 0, 1.

Theorem 6. Arbitrary representation of the conformal algebra AC(2, 2) in thew

class of Lie vector field is equivalent to one of the following representations:

1) Pµ , J?? are of the form (19) with ? = 0,

D = x? ?? + ?1 u?u , ?1 = 0, 1,

K? = 2g?? x? D ? (gµ? xµ x? )?? ;

2) Pµ , J?? are of the form (19) with ? = 0,

D = x? ?? + u?u ,

K? = 2g?? x? D ? (gµ? xµ x? ± u2 )?? ;

3) P? , Jµ? are of the form (19) with ? = 1,

D = x? ?? ,

K1 = 2x1 D ? (gµ? xµ x? )?1 + 2(x2 + x3 cos u ? x4 sin u)?u ,

K2 = 2x2 D ? (gµ? xµ x? )?2 + 2(?x1 + x3 sin u ± x4 cos u)?u ,

K3 = ?2x3 D ? (gµ? xµ x? )?3 + 2(±x4 + x1 cos u ? x2 sin u)?u ,

K4 = ?2x4 D ? (gµ? xµ x? )?4 + 2(?x4 ± x1 sin u ? x2 cos u)?u ,

where µ, ?, ?, ? = 1, 2, 3, 4.

Proofs of the theorems 5 and 6 are similar to the proofs of the theorems 2, 3 that

is why they are omitted.

In conclusion of the Section we adduce all nonequivalent representations of the

extended Poincar? algebra AP (1, 2) [10]

e

1) Pµ , J?? are of the form (9),

D = xµ ?µ + ?u?u , ? = 0, 1;

2) Pµ , J?? are of the form (18),

D = xµ ?µ

On linear and non-linear representations of the generalized Poincar? groups

e 265

and the conformal algebra AC(1, 2) [10]

1) Pµ , J?? are of the form (9),

D = xµ ?µ + ?u?u , ? = 0, 1,

K? = 2g?? x? D ? (gµ? xµ x? )?? ;

2) Pµ , J?? are of the form (9),

D = xµ ?µ + u?u ,

K? = 2g?? x? D ? (gµ? xµ x? ± u2 )?? ;

3) Pµ , J?? are of the form (18),

D = xµ ?µ ,

K1 = 2x1 D ? (gµ? xµ x? )?1 + 2(x2 + x3 cos u)?u ,

K2 = ?2x2 D ? (gµ? xµ x? )?2 + 2(?x1 + x3 sin u)?u ,

K3 = ?2x3 D ? (gµ? xµ x? )?3 ? 2(x1 cos u + x2 sin u)?u .

Here µ, ?, ?, ? = 1, 2, 3.

5 Conclusion

Thus, we have obtained the complete description of nonequivalent representations

of the generalized Poincar? group P (n, m) by operators of the form (3). This fact

e

makes a problem of constructing Poincar?-invariant equations of the form (1) purely

e

algorithmic. To obtain all nonequivalent Poincar?-invariant equations on the order N ,

e

one has to construct complete sets of functionally-independent differential invariants

of the order N for each nonequivalent representation [1, 2].

For example, each P (n, m)-invariant first-order PDE with max{n, m} ? 3 can be

reduced by appropriate change of variables (2) to the eikonal equation

(25)

gµ? uxµ ux? = F (u),

where F (u) is an arbitrary smooth function.

Equation (26) with an arbitrary F (u) is invariant under the algebra AP (n, m)

having the basis elements (9). Provided F (u) = 0, n = m = 2, it admits also the

Poincar? algebra with the basis elements (19) [11].

e

Another interesting example is provided by P (1, n)-invariant PDE (n ? 3). In [16]

a complete basis of functionally-independent differential invariants of the order 2 of

the algebra AP (1, n) with the basis elements (9) has been constructed. Since each

representation of the algebra AP (1, n) with n ? 3 is equivalent to (9), the above

ñòð. 65 |