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Four-vertex quivers supporting twisted graded Calabi-Yau algebras

Jason Gaddis, Thomas Lamkin, Thy Nguyen, Caleb Wright

Abstract

We study quivers supporting twisted graded Calabi-Yau algebras, building on work of Rogalski and the first author. Specifically, we classify quivers on four vertices in which the Nakayama automorphism acts on the degree zero part by either a four-cycle, a three-cycle, or two two-cycles. In order to realize algebras associated to some of these quivers, we show that graded twists of a twisted graded Calabi-Yau algebra is another algebra of the same type.

Four-vertex quivers supporting twisted graded Calabi-Yau algebras

Abstract

We study quivers supporting twisted graded Calabi-Yau algebras, building on work of Rogalski and the first author. Specifically, we classify quivers on four vertices in which the Nakayama automorphism acts on the degree zero part by either a four-cycle, a three-cycle, or two two-cycles. In order to realize algebras associated to some of these quivers, we show that graded twists of a twisted graded Calabi-Yau algebra is another algebra of the same type.
Paper Structure (12 sections, 17 theorems, 57 equations)

This paper contains 12 sections, 17 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Quivers and path algebras
  4. Twisted graded Calabi--Yau algebras
  5. Examples of Calabi--Yau algebras
  6. McKay quivers
  7. Ore extensions
  8. Graded ("Zhang") twists
  9. Quivers
  10. Four cycle
  11. Three cycle
  12. Two two-cycles

Key Result

Theorem 1

Let $Q$ be a quiver with $|Q_0|=4$, let $M$ denote the adjacency matrix of $Q$, and let $\omega$ be a homogeneous superpotential on $Q$ with $s=\deg(\omega)$. Suppose $A=\Bbbk Q/(\partial_a(\omega) : a \in Q_1)$ is a twisted graded Calabi--Yau algebra of dimension 3 and Gelfand--Kirillov dimension 3 Moreover, each matrix $M$ appears as the adjacency matrix of a twisted graded Calabi--Yau algebra s

Theorems & Definitions (33)

  • Theorem 1: Theorems \ref{['thm.4cycle']}, \ref{['thm.3cycle']}, \ref{['thm.22cycle']}
  • Theorem 2: Theorem \ref{['thm.AStwist']}
  • Definition 2.1
  • Theorem 2.2: BSW
  • Theorem 2.4: RR1
  • Lemma 2.5
  • proof
  • Example 3.1
  • Example 3.2
  • Theorem 3.3
  • ...and 23 more