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n(n ? 1) n(n ? 1) n(n + 7)
+ 2n ?
2 =
2 2 2

elements for which we take the following expressions

(1.9)
Rk (ua , uab ), Rk (va , uab ), Sjk (uab , vab ).

The invariance of expressions (1.9) with respect to the operators (1.5) can be easily
proved by their direct substitution to (1.4). To establish their functional independence,
we shall use the following lemma.
Lemma 3. Let

U = (uab )a,b=1,...,n , V = (vab )a,b=1,...,n

be symmetric matrices. Then the expressions

Sjk (uab , vab ) = tr U j V k?j , (1.10)
j = 0, . . . , k; k = 1, . . . , n,

are functionally independent.
Proof. To prove Lemma 3, it is sufficient to show that the generic rank of the Jacobi
matrix of expressions (1.10) is equal to (n(n + 3))/2 that is the difference between the
number of independent elements of U and V and the rank of the operators (1.5). We
shall limit ourselves to the case when uab = 0 if a = b. Then equations (1.10) depend
on (n(n + 3))/2 variables and their independence is equivalent to the nonvanishing of
the Jacobian.
Let us write down the elements of the Jacobian which are needed for further
reasoning

···
1 1
··· 0
2u11 2un
··· ··· ···
· · · nun?1
nun?1 (1.11)
.
nn
11
··· ···
1 0 0 1 1
··· ··· ···
2v11 4v12 4v1n 2v22 2vnn
···

Since, in the first n rows, all the elements besides the first n columns are equal
to j zero, the Jacobian (1.11) is equal to the product of the Jacobian of the elements
tr U k , k = 1, . . . , n, and the Jacobian of all other elements. According to Lemma 2,
the expressions tr U k , k = 1, . . . , n, are independent and their Jacobian is not equal
Second-order differential invariants of the rotation group O(n) 509

to zero; thus, it remains to show the nonvanishing of the Jacobian and the functional
independence only for the elements
j = 0, . . . , k ? 1; k = 1, . . . , n.
tr U j V k?j ,
It follows from (1.11) that it is sufficient to show the nonvanishing of this Jacobian
without the (n + 1)th rows and columns. Thus, to prove the lemma, it is enough to
show that the following expressions are independent
j = 0, . . . , k; k = 1, . . . , n ? 1.
tr U j V k?j V, (1.12)
The above reasoning allows us to make use of the principle of mathematical inducti-
on.
When n = 1, u11 and v11 are independent and the lemma is true. Let us suppose
that it is true for n ? 1 and then prove from this that it is valid for n. Let the
expressions
j = 0, . . . , k; k = 1, . . . , n ? 1,
tr U j V k?j , (1.13)
where U , V are symmetric (n ? 1) ? (n ? 1) matrices and are independent. Then,
we shall prove the independence of (1.12) for the same matrices. The sets (1.12) and
(1.13) coincide with the exception of the following subsets
j = 0, . . . , n ? 1
tr U j V n?j , (1.14)
belong only to (1.12) and
j = 1, . . . , n ? 1
tr U j , (1.15)
belong only to (1.13).
The assumption of validity of the lemma for n ? 1 means that for two symmetric
tensors of order 2, the set (1.13) is a functional basis of invariants of the rotation
algebra. Thus, all the invariants of this algebra can be represented as functions of
(1.13). To prove the functional independence of (1.12), it is sufficient to prove the
nondegeneracy of the Jacobi matrix of the functions expressing the invariants (1.12)
with (1.13). This matrix has the form
? ?
0
1
? ?
···
1
? ?
? ?
..
? ?
.
? ?, (1.16)
? ?
0 1
? ?
? ?
W
? ?
j n?j
?(tr U V )
0 ?(tr U j )

W being the derivative by tr V of the expression
tr V n = F (tr V k , k = 1, . . . , n ? 1).
(We know that from the Hamilton–Cayley theorem); W = 0.
We have only to prove the nonvanishing of the Jacobian of the expressions
tr (U j V n?j ) = F (tr U k , k = 1, . . . , n ? 1, . . .). (1.17)
510 W.I. Fushchych, I.A. Yegorchenko

When V = E, the corresponding quadrant of the matrix (1.16) is the unit matrix
and its determinant does not vanish identically. This fact proves the nondegeneracy of
the matrix (1.16). The expressions (1.17) can be obtained from the Hamilton–Cayley
theorem. They are polynomials and, thus, continuous functions of their arguments.
The functional independence of the expressions (1.12) for (n ? 1) ? (n ? 1) matrices
implies their independence for n ? n matrices. From the above, it follows that the
expressions (1.10) are independent, thus Lemma 3 is proved.
Proof of Theorem 2. It is easy to see from the structure of the set (1.3) that
the invariants involving (u1 ), . . . , (um ), (u2 ), . . . , (um ) depend on the components of
a a ab ab
(u1 ) and of the corresponding vector or tensor, thus it is sufficient to prove the
ab
functions independence of each of the following sets:
Rk (ur , u1 ) for every r = 1, . . . , m;
a ab
Sjk (uab , ur )
1
for every r = 2, . . . , m;
ab

The functional independence of each set of Rk (ur , u1 ) can be proved similarly
a ab
to the proof of Lemma 2. The functional independence of the set Sjk (u1 , ur ) easily
ab ab
follows from Lemma 3, ur are evidently independent of other elements of (1.3).
To make sure that expressions (1.3) are invariants of AO(n), it is sufficient to
substitute them into the condition (1.4).
The set (1.3) consists of
n(n ? 1) n(n ? 1)
n(n + 1)
2mn + m + (m ? 1) +n+1 ?
=m
2 2 2
elements and, thus, it is complete.
So we have proved that this set forms a basis of invariants for the algebra AE(1.n)
(1.1).
1.3. Bases of invariants for the extended Euclid algebra and for the confor-
mal algebra. The extended Euclid algebra AE1 (n) for one scalar function is defined
by the basis operators ?a , Jab (1.1) and D depending on a parameter ?:
(1.18)
D = xa ?a + ?u?u (?u = ?/?u).
The basis of the conformal algebra AC(n) consists of the operators ?a , Jab (1.1)
and D (1.18) and
Ka = 2xa D ? xa xb ?a . (1.19)
Theorem 3. There is a functional basis for the extended Euclid algebra that has
the following form
(1) when ? = 0:
Rk (ua , uab ) Sk (uab )
(1.20)
, ;
uk(1?2/?)+1 uk(1?2/?)
(2) when ? = 0:
Rk (ua , uab ) Sk (uab )
(1.21)
u, , (k = 1);
(uaa )k (uaa )k
a functional basis for the conformal algebra has the following form:
Second-order differential invariants of the rotation group O(n) 511

(1) when ? = 0:
Sk (?ab )uk(2/??1) ; (1.22)
(2) when ? = 0:
Sk (wab )(ua ua )?2k (1.23)
u, (k = n),
where
ua ub uc uc
?ab = ?uab + (1 ? ?) ? ?ab ,
u 2u
(1.24)
?ab
udd ? uc (ua ubc + ub uac ),
wab = uc uc uab +
2?n
?ab being the Kronecker symbol.
Proof. To find absolute differential invariants of the algebra AE1 (n), it is necessary
to add to (1.4) the following condition
2
D F ? xa Fxa + ?uFu + (? ? 1)ua Fua + (? ? 2)uab Fuab = 0. (1.25)
Solving equation (1.25) for
F = F (u, Rk (ua , uab ), Sk (uab )),
we obtain functional bases (1.20), (1.21) for the extended Euclid algebra.
The second-order differential invariants of the algebra AC(n) are defined by the
conditions (1.4), (1.25) and
2
(1.26)
ka K a F = 0,
2
where ka are arbitrary real numbers, K a are the second prolongations of the operators
Ka (1.19):
2 2 2
K a = 2xa D + xb J ab + 2?[u?ua + 2ub ?uab ] + 2ua ?ucc ? 4ub ?uab .
Solving this system for an arbitrary n requires a lot of cumbersome computations.
It is simpler to construct conformally co variant tensors from u, ua , uab and then to
construct invariants of the rotation algebra.
Definition 3. Tensors ?a and ?ab of order 1 and 2 are called covariant with respect
to some algebra L = Jab , Xi if
Xi ?a = ?ab ?b + ? i ?a ,
i
(1.27)
Xi ?ab = ?i ?cb + ?i ?ac + ?i ?ab ,
ab bc

Xi are operators of the form (0.1), ?i , ? i are some functions, ?ab , ?i are some
i
ab
skew-funmetric tensors.
It is easy to show that the expressions Sk (?ab ), Rk (?a , ?ab ), where ?a , ?ab are
tensors covariant with respect to the algebra L are relative invariants of this algebra.
The fact that ?ab and wab (1.24) are covariant with respect to the conformal algebra
AC(n) can be verified by direct substitution of these tensors into the conditions (1.27)
2 2
for the operators D and K a .
512 W.I. Fushchych, I.A. Yegorchenko

The rank of the second prolongation of the algebra AC(n) is equal to the number
of its operators
n(n ? 1) n(n + 3)
+n+n+1= +1
2 2
and, therefore, a functional basis of second-order differential invariants must contain
n invariants.
The functional independence of the expressions (1.22) follows from Lemma 2 if we
notice that the transformation uab > ?ab is nondegenerated. The same is true for the
set (1.23).
The expressions (1.22) and (1.23) satisfy (1.25) and (1.26) for the corresponding ?
and they are invariants of the conformal algebra.
All that is stated above leads to the conclusion that (1.22) and (1.23) form functi-
onal bases for the conformal algebra AC(n) with ? = 0 and ? = 0, respectively.
Note 1. Using condition (1.26), it is easy to show that when ? = 0 covariant tensors
exist for AC(n) of order 2 only; when ? = 0, the tensors wab (l.24) and ua are
conformally covariant but Sk (wab ) and Rk (ua , wab ) are dependent.
Theorem 4. The second-order differential invariants for a vector function u =
(u1 , . . . , um ) and for the algebra AE1 (n) = ?a , Jab , D , the operator D having the
form
D = xa ?a + ?ur ?ur (1.28)
with a summation over r from 1 to m, can be represented as the functions of the
following expressions:
(1) when ? = 0:
ur Sjk (u1 , ur ) Rk (ur , u1 )
a
ab ab ab
(r = 2, . . . , m), , ;
u1 1 )k(1?2/?) 1 )k(1?2/?)+1
(u (u
(2) when ? = 0:
Rk (ur , u1 )(u1 )?k , Sjk (u1 , ur )(u1 )?k
ur , a ab aa ab ab aa

(when r = 1 then k = 1);
the corresponding basis for the conformal algebra AC(n) = ?a , Jab , D, Ka
(Ka = 2xa D ? xb xb ?a ) has the following form:
(1) when ? = 0:
ur
Sjk (?ab , ?ab )(u1 )k(2/??1) ,
r 1
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