Differentiability of continuous homomorphisms between smooth loops
The main result is that every continuous local homomorphism between smooth local \(H\)-spaces is smooth in a neighbourhood of the neutral element. As a consequence, every continuous homomorphism between smooth left or right loops is smooth. This is a far-reaching generalization of a well-known property of Lie groups. In their proofs, though, the authors go different ways than the usual proofs for Lie groups, as those proofs use (local) one-parameter groups which are not available in this generality. They establish differentiability via a Lipschitz condition and then use a strong result by \textit{J. Kozma} [Arch. Math. 55, No. 5, 498-502 (1990; Zbl 0731.22004)] saying that a differentiable homomorphism is locally a linear map in so- called canonical coordinate systems. Since Kozma's result is formulated for (local) loops only, the authors have to ascertain that Kozma's arguments do not need the inverse properties of a loop and hence remain valid for local \(H\)-spaces. In a second section, the authors study kernels of homomorphisms and quotients of smooth left or right loops. The main result of this paper will have many important applications. The authors announce some applications to the theory of differentiable projective planes, and, more generally, of differentiable generalized polygons, to be proved in a forthcoming paper [the authors, On homomorphisms of generalized polygons, Geom. Dedicata (to appear)]. For instance, they assert that, as a consequence of their result, every continuous collineation between two such geometric structures is smooth.
scientific article; zbMATH DE number 554044