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representations. It should be noted that nonlinear representations of the Poincar? and
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, 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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. 64
( 122 .)



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