UDC 519

MATHEMATICS

V. M. MAKSIMOV

ON A GENERALIZATION OF CAUCHY’S FUNDAMENTAL THEOREM

(Presented by Academician A. N. Kolmogorov on 12 XII 1968)

Consider an associative finite-dimensional algebra \(\mathfrak A\) over the field of real numbers. Let \(e_1,e_2,\ldots,e_s\) be some basis of \(\mathfrak A\).

An arc \(\Gamma\) in \(\mathfrak A\) is called rectifiable if it can be given parametrically by
\[ y(t)=\sum_{i=1}^{s} y_i(t)e_i,\qquad t_0\leqslant t\leqslant t_1, \]
where the \(y_i(t)\) are continuous and of bounded variation. Let \(f(y)\) be a continuous mapping of \(\mathfrak A\) into \(\mathfrak A\). The sum
\[ \sum_i f(y(\xi_i))\bigl(y(t_{i+1}^{(n)})-y(t_i^{(n)})\bigr), \]
where \(\xi_i\) is an arbitrary point of \([t_{i+1}^{(n)},t_i^{(n)}]\), as
\[ \max_i\bigl(t_{i+1}^{(n)}-t_i^{(n)}\bigr)\to 0 \]
tends to a limit independent of the sequences \(\{t_i^{(n)}\}\). We shall call this limit, as usual, the integral of \(f(y)\) and denote it by
\[ \int_\Gamma f(y)\,dy. \]
The linear span of the elements \(xy-yx=[x,y]\) for \(x,y\in\mathfrak A\) is called the commutator of \(\mathfrak A\) and is denoted by \([\mathfrak A]\). Let \(\Gamma\) be a rectifiable contour in \(\mathfrak A\). Then the following holds.

Theorem 1. For any integer \(n\geqslant 0\),
\[ \int_\Gamma y^n\,dy\in[\mathfrak A]. \]

Proof. As in the classical case (¹), it is enough to prove Theorem 1 for a triangular contour. Let \(\Gamma\) be a triangular contour with vertices \(A,B,O\). The vertex \(O\) lies at the zero of the algebra. Then
\[ \int_\Gamma y^n\,dy = \int_{\overline{OA}} y^n\,dy + \int_{\overline{AB}} y^n\,dy + \int_{\overline{BO}} y^n\,dy. \]
Put \(\overrightarrow{OA}=a,\ \overrightarrow{AB}=b,\ \overrightarrow{BO}=d\).

Clearly, \(a+b+d=0\). Denote by \(M(n-m,m)\) the set of monomials \(\alpha\) in the expansion of \((a+b)^n\) that contain \(m\) factors \(b\) and \(n-m\) factors \(a\), and put
\[ Q(n-m,m)=\sum_{\alpha\in M(n-m,m)} \alpha. \]
Then
\[ \int_{\overline{OA}} y^n\,dy = \lim_{N\to\infty}\sum_{i=1}^{N}\left(\frac{i}{N}a\right)^n\left(\frac{1}{N}a\right) = a^{n+1}\int_{0}^{1} t^n\,dt = \frac{1}{n+1}a^{n+1}, \]
\[ \int_{\overline{AB}} y^n\,dy = \lim_{N\to\infty}\sum_{i=1}^{N}\left(a+\frac{i}{N}b\right)^n\left(\frac{1}{N}b\right) = \left[\sum_{m=0}^{n}\frac{1}{m+1}Q(n-m,m)b\right], \]
\[ \int_{\overline{BO}} y^n\,dy = \lim_{N\to\infty}\sum_{i=1}^{N}\left\{(a+b)+\frac{i}{N}d\right\}^n\left(\frac{1}{N}d\right) = -(a+b)^{n+1}\lim_{N\to\infty}\sum_{i=1}^{N}\left(1-\frac{i}{N}\right)^n\frac{1}{N} \]
\[ = -(a+b)^{n+1}\int_{0}^{1}(1-t)^n\,dt = -\frac{1}{n+1}(a+b)^{n+1}. \]

After simplification we obtain
\[ \int_\Gamma y^n\,dy = \sum_{m=0}^{n-1} \left\{ \frac{1}{m+1}Q(n-m,m)b - \frac{1}{n+1}Q(n-m,m+1) \right\}. \]
We shall show that the difference
\[ \rho=(n+1)Q(n-m,m)b-(m+1)Q(n-m,m+1)\in[\mathfrak A], \qquad m=0,\ldots,n-1. \]
The number of terms of each sum

\(Q(n-m,m)b\) and \(Q(n-m,m+1)\) are equal respectively to \(\binom{n}{m}\) and \(\binom{n+1}{m+1}\). Therefore the number of monomials in \((n+1)Q(n-m,m)b\) and in \((m+1)Q(n-m,m+1)\) is the same. We shall match the monomials from these sums so that their difference is a commutator.

Let \(\alpha\) be a monomial. Then \(\alpha_i\) is equal to the product of the \(i\) first factors of \(\alpha\) in the order in which they occur, and \(\beta_j\), respectively, to the product of the \(j\) last factors, i.e. \(\alpha=\alpha_i\beta_{n-i}\), if \(\alpha\) has \(n\) factors. Let, further,

\[ R_i(\alpha)=\alpha_i b\beta_{n-i},\qquad R_i(\alpha+\alpha'+\ldots)=R_i(\alpha)+R_i(\alpha')+\ldots . \]

Lemma 1.

\[ R(Q(n,m))=bQ(n,m)+\sum_{i=1}^{n+m-1} R_i(Q(n,m))+Q(n,m)b= \]

\[ =(m+1)Q(n,m+1). \tag{1} \]

Proof. If (1) holds for \(Q(n-1,m)\) and \(Q(n,m-1)\), then it also holds for \(Q(n,m)\). Indeed,

\[ Q(n,m)=Q(n-1,m)a+Q(n,m-1)b. \tag{2} \]

Then
\[ R(Q(n,m))=R(Q(n-1,m)a)+R(Q(n,m-1)b). \]
But
\[ R(Q(n-1,m)a)=R(Q(n-1,m))a+Q(n-1,m)ab \]
and
\[ R(Q(n,m-1)b)=R(Q(n,m-1))b+Q(n,m-1)b^2. \]
Taking now (1) and (2) into account, we obtain
\[ \begin{aligned} R(Q(n,m))&=(m+1)Q(n-1,m+1)a+mQ(n,m)b\\ &\quad+[Q(n-1,m)a+Q(n,m-1)b]b\\ &=(m+1)Q(n-1,m+1)a+mQ(n,m)b+Q(n,m)b\\ &=(m+1)[Q(n-1,m+1)a+Q(n,m)b]\\ &=(m+1)Q(n,m+1). \end{aligned} \]

To prove the lemma it remains to show that (1) holds for \(Q(0,m)\) and \(Q(n,0)\) for arbitrary \(n\) and \(m\). But from the definition of \(R\) we have
\[ R(Q(0,m))=(m+1)b^{m+1}=(m+1)Q(0,m+1), \]
\[ R(Q(n,0))=ba^n+aba^{n+1}+\ldots+a^n b=Q(n,1). \]
Put now \(\pi_j(\alpha)=\beta_{n-j}\alpha_j\) for \(\alpha\in M(n-m,m)=M\),
\[ \pi_j(\alpha+\alpha'+\ldots)=\pi_j(\alpha)+\pi_j(\alpha')+\ldots . \]
Since from \(\pi_j(\alpha)=\pi_j(\alpha')\) it follows that \(\alpha=\alpha'\) and conversely (here equality is understood in the sense of coincidence of the factors in the same positions), we have

\[ \pi_j\left(\sum_{\alpha\in M}\alpha\right) =\sum_{\alpha\in M}\pi_j(\alpha) =\sum_{\alpha\in M}\beta_{n-j}\alpha_j =Q(n-m,m). \]

Therefore

\[ \pi_j(Q(n-m,m)b)=\sum_{\alpha\in M}\pi_j(\alpha b) =\sum_{\alpha\in M}\beta_{n-j}b\alpha_j= \]

\[ =R_{n-j}\left(\sum_{\alpha\in M}\beta_{n-j}\alpha_j\right) =R_{n-j}(Q(m-m,m)). \tag{3} \]

By virtue of the lemma and (3) we have

\[ (m+1)Q(n-m,m)=bQ(n-m,m)+\sum_{i=1}^{n-1}R_i(Q(n-m,m))+ \]

\[ +Q(n-m,m)b=bQ(n-m,m)+\sum_{j=1}^{n-1}\pi_j(Q(n-m,m)b)+Q(n-m,m)b. \]

Consequently

\[ \varphi=Q(n-m,m)b-bQ(n-m,m)+\sum_{j=1}^{n-1}\{Q(n-m,m)b-\pi_j(Q(n-m,m)b)\}= \]

\[ =\sum_{\alpha\in M}[\alpha,b]+\sum_{j=1}^{n-1}\left(\sum_{\alpha\in M}(\alpha b-\pi_j(\alpha b))\right). \]

Since from the definition of \(\pi_j\) it follows that
\[ \alpha b-\pi_j(\alpha b)=[\alpha_j,\beta_{n-j}b], \]
then

\[ \rho=\sum_{\alpha\in M}[\alpha,b]+\sum_{j=1}^{n-1}\left(\sum_{\alpha\in M}[\alpha_j,\beta_{n-j}b]\right)\in[\mathfrak A]. \]

The theorem is proved.

Considering \([\mathfrak A]\) as a linear subspace of \(\mathfrak A\), define \(B\), \(B\perp[\mathfrak A]\). If \(L=a_1e_1+\cdots+a_se_s\perp[\mathfrak A]\), then for \(x=x_1e_1+\cdots+x_se_s\in[\mathfrak A]\) we have \(L(x)=a_1x_1+\cdots+a_sx_s=0\). The totality of such forms has dimension equal to the dimension of \(B\). According to the theorem, the integrals \(\int_a^b y^n\,dy\), taken along different paths from \(a\) to \(b\), differ by an element of \([\mathfrak A]\). Therefore the following generalization of the fundamental theorem of Cauchy holds.

Theorem 2. The value
\[ L\left(\int_a^b y^n\,dy\right) \]
does not depend on the path connecting the points \(a\) and \(b\).

Suppose now that \(\mathfrak A\) has a unit. We shall assume that the basis element \(e_1\) is the unit of \(\mathfrak A\).

Theorem 3.
\[ L\left(\int_\Gamma y^{-1}dy\right)=0, \]
if the contour \(\Gamma\) belongs to some simply connected domain \(D\) in which \(y^{-1}\) is defined.

Proof. Since \(D\) is simply connected, for arbitrary \(d>0\) there exist in \(D\) contours \(\Gamma_i\) of diameter \(<d\) such that
\[ \int_\Gamma y^{-1}dy=\sum_i\int_{\Gamma_i}y^{-1}dy. \]
Let \(y_i\in D\) be such that \(|x-y_i|\le d\) for \(x\in\Gamma_i\), and let \(d\) be chosen so that from \(|y|<d\) it follows that
\[ (e_1+y)^{-1}=\sum_{n=0}^{\infty}(-y)^n . \]
Then
\[ \int_{\Gamma_i}y^{-1}dy = \int_{\Gamma_i-y_i}(y_i+z)^{-1}dz = \int_{\Gamma_i-y_i}\bigl(y_i(e_1+y_i^{-1}z)\bigr)^{-1}dz = \int_{y_i^{-1}(\Gamma_i-y_i)}(e_1+y)^{-1}dy . \]
Since for \(|y|<d\)
\[ (e_1+y)^{-1}=\sum_{n=0}^{\infty}(-y)^n, \]
Theorem 2 is fulfilled for it. Theorem 3 is proved.

The function
\[ \int_{e_1}^{x}y^{-1}dy \]
has an application in probability theory on finite groups.

Lemma 2.
\[ L\left(\int_{e_1}^{x}y^{-1}dy\right)=\varphi(x) \]
is a one-dimensional representation of a neighborhood of the unit of \(\mathfrak A\).

Indeed, since
\[ \int_{x_1}^{x_1x_2}y^{-1}dy = \int_{e_1}^{x_2}(x_1y)^{-1}\,d(x_1y) = \int_{e_1}^{x_2}y^{-1}dy, \]
we have
\[ \varphi(x_1x_2)L\left(\int_{e_1}^{x_1x_2}y^{-1}dy\right) = L\left(\int_{e_1}^{x_1}y^{-1}dy+\int_{x_1}^{x_1x_2}y^{-1}dy\right) = \varphi(x_1)+\varphi(x_2). \]

Let now \(\mathfrak A\) be the group algebra of some group \(G\). Then one may assume that the basis \(e_1,e_2,\ldots,e_s\) is the group \(G\). In this case compute
\[ \varphi_j(x)=L_j\left(\int_{e_1}^{x}y^{-1}dy\right), \]
where \(L_j,\ j=1,\ldots,r,\) is a basis of linear forms,

corresponding to some basis \(\{\gamma^{(j)}\}\) of the orthogonal complement to \(\mathfrak A\). It is not difficult to show that in the case of a group algebra one may take for \(\{\gamma^{(j)}\}\) vectors consisting of zeros and ones, the ones in \(\gamma^{(j)}\) occurring only in those places \(\alpha\) for which \(e_\alpha\) belongs to the \(j\)-th class of conjugate elements of \(G\). Here a fixed numbering of these classes is meant; denote them by \(T_j,\ j=1,\ldots,r\). Then

\[ \varphi_i(x)=\sum_{j=1}^{r}\frac{h_j}{s}\chi_i(T_j^{-1})\ln |R_i|, \tag{4} \]

where \(|R_i|\) is the determinant of the \(i\)-th irreducible part of the group matrix of the element \(x\); \(\chi_i\) is the character of the \(i\)-th irreducible representation; \(h_j\) is the number of elements of \(T_j\); \(s\) is the order of the group \(G\).

Let \(y(t)\), \(t\geqslant 0\), \(y(0)=e_1\), be a continuous curve in \(\mathfrak A\) such that for each \(t\) there exists \(y^{-1}(t)\), and for the elements \(y^{-1}(t_1)y(t_2)\), denoted by
\(y(t_1,t_2)=p_1(t_1,t_2)e_1+\cdots+p_s(t_1,t_2)e_s\), one has \(p_i(t_1,t_2)\geqslant 0\),

\[ \sum_{i=1}^{s}p_i(t_1,t_2)=1. \]

Then \(y(t_1,t_2)\) may be regarded as a probability distribution on the group \(G\). From the distributions \(y(t)\) one can construct a process with independent increments on the group \(G\) such that the distribution of the increment of the process on the interval \([t_1,t_2]\) is equal to \(y(t_1,t_2)\). The number of jumps of such a process is finite almost everywhere. Using expression (1) from (2), one can show that the curve \(y(t)\) on \([0,t]\) is rectifiable and, for

\[ \Delta_n=\max_i\bigl(t_{i+1}^{(n)}-t_i^{(n)}\bigr)\to 0,\qquad \sum_{i=1}^{n-1}p_j\bigl(t_i^{(n)},t_{i+1}^{(n)}\bigr)\to m_j(t)\geqslant 0,\quad j\geqslant 2, \]

\[ \sum_{i=1}^{n-1}\{p_i(t_i^{(n)},t_{i+1}^{(n)})-1\}\to m_1(t). \]

The functions \(m_j(t)\), \(j\geqslant 2\), are nondecreasing and continuous. From (3), the number of jumps of the process under consideration on the interval \([t_1,t_2]\) into the set \(A\subset G/e_1\) is distributed according to the Poisson law. The parameter of this Poisson distribution is equal to

\[ \sum_{i,\,e_i\in A} m_i(t_2)-m_i(t_1). \]

Let us integrate the function \(y_{n-1}^{-1}\) along the curve \(y(t)\). We have

\[ L_j\left(\int_{e_1}^{y(t)} y^{-1}\,dy\right) = L_j\left(\lim_{\Delta_n\to 0}\sum_{i=1}^{n-1}y^{-1}(t_i^{(n)}) \bigl(y(t_{i+1}^{(n)})-y(t_i^{(n)})\bigr)\right) = \]

\[ = L_j\left(\lim_{\Delta_n\to 0}\sum_{i=0}^{n-1}y^{-1}(t_i^{(n)})y(t_i^{(n)}) \bigl(y(t_i^{(n)},t_{i+1}^{(n)})-1\bigr)\right) = \sum_{i,\,e_i\in T_j} m_i(t). \]

By Theorem 3 this integral does not depend on the path of integration. Therefore the quantity

\[ \sum_{i,\,e_i\in T_j} m_i(t) \]

will be the same for different curves \(y_1,y_2\), provided \(y_1(t)=y_2(t)\).

We shall say that a set \(A\subset G\setminus e_1\) is invariant if the distribution of jumps into this set on any interval \([t_1,t_2]\) depends only on the distribution of the increment of the process, i.e., on \(y(t_1,t_2)\). Then, on the basis of what has been said, the following holds.

Theorem 4. A set \(A\subseteq G\setminus e_1\) is invariant if and only if it is the union of some number of conjugacy classes \(T_j\).

Institute of Chemical Physics
Academy of Sciences of the USSR
Moscow

Received
12 XII 1968

REFERENCES

\(^{1}\) S. Stoilov, Theory of Functions of a Complex Variable, 1, IL, 1962.
\(^{2}\) V. M. Maksimov, Probability Theory and Its Applications, 12, no. 4 (1967).
\(^{3}\) V. M. Maksimov, DAN, 182, no. 1 (1968).