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e

conformal groups often occur as realizations of symmetry groups of nonlinear PDE

such as eikonal, Born–Infeld and Monge–Amper? equations (see [3] and references

e

therein). On sets of solutions of some nonlinear heat equations nonlinear represen-

tations of the Galilei group are realized [3]. So, nonlinear representations of the

transformations groups are intimately connected with nonlinear PDE, and systematic

study of these is of great importance.

J. Nonlinear Math. Phys., 1994, 1, ¹ 3, P. 295–308.

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

In the present paper we obtain the complete description of the Poincar? group

e

P (n, m) (called for bravity the Poincar? group) and of its extensions — the extended

e

Poincar? group P (n, m) and conformal group C(n, m) acting as Lie transformation

e

groups in the space R(n, m) ? R1 , where R(n, m) is the pseudo-Euclidean space with

the metric tensor

?

? 1, ? = ? = 1, n,

?1, ? = ? = n + 1, n + m,

g?? =

?

0, ? = ?.

The paper is organized as follows. In Section 2 we give all necessary notations and

definitions. In Section 3 we investigate representations of groups P (n, m), P (n, m),

C(n, m) with max{n, m} ? 3 and prove, in particular, that each representation of

the Poincar? group P (n, m) with max{n, m} ? 3 is equivalent to the standard li-

e

near representation. In Section 3 we study representations of the above groups with

max{n, m} < 3 and show that groups P (1, 2), C(1, 2), P (2, 2), P (2, 2), C(2, 2) have

nontrivial nonlinear representations. It should be noted that nonlinear representati-

ons of the groups P (1, 1), P (1, 1), C(1, 1) were constructed in [9] and of the group

P (1, 2) — in [10].

2 Notations and definitions

Saying about a representation of the Poincar? group P (n, m) in the class of Lie

e

transformation groups we mean the transformation group

(2)

xµ = fµ (x, u, a), µ = 1, n + m, u = g(x, u, a),

where a = {aN , N = 1, 2, . . . , n + m + Cn+m } are group parameters preserving the

2

quadratic form S(x) = g?? x? x? . Here and below summation over the repeated indices

is understood.

It is common knowledge that a problem of description of inequivalent representa-

tions of the Lie transformation group (2) can be reduced to a study of inequivalent

representations of its Lie algebra [1, 2, 12].

Definition 1. Set of n + m + Cn+m differential operators Pµ , J?? = ?j?? , µ, ?, ? =

2

1, n + m of the form

(3)

Q = ?µ (x, u)?µ + ?(x, u)?u

satisfying the commutational relations

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

(4)

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

is called a representation of the Poincar? algebra AP (n, m) in the class of Lie vector

e

fields.

In the above formulae

? ?

[Q1 , Q2 ] = Q1 Q2 ? Q2 Q1 ,

?µ = , ?u = , ?, ?, ?, µ, ? = 1, n + m.

?xµ ?u

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

e 257

Definition 2. Set of 1 + n + m + Cn+m differential operators Pµ , J?? = ?J?? , D

2

(µ, ?, ? = 1, n + m) of the form (3) satisfying the commutational relations (4) and

(5)

[D, J?? ] = 0, [P? , D] = P? (?, ? = 1, n + m)

is called a representation of the extended Poincar? algebra AP (n, m) in the class of

e

Lie vector fields.

2

Using the Lie theorem [1, 2] one can construct the (1 + n + m + Cn+m )-parameter

Lie transformation group corresponding to the Lie algebra {Pµ , J?? , D}. This trans-

formation group is called a representation of the extended Poincar? group P (n, m).

e

Definition 3. Set of 1 + 2(n + m) + Cn+m differential operators Pµ , J?? = ?J?? , D,

2

Kµ (µ, ?, ? = 1, n + m) of the form (3) satisfying the commutational relations (4),

(5) and

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

(6)

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

is called a representation of the conformal algebra AC(n, m) in the class of Lie

vector fields.

2

(1 + 2(n + m) + Cn+m )-parameter transformation group corresponding to the Lie

algebra {Pµ , J?? , D, Kµ } is called a representation of the conformal group C(n, m).

Definition 4. Representation of the Lie transformation group (2) is called linear if

functions fµ , g satisfy conditions fµ = fµ (x, a) (µ = 1, n + m), g = g(x, a)u. If these

conditions are not satisfied, representation is called nonlinear.

Definition 5. Representation of the Lie algebra in the class of Lie vector fields (3)

is called linear if coefficients of its basis elements satisfy the conditions

(7)

?? = ?? (x), ? = 1, n + m, ? = ?(x)u,

otherwise it is called nonlinear.

Using the Lie equations [1, 2] it is easy to establish that if a Lie algebra has a

nonlinear representation, its Lie group also has a nonlinear representation and vice

versa.

Since commutational relations (4)–(6) are not altered by the change of variables

(8)

x? = F? (x, u), u = G(x, u),

two representations {P? , J?? , D, K? } and {P? , J?? , D , K? } are called equivalent pro-

vided they are connected by relations (8).

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

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

Theorem 1. Arbitrary representation of the Poincar? algebra AP (n, m) with

e

max{n, m} ? 3 in the class of Lie vector fields is equivalent to the standard repre-

sentation

J?? = g?? x? ?? ? g?? x? ?? (9)

P ? = ?? , (?, ? = 1, n + m).

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

Proof. By force of the fact that operators P? commute, there exists the change of

variables (8) reducing these to the form P? = ?? , ? = 1, n + m (a rather simple

proof of this assertion can be found in [1, 3]). Substituting operators P? = ?? , J?? =

???? (x, u)?? + ??? (x, u)?u into relations [P? , J?? ] = g?? P? ? g?? P? and equating

coefficients at the linearly-independent operators ?? , ?u we get a system of PDE for

unknown functions ???? , ???

????xµ = gµ? g?? ? gµ? g?? , ???xµ = 0, ?, ?, ?, µ = 1, n + m,

whence

???? = x? g?? ? x? g?? + F??? (u), ??? = G?? (u).

Here F??? = ?F??? , G?? = ?G?? are arbitrary smooth functions, ?, ?, ? = 1, n + m.

Consider the third commutational relation from (4) under 1 ? ?, ?, µ, ? ? n,

? = µ. Equating coefficients at the operator ?u , we get the system of nonlinear

ordinary differential equations for Gµ? (u)

? ?

G?? = G?? G?? ? G?? G?? (11a)

(no summation over ?), where a dot means differentiation with respect to u.

Since (11a) holds under arbitrary ?, ?, ? = 1, n, we can redenote subscripts in

order to obtain the following equations

? ?

G?? = G?? G?? ? G?? G?? , (11b)

? ?

G?? = G?? G?? ? G?? G?? (11c)

(no summation over ? and ?).

Multiplying (11a) by G?? , (11b) by G?? , (11c) by G?? and summing we get

G2 + G2 + G2 = 0,

?µ ?µ ??

whence G?? = G?? = G?? = 0.

Since ?, ?, ? are arbitrary indices satisfying the restriction 1 ? ?, ?, ? ? n, we

conclude that G?? = 0 for all ?, ? = 1, 2, . . . , n.

Furthermore, from commutational relations for operators J?? , ?, ? = 1, n we get

the homogeneous system of linear algebraic equations for functions F??? (u), which

general solution reads

F??? = F? (u)g?? ? F? (u)g?? , ?, ?, ? = 1, n,

where F? (u) are arbitrary smooth functions.

Consequently, the most general form of operators Pµ , J?? with 1 ? ?, ? ? n

satisfying (4) is equivalent to the following:

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

P µ = ?µ ,

Making in the above operators the change of variables

xµ = xµ + Fµ (u), µ = 1, n, xA = xA , A = n + 1, n + m, u =0

and omitting primes we arrive at the formulae (9) with 1 ? ?, ? ? n.

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

e 259

Consider the commutator of operators J?? , J?A under 1 ? ?, ? ? n, n + 1 ? A ?

n+m

J?? , J?A = [x? ?? ? x? ?? , g?? x? ?A ? gA? x? ?? +

(12a)

+ F?A? (u)?? + G?A (u)?u ] = xA ?? ? x? ?A .

On the other hand, by force of commutational relations (4) an equality

(12b)

[J?? , J?A ] = J?A

holds. Comparing right-hand sides of (12a) and (12b) we come to conclusion that

F?A? = 0, G?A = 0. Consequently, operators J?A = ?JA? with ? = 1, n, A =

n + 1, n + m have the form (9).

Analogously, computing the commutator of operators J?A , JAB under 1 ? ? ? n,

n + 1 ? A, B ? n + m and taking into account commutational relations (4) we get

FAB? = 0, A, B = n + 1, n + m, ? = 1, n. Consequently, operators JAB are of the

form

JAB = xB ?A ? xA ?B + GAB (u)?u , A, B = n + 1, n + m.

At last, substituting the results obtained into commutational relations

[J?A , J?B ] = ?JAB

(no summation over ?), where ? = 1, n, A, B = n + 1, n + m, we get

GAB = 0, A, B = n + 1, n + m.

Thus, we have proved that there exists the change of variables (8) reducing an

arbitrary representation of the Poincar? algebra AP (n, m) with max{n, m} ? 3 to the

e

standard representation (9). Theorem is proved.

Note 1. Poincar? algebra AP (n, m) contains as a subalgebra the Euclid algebra AE(n)

e

with basis elements P? , J?? , ?, ? = 1, n. When proving the above theorem we have

established that arbitrary representations of the algebra AE(n) with n ? 3 in the

class of Lie vector fields are equivalent to the standard representation

J?? = x? ?? ? x? ?? ,

P µ = ?µ , µ, ?, ? = 1, n.

Theorem 2. Arbitrary representation of the extended Poincar? algebra AP (n, m)

e

with max{n, m} ? 3 in the class of Lie vector fields is equivalent to the following

representation:

J?? = g?? x? ?? ? g?? x? ?? , (13)

P ? = ?? , D = x? ?? + ?u?u ,

where ? = 0, 1; ?, ?, ? = 1, n + m.

Proof. From theorem 1 it follows that a representation of the Poincar? algebra e

AP (n, m) = Pµ , J?? can always be reduced to the form (9). To find the explicit

form of the dilatation operator D = ?µ (x, u)?µ + ?(x, u)?u we use the commutational

relations [P? , D] = P? . Equating coefficients at linearly-independent operators ?µ , ?u ,

we get

?µx? = ?µ? , ?x? = 0,

where ?µ? is a Kronecker symbol; µ, ? = 1, n + m.

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

Integrating the above equations we have

?µ = xµ + Fµ (u), ? = G(u),

where Pµ (u), G(u) are arbitrary smooth functions.

Using commutational relations [Jµ? , D] = 0 we arrive at the following equalities:

gµ? F? ?? ? g?? F? ?µ = 0; µ, ? = 1, n + m,

whence F? = 0, ? = 1, n + m.

Thus, the most general form of the operator D is the following:

D = xµ ?µ + G(u)?u .

Provided G(u) = 0, we get the formulae (13) under ? = 0. If G(u) = 0, then after

making the change of variables

?1

xµ = xµ , µ = 1, n + m, u= G(u) du

we obtain the formulae (8) under ? = 1. Theorem is proved.

Theorem 3. Arbitrary representation of the conformal algebra AC(n, m) with

max{n, m} ? 3 in the class of Lie vector fields is equivalent to one of the following

representations:

1) operators Pµ , J?? , D are given by (13), and operators K? have the form

K? = 2g?? x? D ? (gµ? xµ x? )?? ; (14)

2) operators Pµ , J?? , D are given by (13) with ? = 1, and operators K? have the

form

K? = 2g?? x? D ? (gµ? xµ x? ± u2 )?? . (15)

Proof. From theorem 2 it follows that the basis of the algebra AP (n, m) up to the

change of variables (8) can be chosen in the form (13).

From the commutational relations for operators P? = ?? and K? = ??µ (x, u)?µ +

?? (x, u)?u we get the following system of PDE:

??µx? = 2g?? xµ ? 2g?? x? ??µ + 2g?? x? ?µ? , ??x? = 2?g?? u.

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