Multidimensional smooth loops with universal elasticity

Let (E) over tilde be a universal (isotopically invariant) identity that is derived from the elasticity identity E: (xy)x = x(yx). One of the authors has previously shown that a) each local loop of dimension r with identity (E) over tilde (briefly, a loop (E) over tilde) is a smooth middle Bol loop of dimension r; b) smooth two-dimensional loops (E) over tilde are Lie groups; c) up to isotopy, there exist only two three-dimensional loops (E) over tilde: the loops E-1 and E-2. In this paper, the loops E-1 and E-2 are extended to the multidimensional case. The fact that each smooth loop (E) over tilde of dimension r corresponds to a unique multidimensional three-web on a manifold of dimension 2r is key to our work. In addition, the class of loops under investigation is characterized by the fact that the torsion tensor of the corresponding web has rank 1 (that is, the algebra generated by this tensor has a one-dimensional derived algebra). This enables us to express the differential equations of the problem in an invariant form. The system of equations thus obtained was found to be amenable to integration in the most general case, and the equations of the required loops have been obtained in local coordinates.