Differentiability of continuous homomorphisms between smooth loops
type: Item
instance of: scholarly article
title: Differentiability of continuous homomorphisms between smooth loops
zbMATH Open document ID: 0817.22004
DOI: 10.1007/BF03323137
author: Richard Bödi
author: Linus Kramer
published in: Results in Mathematics
publication date: 1994-05-16T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 22A22
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 58H05
Mathematics Subject Classification ID: 51H25
zbMATH DE Number: 554044
zbMATH Keywords: continuous local homomorphism
zbMATH Keywords: smooth local \(H\)-spaces
zbMATH Keywords: smooth left or right loops
zbMATH Keywords: differentiability
zbMATH Keywords: differentiable homomorphism
zbMATH Keywords: differentiable projective planes
reviewed by: Hermann Hähl
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 554044