UDC 513.838+519.4

MATHEMATICS

M. A. AKIVIS

ON CANONICAL DECOMPOSITIONS OF EQUATIONS OF A LOCAL ANALYTIC QUASIGROUP

(Presented by Academician A. D. Aleksandrov on 6 III 1969)

1. Let \(X, Y, Z\) be three analytic manifolds of the same dimension \(r\). An analytic mapping \(f: X \times Y \to Z\) is called a local analytic quasigroup \(Q_r\) if, for \(a \in X\), \(b \in Y\), and \(c = f(a,b) \in Z\), the following conditions are satisfied:

a) For every neighborhood \(U_c\) of the point \(c\) there exist neighborhoods \(U_a\) and \(U_b\) of the points \(a\) and \(b\) such that, for any \(x \in U_a\), \(y \in U_b\), the function \(f(x,y)\) is defined and \(f(x,y) \in U_c\).

b) For every neighborhood \(U_a\) of the point \(a\) there exist neighborhoods \(U_b\) and \(U_c\) of the points \(b\) and \(c\) such that, for any \(y \in U_b\) and \(z \in U_c\), the equation \(f(x,y)=z\) has a unique solution with respect to \(x\), and \(x \in U_a\).

c) For every neighborhood \(U_b\) of the point \(b\) there exist neighborhoods \(U_a\) and \(U_c\) of the points \(a\) and \(c\) such that, for any \(x \in U_a\) and \(z \in U_c\), the equation \(f(x,y)=z\) has a unique solution with respect to \(y\), and \(y \in U_b\).

If the manifolds \(X, Y, Z\) coincide, \(X \equiv Y \equiv Z\), and there exists an element \(e\) such that \(f(x,e)=f(e,x)=x\), then the quasigroup \(Q_r\) becomes a local analytic loop \((^1)\). If, moreover, the associativity condition \(f[x,f(y,z)]=f[f(x,y),z]\) is satisfied on the quasigroup \(Q_r\), then it becomes a local \(r\)-parameter Lie group.

In the present paper canonical decompositions are obtained for the analytic function \(z=f(x,y)\) defining the quasigroup \(Q_r\), which pass into the Campbell–Hausdorff formula \((^2)\) when the quasigroup \(Q_r\) becomes a local Lie group. The constructed canonical decompositions are applied to the study of certain special classes of local analytic quasigroups.

1. Let \(a\) and \(b\) be arbitrary points of the manifolds \(X\) and \(Y\), and let \(c=f(a,b)\) be a point belonging to the manifold \(Z\). Introduce local coordinates in neighborhoods \(U_a, U_b\), and \(U_c\) of the points \(a, b\), and \(c\). Then in these neighborhoods the mapping \(f: X \times Y \to Z\) is written in the form:

\[ z^i=f^i(x^j,y^k), \quad i,j,k=1,\ldots,r, \tag{1} \]

where \(x^i\) are the coordinates of the point \(x \in U_a\), \(y^i\) are the coordinates of the point \(y \in U_b\), \(z^i\) are the coordinates of the point \(z \in U_c\), and \(f^i\) are analytic functions.

Suppose that the points \(a\) and \(b\) of the manifolds \(X\) and \(Y\) are determined by the coordinates \(x^i=0\) and \(y^i=0\). Then in the neighborhoods \(U_a\) and \(U_b\) the right-hand sides of equations (1) can be represented in the form of convergent power series:

\[ z^i=\sum_{s=0}^{\infty}\underset{(s)}{\Lambda^i}(x^j,y^k), \tag{2} \]

where

\[ \underset{(s)}{\Lambda^i}(x^j,y^k) = \frac{1}{s!}\sum_{p=0}^{s} C_s^p \lambda^i_{j_1\ldots j_p\, r+1\, j_{p+1}\ldots r+j_s} x^{j_1}\ldots x^{j_p}y^{j_{p+1}}\ldots y^{j_s}. \tag{3} \]

homogeneous polynomials of degree \(s\) in \(x^j, y^k\). Their coefficients \(\lambda^i_{j_1\ldots j_p\, r+j_{p+1}\ldots r+j_s}\) are symmetric in the indices \(j_1\ldots j_p\) and \(j_{p+1}\ldots j_s\), and \(C_s^p\) are binomial coefficients.

The local coordinates of points of the neighborhoods \(U_a, U_b\), and \(U_c\) of the analytic manifolds \(X, Y, Z\) admit transformations of the form
\[ \tilde{x}^i=\varphi^i(x^j),\qquad \tilde{y}^i=\psi^i(y^j),\qquad \tilde{z}^i=\chi^i(z^j), \tag{4} \]
where \(\varphi^i,\psi^i,\chi^i\) are uniquely invertible analytic functions such that \(\varphi^i(0)=0,\ \psi^i(0)=0\). Such transformations are called isotopic coordinate transformations of the quasigroup \(Q_r\). These transformations make it possible to reduce the expansions (2) to a certain simplest form, namely:

Theorem 1. With the aid of transformations (4), the polynomials \(\Lambda^i_{(s)}(x^j,y^k)\) occurring in the expansions (2) can be reduced to the following canonical form:
\[ \Lambda^i_{(0)}=0,\qquad \Lambda^i_{(1)}=x^i+y^i,\qquad \Lambda^i_{(2)}=a^i_{jk}x^j y^k, \]
\[ \Lambda^i_{(s)}=\frac{1}{s!}\sum_{p=1}^{s-1} C_s^p a^i_{j_1\ldots j_p\, r+j_{p+1}\ldots r+j_s} x^{j_1}\ldots x^{j_p}y^{j_{p+1}}\ldots y^{j_s}, \tag{5} \]
where the coefficients of these polynomials satisfy the relations
\[ a^i_{(jk)}=0,\qquad b^i_{(j_1\ldots j_s)}=0, \tag{6} \]
where
\[ b^i_{(j_1\ldots j_s)} = \sum_{p=1}^{s-1} C_s^p\sigma^{\,s-p} a^i_{j_1\ldots j_p\, r+j_{p+1}\ldots r+j_s}, \]
and \(\sigma\) is some fixed real number, \(\sigma\ne 0,-1\).

Expansions of equations (2) satisfying the conditions of this theorem will be called canonical expansions of the equations of the quasigroup \(Q_r\). Thus, the quasigroup \(Q_r\) has not one but an entire pencil of canonical expansions, depending on the parameter \(\sigma\). The variables \(x^i\) and \(y^i\) occurring in the canonical expansions of the quasigroup \(Q_r\) will be called its canonical parameters.

The uniqueness of the canonical expansions is confirmed by the following theorem:

Theorem 2. The canonical expansion of the equations of a local analytic quasigroup preserves its form if and only if the variables entering into it are transformed according to the formulas
\[ \tilde{x}^i=\alpha^i_j x^j,\qquad \tilde{y}^i=\alpha^i_j y^j,\qquad \tilde{z}^i=\alpha^i_j z^j. \tag{7} \]
In this case the coefficients \(a^i_{jk}\) and \(a^i_{j_1\ldots j_p\, r+j_{p+1}\ldots r+j_s}\) of the canonical expansion are transformed according to the tensor law.

The proof of Theorems 1 and 2 is carried out by the method of complete induction on the degree \(s\) of the polynomials \(\Lambda_{(s)}(x,y)\).

1. Let us identify the points of the neighborhoods \(U_a, U_b\), and \(U_c\) that have identical coordinates, and denote by the letter \(e\) the point obtained by identifying the points \(a,b,c\). Then the local quasigroup \(Q_r\) will be defined in some neighborhood \(U\) of this point, and the binary quasigroup operation will be an analytic mapping \(f: U\times U\to U\). In this case the point \(e\) will correspond to the identity element of the quasigroup \(Q_r\), since by virtue of relations (5)
   \[ f(x,e)=f(e,x)=x. \]
   Consequently, the quasigroup \(Q_r\) becomes a local analytic loop. Moreover, by virtue of relations (5) and (6),
   \[ f(x,\sigma x)=(1+\sigma)x, \tag{8} \]

where the latter relation is satisfied for one fixed value of the parameter \(\sigma \ne 0,-1\). We shall call condition (8) the normalization condition for the equations of the quasigroup.

1. Let \(Q_1\) be a one-parameter quasigroup. By virtue of the first of conditions (6), its canonical decompositions are written in the form

\[ z=x+y+\sum_{s=3}^{\infty}\underset{(s)}{\Lambda}(x,y). \]

If we introduce the notation

\[ \underbrace{a_{1\ldots 1}}_{p}\underbrace{\,{}_{2\ldots 2}}_{q}=a_{pq}, \]

then the polynomials \(\underset{(s)}{\Lambda}(x,y)\), for \(s \geqslant 3\), take the form

\[ \underset{(s)}{\Lambda}(x,y)=\frac{1}{s!}\sum_{p=1}^{s-1} C_s^p a_{p,s-p}x^p y^{s-p}, \tag{9} \]

where

\[ \sum_{p=1}^{s-1} C_s^p \sigma^{s-p} a_{p,s-p}=0. \]

A direct verification shows that if the quasigroup \(Q_1\) becomes a one-parameter Lie group \(G_1\), then all its canonical decompositions take the form

\[ z=x+y. \]

Consequently, if the quasigroup \(Q_r\) becomes a Lie group, then the canonical parameters introduced on \(Q_r\) pass into the ordinary canonical parameters of the local Lie group \(G_r\). All canonical decompositions of the quasigroup \(Q_r\) then pass into the Campbell—Hausdorff formula for the Lie group \(G_r\).

1. Consider on the quasigroup \(Q_r\) a one-parameter subquasigroup \(Q_1\). It is defined on \(Q_r\) by the equations:

\[ x^i=x^i(u), \qquad y^i=y^i(v), \qquad f^i[x^j(u),y^h(v)]=z^i(w), \tag{10} \]

where \(w=\varphi(u,v)\). From these relations the following theorem follows easily:

Theorem 3. In order that the first two of equations (10) define on the quasigroup \(Q_r\) a one-parameter subquasigroup \(Q_1\), it is necessary and sufficient that the relations

\[ \lambda \frac{\partial z^i}{\partial x^j}\frac{dx^j}{du}+\mu \frac{\partial z^i}{\partial y^j}\frac{dy^j}{dv}=0, \tag{11} \]

hold, where \(\lambda\) and \(\mu\) are analytic functions of \(u\) and \(v\).

Relations (11) are differential equations of one-parameter subquasigroups on \(Q_r\). These relations make it possible to prove the following theorem:

Theorem 4. In order that on the loop \(Q_r\), for each direction issuing from the point \(e\), there exist a one-parameter subloop \(Q_1\) tangent to this direction, it is necessary and sufficient that the coefficients of at least one of its canonical decompositions satisfy the relations:

\[ \underset{p}{b^i_{(j_1\ldots j_s)}}=\delta^i_{(j_1}\underset{p}{b}_{j_2\ldots j_s)}, \tag{12} \]

where

\[ \underset{p}{b^i_{j_1\ldots j_s}}=a^i_{j_1\ldots j_p\, r+j_{p+1}\ldots r+j_s}. \]

We note that if the condition of Theorem 4 is fulfilled, the coefficients of any of the canonical decompositions of the quasigroup \(Q_r\) will satisfy relations (12).

The three-web \((^3)\) corresponding to the quasigroup \(Q_r\) defined by Theorem 4 was considered by us in \((^4)\) under the name of a transversal-geodesic web.

1. The following theorem answers the question of when the one-parameter subloops \(Q_1\) of the loop \(Q_r\) become groups.

Theorem 5. In order that on the loop \(Q_r\), for every direction issuing from the point \(e\), there exist a one-parameter subgroup \(G_1\) tangent to this direction, it is necessary and sufficient that the coefficients of at least one of its canonical decompositions satisfy the conditions

\[ \underset{p}{b}{}^{\,i}_{(j_1\ldots j_s)}=0. \tag{13} \]

The condition contained in Theorem 5 is equivalent to the monoassociativity \((^3)\) of the loop \(Q_r\). Therefore condition (13) is a necessary and sufficient condition for the monoassociativity of the loop \(Q_r\). Note that in \((^1)\) the existence of one-parameter subgroups of a local loop \(Q_r\) was proved only for alternative loops, which form a narrower class than monoassociative loops.

When condition (13) is fulfilled, for arbitrary \(x\) belonging to a sufficiently small neighborhood of the point \(e\) of the loop \(Q_r\), and for sufficiently small real \(\lambda\) and \(\mu\), the relation

\[ f(\lambda x,\mu x)=(\lambda+\mu)x \]

will hold, and this means that the normalization condition (8) is fulfilled on the quasigroup \(Q_r\) for every \(\sigma\ne 0,-1\). Hence it follows:

Theorem 6. In order that all the canonical decompositions of the equations of the quasigroup \(Q_r\) introduced in Theorem 1 coincide with one another, it is necessary and sufficient that this quasigroup be monoassociative.

Moscow Institute
of Steel and Alloys

Received
24 I 1969

CITED LITERATURE

\(^1\) A. I. Mal’tsev, Matem. sborn., 36 (78), No. 3 (1955).
\(^2\) E. B. Dynkin, DAN, 57, No. 4 (1947).
\(^3\) J. Aczel, Advances Math., 1, No. 3 (1965).
\(^4\) M. A. Akivis, Abstracts of Reports, III Baltic Geom. Conf., Palanga, 1968.