ñòð. 120 |

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

ñòð. 120 |