An elementary proof of the fundamental theorem of projective geometry. (English) Zbl 0996.51001
The author proves the following generalization of the classical result of projective geometry which states that every isomorphism between Desarguesian projective spaces is induced by some semilinear map between the associated vector spaces. Theorem: Let \(V\), \(W\) denote vector spaces over division rings \(K\) and \(L\), respectively. Let \(g:{\mathcal P}(V) \to {\mathcal P}(W)\) be a morphism between the associated projective spaces. If the image of \(g\) is not contained in some line of \({\mathcal P}(W)\), then there exists a semilinear map \(f:V\to W\) such that \(g\) is induced by \(f\), i.e. \(g(\langle x\rangle) = \langle f(x)\rangle\) for any admissible \(x\). Moreover, as in the classical case, the map \(f\) is unique up to scalar multiplication.
As an application, the author gives a short proof of Wigner’s theorem.
As an application, the author gives a short proof of Wigner’s theorem.
Reviewer: Richard Bödi (Adliswil)
MSC:
51A05 | General theory of linear incidence geometry and projective geometries |
51A10 | Homomorphism, automorphism and dualities in linear incidence geometry |
81P10 | Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) |
46C05 | Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) |