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SDYME (1) are invariant under the conformal group C(1, 3), we can restrict ourselves
?
to the three-dimensional subalgebras of AP (1, 3) determined up to conformal conjuga-
?
tion. Such subalgebras of the AP (1, 3) algebra are known [17, 18]. Since the case of
188 V.I. Lahno, W.I. Fushchych

the Poincar? algebra AP (1, 3) has been considered in [14, 15], we limit ourselves to
e
?
those subalgebras of AP (1, 3) that are not C(1, 3)-conjugates to the subalgebras of
AP (1, 3). We use the results and notation of [18], in particular, the fact that the
?
list of three-dimensional subalgebras of AP (1, 3) that are not conjugate to the three-
dimensional subalgebras of AP (1, 3) is exhausted, up to C(1, 3)-conjugation, by the
following algebras:
L1 = D, P0 , P3 , L2 = J12 + ?D, P0 , P3 ,
L3 = J12 , D, P0 , L4 = J12 , D, P3 ,
L5 = J03 + ?D, P0 , P3 , L6 = 2 J03 + ?D, P1 , P2 ,
L7 = J03 + ?D, M, P1 (? = 0), L8 = J03 + D + 2T, P1 , P2 ,
L9 = J02 + D + 2T, M, P1 , L10 = J03 , D, P1 ,
L11 = J03 , D, M , L12 = J12 + ?J03 + ?D, P0 , P3 ,
L13 = J12 + ?J03 + ?D, P1 , P2 ,
(3)
L14 = J12 + ?(J03 + D + 2T ), P1 , P2 , L15 = J12 + ?J03 , D, M ,
L16 = J03 + ?D, J12 + ?D, M , (0 ? |?| ? 1, ? ? 0, |?| + |?| = 0),
L17 = J03 + D + 2T, J12 + ?T, M (? ? 0),
L18 = J03 + D, J12 + 2T, M , L19 = J03 , J12 , D ,
L20 = G1 , J03 + ?D, P2 (0 < |?| ? 1), L21 = J03 + D, G1 + P2 , M ,
L22 = J03 ? D + M, G1 , P2 , L23 = J03 + 2D, G1 + 2T, M ,
L24 = J03 + 2D, G1 + 2T, P2 .

Here, M = P0 +P3 , G1 = J01 ?J13 , and T = 1 (P0 ?P3 ); also, ?, ? > 0 unless explicitly
2
stated otherwise. In what follows, ? and ? take on the values given in list (3).
Note that all of the subalgebras Lj (j = 1, 24) satisfy the condition r = r(1) = 3.
Let us demonstrate that, similar to [14, 15, 19], the ansatz for the Aµ ?elds can
be taken, without any loss of generality, in the linear form
(4)
Aµ (x) = ?(x)Bµ (?),

where ?(x) is a known square nondegenerate order-12 matrix and Bµ (?) are new
unknown vector-functions of the independent variable ? = ?(x), with x = (x0 , x1 , x2 ,
x3 ) ? R(1, 3).
Obviously, the fact that the sought for ansatz is linear requires that the algebra
Lj contain an invariant ?(x) independent of Aµ , as well as twenty linear invariants
of the form
fµ0 (x)Am + fµ1 (x)Am + fµ2 (x)Am + fµ3 (x)Am ,
m m m m
0 1 2 3

which are functionally dependent as functions of Am , Am , Am , and Am . These invari-
0 1 2 3
m
ants can be considered as components of a vector F A, where F = (fµ? (x)), while
? ?
A0
? ?
? A1 ?
A=? ?.
? A2 ?
A3
Reduction of self-dual Yang–Mills equations 189

Here, the matrix F is nondegenerate in some domain in R(1, 3). According to the
theorem on the conditional existence of invariant solutions [11], the ansatz F A = B(?)
results in a reduction of system (1) to a system of ODE that relates the independent
m
variable ?, the sought for functions Bµ , and the ?rst derivatives thereof. Setting
? = F ?1 (x), we arrive at ansatz (4).
?
Let L = X1 , X2 , X3 be one of the subalgebras of AP (1, 3) from list (3), with Xk
being an operator of form (2), i.e.,
?
Xk = ?km (x)?µ + ?m?? (x)Am (k = 1, 2, 3).
?
?Am?

The function f?? (x)An is an invariant of the operator Xk if and only if
n
?
n
?f?? (x) n
A? + ?k?? (x)An f?? (x) = 0
n
?kµ (x) ?
?xµ
or
n
?f?? (x) n
(5)
?kµ (x) + f?? (x)?k?? (x) = 0
?xµ
n
for all values of ?. Let F (x) = (f?? (x)) and ?k (x) = (?k?? (x)) be square matrices
of order 12. Then the second term on the left-hand side of (5) is an element of the
matrix F (x)?k (x).
These observations lead us to the following theorem.
Theorem 1. The system of functions f?? (x)An is a system of functional invariants of
n
?
n
a subalgebra L if and only if F = (f?? (x)) is a nondegenerate matrix in some domain
of R(1, 3) and satis?es the system of equations
?F (x)
(6)
?kµ (x) + F (x)?k (x) = 0 (k = 1, 2, 3).
?xµ
Similarly, the function ?(x) is an invariant of the operator Xk if and only if Xk ? = 0,
i.e.,
??
(7)
?kµ (x) = 0.
?xµ
Since all of the algebras Lj satisfy the condition
rank ?kµ (x) = 3,
systems (6) and (7) are compatible.
Theorem 1 assigns a matrix ?k to every generator Xk of the subalgebra L of
? (1, 3). Let us indicate the explicit form of these matrices for all generators (2) of
AP
?
the algebra AP (1, 3).
?
Since the operator Pµ is independent of ?Am , the corresponding ? is a zero matrix.
µ
Denote by ?Sµ? the ?-matrix that corresponds to the operator Jµ? . It is easy to verify
that
? ? ? ?
0 ?I 0 0 0 0 ?I 0
? ?I 0 0 0 ? ? ?
? , S02 = ? 0 0 0 0 ? ,
S01 = ? ?0 0 0 0? ? ?I 0 0 0 ?
0 0 00 0000
190 V.I. Lahno, W.I. Fushchych
? ? ? ?
0 0 0 ?I 00 0 0
? ? ? 0 0 ?I 0?
0 00 0?
S03 = ? , S12 = ? ?,
? 0 00 0? ?0 I 0 0?
?I 0 0 0 00 0 0
? ? ? ?
000 0 000 0
? ? ?0 0 0 0 ?
0 0 0 ?I ?
S13 = ? , S23 = ? ?,
? 000 0? ? 0 0 0 ?I ?
0I0 0 00I 0

where 0 is the zero and I is the unit matrix of order 3.
The D operator corresponds to the matrix ?E, where E is the unit order-12
matrix.
?
The above matrices determine a matrix representation of the algebra AQ(1, 3) =
AQ(1, 3) ? D , because

[Sµ? , S?? ] = gµ? S?? + g?? Sµ? ? gµ? S?? ? g?? Sµ? , [E, Sµ? ] = 0.

Let a = (1, 0, 0, 0), b = (0, 1, 0, 0), c = (0, 0, 1, 0), d = (0, 0, 0, 1), and k = a + d.
Denote by aµ , bµ , cµ , and dµ , the µth component of the vectors a, b, c, and d,
respectively. Then,
x0 = ax = aµ xµ , x1 = ?bx = ?bµ xµ ,
x2 = ?cx = ?cµ xµ , x3 = ?dx = ?dµ xµ .

Theorem 2. For every subalgebra Lj (j = 1, . . . , 24) from list (3), there exists a linear
ansatz (4), in which ? is a solution to system (7) and

??1 = exp{? log ?E} exp{?0 S03 } exp{??1 S12 } exp{?2?2 (S01 ? S13 )}.

Moreover, the functions ?, ?0 , ?1 , ?2 and ? can be represented as follows:
? = |bx|?1 , ? = cx(bx)?1 ,
L1 : ?0 = ?1 = ?2 = 0,
?1
? = ?1 2 ,
L2 : ?0 = ?2 = 0, ?1 = ?, ? = log ?1 + 2?,
?1
? = ?1 (dx)?2 ,
? = |dx|
L3 : , ?0 = ?2 = 0, ?1 = ?,
? = |ax|?1 , ? = ?1 (ax)?2 ,
L4 : ?0 = ?2 = 0, ?1 = ?,
? = |bx|?1 , ?0 = ??1 log |bx|, ?1 = ?2 = 0, ? = cx(bx)?1 ,
L5 :
1
1
? = |?2 |? 2 , ?0 = log |(ax ? dx)(kx)?1 |, ?1 = ?2 = 0,
L6 :
2
? = (1 ? ?) log |ax ? dx| + (1 + ?) log |kx|,
? = |cx|?1 , ?0 = ??1 log |cx|, ?1 = ?2 = 0, ? = |kx|? |cx|1?? ,
L7 :
1
1
? = |ax ? dx|? 2 , ?0 = log |ax ? dx|, ?1 = ?2 = 0,
L8 :
2
? = kx ? log |ax ? dx|,
? = |cx|?1 , ?0 = log |cx|, ? = kx ? 2 log |cx|,
L9 : ?1 = ?2 = 0,
? = |cx|?1 , ?0 = log |(ax ? dx)(cx)?1 |,
L10 : ?1 = ?2 = 0,
? = ?2 (cx)?2 ,
? = |cx|?1 , ?0 = ? log |(kx(cx)?1 |, ? = cx(bx)?1 ,
L11 : ?1 = ?2 = 0,
Reduction of self-dual Yang–Mills equations 191

?1
?0 = ???, ?1 = ?, ?2 = 0, ? = log ?1 + 2??,
? = ?1 2 ,
L12 :
1
1
? = |?2 |? 2 , ?0 = log |(ax ? dx)(kx)?1 |,
L13 :
2
1
log |(ax ? dx)(kx)?1 |, ?2 = 0,
?1 = ?
2?
? = (? ? ?) log |ax ? dx| + (? + ?) log |kx|,
1 1
1
? = |ax ? dx|? 2 , ?0 = log |ax ? dx|, ?1 = ? log |ax ? dx|,
L14 :
2 2
?2 = 0, ? = kx ? log |ax ? dx|,
?1
?0 = ???, ?1 = ?, ?2 = 0, ? = log[?1 (kx)?2 ] + 2??,
? = ?1 2 ,
L15 :
1
?1
? = ?1 2 , ?0 = log |?1 (kx)?2 ], ?1 = ?, ?2 = 0,
L16 :
2
1??
? = log[?1 (kx)2? ] + 2??,
1
?1
? = ?1 2 , ?0 = log ?1 , ?1 = ?, ?2 = 0, ? = kx ? log ?1 + 2??,
L17 :
2
1
?1
? = ?1 2 , ?0 = log ?1 , ?1 = ?, ?2 = 0, ? = kx + 2?,
L18 :
2
1
?1
? = ?1 2 , ?0 = ? log |kx(ax ? dx)?1 |, ?1 = ?, ?2 = 0,
L19 :
2
?1
? = ?1 |?2 | ,
1 1
1
? = |?3 |? 2 , ?0 = log |?3 |, ?1 = 0, ?2 = bx(kx)?1 ,
L20 :
2? 2
? = |kx| |?3 |
2? 1??
,
1
? = |cxkx ? bx|?1 , ?0 = log |cxkx ? bx|?1 , ?1 = 0, ?2 = cx,
L21 :
2
? = kx,
1 1
1
? = |kx|? 2 , ?0 = ? log |kx|, ?1 = 0, ?2 = bx(kx)?1 ,
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