The Poincaré and related groups are algebraically determined Polish groups
An important problem for Polish groups, that is, complete separable metric topological groups, is whether or not such a group admits a unique Polish topology compatible with its group structure. There are many Polish groups that are known to have this property; for example, compact simple Lie groups or the group of diffeomorphisms of a smooth manifold, among many others. The authors use a different but equivalent notion to describe this uniqueness property. They call a Polish group \(G\) algebraically determined if every Polish group \(H\) and every group isomorphism from \(H\) to \(G\) is already a homeomorphism. In the paper under review the authors deal with groups that are important in physics: the Heisenberg group \({\mathcal H}\) of all unipotent upper triangular real \(3\times 3\) matrices, the Lorentz group \({\mathcal L}\simeq O_4(\mathbb R,1)\), the Poincaré group \({\mathcal P}\), which is the semi-direct product of \({\mathcal L}\) with \(\mathbb R^4\) under the natural action of \({\mathcal L}\) on \(\mathbb R^4\), and close relatives thereof, the orthochronous Lorentz group \({\mathcal L}_t\), the proper Lorentz group \({\mathcal L}_s\) (determinant 1), the proper orthochronous Lorentz group \({\mathcal L}_e={\mathcal L}_t\cap{\mathcal L}_t\) and the semi-direct products of these subgroups of \({\mathcal L}\) with \(\mathbb R^4\), denoted by \({\mathcal P}_s\), \({\mathcal P}_t\) and \({\mathcal P}_e\), respectively. The authors show that each of \({\mathcal L}\), \({\mathcal L}_t\), \({\mathcal P}\), \({\mathcal P}_s\), \({\mathcal P}_t\) and \({\mathcal P}_e\) is algebraically determined. On the other hand it is observed that \({\mathcal L}_s\), \({\mathcal L}_e\) and \({\mathcal H}\) are Polish groups that are not algebraically determined. For the first two groups this is a consequence of the fact that the complex numbers admit discontinuous automorphisms, and for \({\mathcal H}\) it follows from the existence of discontinuous additive mappings from \(\mathbb R\) to itself. Furthermore, a partial continuity result for \({\mathcal H}\) is obtained: if \(\varphi\) is an algebraic isomorphism from a Polish group \(H\) to \({\mathcal H}\), then its restriction to the center \(Z(H)\) and the induced quotient isomorphism \(\tilde\varphi: H/Z(H) \to {\mathcal H}/Z({\mathcal H})\) are topological isomorphisms of Polish groups. A key tool used in the proof is the well-known result that an algebraic isomorphism between Polish groups which is a Borel mapping is a topological isomorphism. The methods used are a combination of descriptive set theory, Lie group theory, algebraic manipulations involving certain one-parameter subgroups and suitable generating subsets for the various groups considered and a careful analysis on how various groups act on certain naturally associated subspaces of \(\mathbb R^4\).
scientific article; zbMATH DE number 5939724
scientific article