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3-Preprojective Algebras of Type D

Jordan Haden

TL;DR

The work studies type $D$ selfinjective algebras as $\mathbb{Z}_3$-quotients of type $A$ 3-preprojective algebras, proving Morita equivalence with $\Pi_\mathcal{A}^s\#\mathbb{Z}_3$ and identifying a realization as Evans–Pugh type $D$ quivers. It shows a third of these algebras are themselves 3-preprojective and that the associated 2-representation-finite algebras are fractional Calabi–Yau, with a precise CY-dimension $\frac{2t}{t+1}$ when $s=3t+1$. The paper also provides explicit descriptions of $2$-Auslander–Reiten quivers and connects the construction to SU(3) modular invariants via operator-algebra viewpoints. It ends with criteria for cuts, derived-equivalences, and several natural questions for higher $d$-analogues and Auslander-theoretic aspects.

Abstract

We present a family of selfinjective algebras of type D, which arise from the 3-preprojective algebras of type A by taking a $\mathbb{Z}_3$-quotient. We show that a subset of these are themselves 3-preprojective algebras, and that the associated 2-representation-finite algebras are fractional Calabi-Yau. In addition, we show our work is connected to modular invariants for SU(3).

3-Preprojective Algebras of Type D

TL;DR

The work studies type selfinjective algebras as -quotients of type 3-preprojective algebras, proving Morita equivalence with and identifying a realization as Evans–Pugh type quivers. It shows a third of these algebras are themselves 3-preprojective and that the associated 2-representation-finite algebras are fractional Calabi–Yau, with a precise CY-dimension when . The paper also provides explicit descriptions of -Auslander–Reiten quivers and connects the construction to SU(3) modular invariants via operator-algebra viewpoints. It ends with criteria for cuts, derived-equivalences, and several natural questions for higher -analogues and Auslander-theoretic aspects.

Abstract

We present a family of selfinjective algebras of type D, which arise from the 3-preprojective algebras of type A by taking a -quotient. We show that a subset of these are themselves 3-preprojective algebras, and that the associated 2-representation-finite algebras are fractional Calabi-Yau. In addition, we show our work is connected to modular invariants for SU(3).
Paper Structure (8 sections, 25 theorems, 47 equations, 2 figures, 2 tables)

This paper contains 8 sections, 25 theorems, 47 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. 3-preprojective algebras of type A
  3. Morita equivalence of $\Pi_\mathcal{D}^s$ and $\Pi_\mathcal{A}^s\#\mathbb{Z}_3$
  4. Connection with operator algebras
  5. Cuts of $\Pi_\mathcal{D}^s$
  6. $\Pi_\mathcal{D}^s/\langle C\rangle$ is fractional Calabi-Yau
  7. 2-Auslander-Reiten quivers
  8. Further questions

Key Result

Theorem 1

For each $s\ge2$, $\Pi^s_\mathcal{D}$ is Morita equivalent to $\Pi^s_\mathcal{A}\#\mathbb{Z}_3$.

Figures (2)

  • Figure 1: Top to bottom: $\mathcal{D}^4,\mathcal{D}^5,\mathcal{D}^6,\mathcal{D}^7$.
  • Figure 2: The $\mathbb{Z}_3$-invariant cut $K$ of $\mathcal{A}^7$ and the induced cut $K'$ of $\mathcal{D}^7$, indicated by the dashed arrows.

Theorems & Definitions (64)

  • Definition 1.2
  • Definition 1.4
  • Example 1.5
  • Theorem 1: \ref{['Theorem 1 body']}
  • Theorem 2: \ref{['Theorem 2 body']}
  • Theorem 3: \ref{['Theorem 3 body']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • ...and 54 more