Combinatorial classification of $(\pm 1)$skew projective spaces
Abstract
The noncommutative projective scheme $\operatorname{\mathsf{Proj_{nc}}} S$ of a $(\pm 1)$skew polynomial algebra $S$ in $n$ variables is considered to be a $(\pm 1)$skew projective space of dimension $n1$. In this paper, using combinatorial methods, we give a classification theorem for $(\pm 1)$skew projective spaces. Specifically, among other equivalences, we prove that $(\pm 1)$skew projective spaces $\operatorname{\mathsf{Proj_{nc}}} S$ and $\operatorname{\mathsf{Proj_{nc}}} S'$ are isomorphic if and only if certain graphs associated to $S$ and $S'$ are switching (or mutation) equivalent. We also discuss invariants of $(\pm 1)$skew projective spaces from a combinatorial point of view.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 DOI:
 10.48550/arXiv.2107.12927
 arXiv:
 arXiv:2107.12927
 Bibcode:
 2021arXiv210712927H
 Keywords:

 Mathematics  Rings and Algebras;
 Mathematics  Algebraic Geometry;
 Mathematics  Combinatorics;
 Mathematics  Representation Theory
 EPrint:
 14 pages, v2: minor modifications, v3: removed an example