MATHEMATICS

L. E. EVTUSHIK

ON ONE CONSTRUCTION OF INVARIANT FORMS AND STRUCTURAL EQUATIONS OF INFINITE GROUPS

(Presented by Academician P. S. Aleksandrov on 31 III 1962)

Let

\[ \partial_k x^i = h_k^i (X^l, x^l, x^{\alpha_1}), \ldots,\, \partial_{k_1\ldots k_p}x^i = h_{k_1\ldots k_p}^i (X^l, x^l, x^{\alpha_1}, \ldots, x^{\alpha_p}) \]
\[ (i,k,l=1,\ldots,n) \tag{1} \]

be an involutive system of differential equations of order \(p\), determining an infinite group of transformations

\[ x^i=\varphi^i(X^k). \tag{2} \]

Here \(\partial_{k_1\ldots k_s}x^i\) are the partial derivatives of the variables \(x^i\) with respect to \(X^k\), and \(x^{\alpha_1},\ldots,x^{\alpha_p}\) are parametric derivatives from the first to the \(p\)-th order, respectively.

In solving the problem of constructing invariant forms of infinite groups, É. Cartan \((^1)\) reduces equations (1) to a Pfaffian system and gradually transforms it into a system written by means of invariant forms. However, one can avoid the calculations and arguments that are often repeated in this procedure if one first constructs the invariant forms of the group of all analytic transformations. Such a construction is needed in itself, since this group underlies various geometric theories.

Let now

\[ Y^i = a^i(X^k)=x^i+x_k^iX^k+\ldots+\frac{1}{p!}x_{k_1\ldots k_p}^iX^{k_1}\ldots X^{k_p}+\ldots \tag{3} \]

be the totality of all transformations analytic at the point \(X^k=0\). The space of the variables \((x^i,x_k^i,\ldots,x_{k_1\ldots k_p}^i)\) is the representation space of the group of all analytic transformations, showing how the derivatives \(\partial x^i/\partial X^k,\ldots,\partial^p x^i/\partial X^{k_1}\ldots\partial X^{k_p}\) transform under transformations \(x^i\) and with \(X^k\) preserved. We shall for the time being regard \(Y^i\) as arbitrary constants, and \(x^i,x_k^i,\ldots,x_{k_1\ldots k_p}^i\) as functions of a sufficiently large number of parameters. Differentiating (3) under this assumption, we find

\[ dX^i=\omega^i+\omega_k^iX^k+\ldots+\frac{1}{p!}\omega_{k_1\ldots k_p}^iX^{k_1}\ldots X^{k_p}+\ldots, \tag{4} \]

where by \(\omega^i,\omega_k^i,\ldots,\omega_{k_1\ldots k_p}^i\) the following differential forms are denoted:

\[ \omega^i=\overset{*}{x}{}^i_k dx^k \qquad (\overset{*}{x}{}^i_k x_l^k=\delta_l^i), \]
\[ \omega_k^i=\overset{*}{x}{}^i_l (dx_k^l-x_{km}^l\omega^m), \tag{5} \]
\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots \]
\[ \omega_{k_1\ldots k_p}^i= \overset{*}{x}{}^i_l \left( dx_{k_1\ldots k_p}^l - x_{lm\{k_1\ldots k_\alpha} \omega_{k_{\alpha+1}\ldots k_p\}}^m \right), \]

and the braces denote the sum over all possible permutations of pairs of indices taken one from different groups \((k_1\ldots k_\alpha)\) and \((k_{\alpha+1}\ldots k_p)\), with subsequent summation over \(\alpha=1,\ldots,p\). Formula (4) shows that the forms (5) remain invariant when \(X^k\) is preserved and under all analytic transformations over \(Y^i=a^i(X^k)\) and the transformations induced by them-offsetof

formations \((x^i, x_k^i, \ldots, x_{k_1\ldots k_p}^i)\). The structural equations of the invariant forms \(\omega^i, \omega_k^i, \ldots, \omega_{k_1\ldots k_p}^i\) are easily obtained by exterior differentiation of (4)

\[ D\omega^i=[\omega^k\omega_k^i],\qquad D\omega_k^i=[\omega_k^l\omega_l^i]+[\omega^l\omega_{kl}^i], \]

\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \tag{6} \]

\[ D\omega_{k_1\ldots k_p}^i= [\omega_{\{k_1\ldots k_\alpha}^{\,l}\omega_{k_{\alpha+1}\ldots k_p\}l}^{\,i}] +[\omega^l\omega_{k_1\ldots k_p l}^{\,i}]. \]

It is necessary to note the following property of the forms (5): the invariant forms \(\omega^i, \omega_k^i, \ldots, \omega_{k_1\ldots k_p}^i\) take the values

\[ \omega^i=dX^i,\qquad \omega_k^i=0,\ldots,\qquad \omega_{k_1\ldots k_p}^i=0 \]

if and only if

\[ x^i=a^i(X^k),\qquad x_k^i=\partial_k a^i,\ldots,\qquad x_{k_1\ldots k_p}^i=\partial_{k_1\ldots k_p}a^i. \]

Passing now to the group with defining equations (1), put in the forms \(\omega^i, \omega_k^i, \ldots, \omega_{k_1\ldots k_{p-1}}^i\)

\[ x_k^i=h_k^i(X^l,x^l,x^{\alpha_1}), \]

\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots \]

\[ x_{k_1\ldots k_p}^i = h_{k_1\ldots k_p}^i(X^l,x^l,x^{\alpha_1},\ldots,x^{\alpha_p}). \]

Denoting by \(\omega^{\alpha_1},\ldots,\omega^{\alpha_{p-1}}\) the forms, linearly independent among \(\omega_k^i,\ldots,\omega_{k_1\ldots k_{p-1}}^i\), we obtain

\[ \omega_k^i=C_{k\alpha_1}^i\omega^{\alpha_1}+\overline{C}_{kl}^i(\omega^l-dX^l), \]

\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \tag{7} \]

\[ \omega_{k_1\ldots k_{p-1}}^i = C_{k_1\ldots k_{p-1}\alpha_{p-1}}^i\omega^{\alpha_{p-1}} +\cdots+ C_{k_1\ldots k_{p-1}\alpha_1}^i\omega^{\alpha_1} + C_{k_1\ldots k_{p-1}l}^i(\omega^l-dX^l), \]

where all coefficients are invariants of our group.

It can be shown that the defining equations (1) are equivalent to the system of Pfaffian equations

\[ \omega^i=dX^i,\qquad \omega^{\alpha_1}=0,\ldots,\qquad \omega^{\alpha_{p-1}}=0. \]

On this basis one can prove the main assertion: the group with defining equations (1) can be defined as the group of transformations of the variables \(x^i,x^{\alpha_1},\ldots,x^{\alpha_{p-1}}\) \((X^k=0)\) leaving invariant the forms

\[ \omega^i,\omega^{\alpha_1},\ldots,\omega^{\alpha_{p-1}}. \]

Taking into account equations (6) and (7), we obtain the structural equations for the invariant forms \(\omega^i,\omega^{\alpha_1},\ldots,\omega^{\alpha_{p-1}}\)

\[ D\omega^i= \frac12 C_{kl}^i[\omega^k\omega^l]+C_{k\alpha_1}^i[\omega^k\omega^{\alpha_1}], \]

\[ D\omega^{\alpha_1} = \frac12 C_{A_1B_1}^{\alpha_1}[\omega^{A_1}\omega^{B_1}] + C_{k\alpha_2}^{\alpha_1}[\omega^k\omega^{\alpha_2}], \tag{8} \]

\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \]

\[ D\omega^{\alpha_{p-1}} = \frac12 C_{A_{p-1}B_{p-1}}^{\alpha_{p-1}} [\omega^{A_{p-1}}\omega^{B_{p-1}}] + C_{k\alpha_p}^{\alpha_{p-1}}[\omega^k\omega^{\alpha_p}], \]

where the indices \(A_s,B_s\) run through the group of indices \(i,\alpha_1,\ldots,\alpha_s\). It also follows from equations (6) and (7) that some of the coefficients of equations (8) are known in advance to be zero. Thus,

\[ C_{\beta_s\gamma_t}^{\alpha_r}=0\quad \text{for } s+t>r+1. \]

Moscow State University
named after M. V. Lomonosov

Received
27 III 1962

CITED LITERATURE

1. E. Cartan, Sur la structure des groupes infinis de transformations, Oeuvres complètes, partie II, 2, Paris, 1953, p. 571; Les groupes de transformations continus, infinis, simples, ibid., p. 857.