On analytical diassociative composition laws (Q1896939)

Let \(M\) be an analytical manifold with a fixed point \(e\). Assume that \(M\) is endowed with a local analytical composition law \(M\times M\to M\), \((x,y)\mapsto xy\) such that \(xe=ex=x\) for all \(x\in M\) that are sufficiently close to \(e\). Suppose that the identities \(x(xy)=x^2y\), \((yx)x=yx^2\), hold in \(M\). The composition law induces a structure of an anticommutative algebra on the tangent space \(T_e (M)\). This algebra satisfies a homogeneous identity of degree 5. Some calculations in the algebra \(T_e (M)\) are made. These calculations are used in the proof of the main result of the paper which states that the following are equivalent: (i) \(M\) satisfies the identity \(x(xy)=(xy)x\); (ii) an arbitrary pair of elements in \(M\) generates a subgroup in \(M\) provided these elements are sufficiently close to \(e\).