Combinatorics of graded module categories over skew polynomial algebras at roots of unity

TL;DR

This work addresses the problem of classifying equivalence classes of graded module categories over standard graded skew polynomial algebras at a primitive $\ell$-th root of unity by encoding the defining commutation data in skew-symmetric matrices $M_{\omega} \in \operatorname{Alt}_n(\mathbb{Z}/\ell\mathbb{Z})$ and introducing the switching operation. It proves that graded module category equivalence coincides with switching equivalence of the associated $M_{\omega}$, and defines modular Eulerian matrices to capture canonical representatives when $\gcd(n,\ell)=1$, with a prime-$\ell$ counting result. In the cube-root case ($\ell=3$), the point simplicial complex $\Delta_{\omega}$ determines equivalence for $n\le 5$, with explicit classifications via vertex isolations and several counterexamples for larger $n$, and the framework extends to $\ell \ge 4$. Overall, the paper connects noncommutative projective geometry invariants to combinatorial matrix theory, reducing a geometric classification problem to a tractable algebraic-switching and enumeration problem on matrices and digraphs.

Abstract

We introduce an operation on skew-symmetric matrices over $\mathbb{Z}/\ell\mathbb{Z}$ called switching, and also define a class of skew-symmetric matrices over $\mathbb{Z}/\ell\mathbb{Z}$ referred to as modular Eulerian matrices. We then show that these are closely related to the graded module categories over skew polynomial algebras at $\ell$-th roots of unity. As an application, we study the point simplicial complexes of skew polynomial algebras at cube roots of unity.

Theorems & Definitions (59)

• Theorem 1.1: HU
• Theorem 1.2: Theorem \ref{['thm.grmodl']}
• Theorem 1.3: Theorem \ref{['thm.Eulermodl']}
• Theorem 1.4: Theorem \ref{['thm.classps']}, Example \ref{['ex.simpcomp']}(1)
• Definition 2.1
• Definition 2.2
• Proposition 2.3: Vi, G
• Definition 2.4
• Lemma 2.5
• proof