Combinatorial classification of (±1)-skew projective spaces

Abstract

Abstract The non-commutative 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 $\operatorname{\mathsf{Proj_{nc}}} S^{\prime}$ are isomorphic if and only if certain graphs associated with S and Sʹ are switching (or mutation) equivalent. We also discuss invariants of $(\pm 1)$-skew projective spaces from a combinatorial point of view.