Year: 2023
Type: preprint
Abstract: Let $\mathcal{P}$ be the set of points of a finite-dimensional projective space over a local field $F$, endowed with the topology $τ$ naturally induced from the canonical topology of $F$. Intuitively, continuous incidence abelian group structures on $\mathcal{P}$ are abelian group structures on $\mathcal{P}$ preserving both the topology $τ$ and the incidence of lines with points. We show that the real projective line is the only finite-dimensional projective space over an Archimedean local field which admits a continuous incidence abelian group structure. The latter is unique up to isomorphism of topological groups. In contrast, in the non-Archimedean case we construct continuous incidence abelian group structures in any dimension $n \in \mathbb{N}$. We show that if $n>1$ and the characteristic of $F$ does not divide $n+1$, then there are finitely many possibilities up to topological isomorphism and, in any case, countably many.
Source: arXiv (Cornell University)
Authors: Cangiotti, Nicolò, Linzi, Alessandro
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: From Models to Decisions, G15967578
OpenAlex ID: W4375957268