Combinatorial classification of $(\pm 1)$-skew projective spaces

TL;DR

This paper classifies $( ext{±}1)$-skew projective spaces using combinatorial graph methods, establishing criteria for their isomorphism and exploring related invariants.

Contribution

It provides a combinatorial classification theorem for $( ext{±}1)$-skew projective spaces, linking algebraic isomorphisms to graph switching equivalences.

Findings

Spaces are isomorphic iff their associated graphs are switching equivalent.

Introduces invariants of skew projective spaces from a combinatorial perspective.

Establishes a classification criterion based on graph mutations.

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 $n-1$. 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.