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Quiver down-up algebras

Jason Gaddis, Dennis Keeler

TL;DR

The paper generalizes Benkart–Roby down-up algebras to quivers by introducing quiver down-up algebras, and develops a comprehensive structural theory. It proves noetherian and piecewise-domain properties under nondegeneracy conditions via a generalized Weyl algebra realization, and establishes twisted graded Calabi–Yau status for the graded family. It further adopts superpotential and skew group algebra frameworks to illuminate homological properties and Morita equivalences, culminating in a complete isomorphism classification for graded quiver down-up algebras when the quiver has at least three vertices. The results connect to preprojective algebras and highlight the interplay between quiver geometry, derivation-quotient presentations, and CY–type homological behavior, with clear criteria for when isomorphisms occur and how the parameters transform.

Abstract

We present a generalization of down-up algebras, originally defined by Benkart and Roby. These quiver down-up algebras arise as quotients of the double of the extended Dynkin quiver of type $A$. Under a certain non-degeneracy condition, we show that quiver down-up algebras are noetherian piecewise domains. The $\NN$-graded members of the family are proved to be, generically, twisted Calabi--Yau. Finally, we consider the isomorphism problem for graded quiver down-up algebras.

Quiver down-up algebras

TL;DR

The paper generalizes Benkart–Roby down-up algebras to quivers by introducing quiver down-up algebras, and develops a comprehensive structural theory. It proves noetherian and piecewise-domain properties under nondegeneracy conditions via a generalized Weyl algebra realization, and establishes twisted graded Calabi–Yau status for the graded family. It further adopts superpotential and skew group algebra frameworks to illuminate homological properties and Morita equivalences, culminating in a complete isomorphism classification for graded quiver down-up algebras when the quiver has at least three vertices. The results connect to preprojective algebras and highlight the interplay between quiver geometry, derivation-quotient presentations, and CY–type homological behavior, with clear criteria for when isomorphisms occur and how the parameters transform.

Abstract

We present a generalization of down-up algebras, originally defined by Benkart and Roby. These quiver down-up algebras arise as quotients of the double of the extended Dynkin quiver of type . Under a certain non-degeneracy condition, we show that quiver down-up algebras are noetherian piecewise domains. The -graded members of the family are proved to be, generically, twisted Calabi--Yau. Finally, we consider the isomorphism problem for graded quiver down-up algebras.

Paper Structure

This paper contains 7 sections, 18 theorems, 40 equations.

Table of Contents

  1. Quivers and a basis for quiver down-up algebras
  2. Generalized Weyl algebras
  3. Skew group algebras and twisted CY algebras
  4. Superpotential algebras
  5. Skew group algebras
  6. Main proof
  7. The isomorphism problem

Key Result

Theorem A

Suppose $\beta_i \neq 0$ for all $i \in Q_0$. Then $\mathcal{H}=\mathcal{H}(\alpha,\beta,0)$ is twisted-graded Calabi--Yau of global dimension three with Nakayama automorphism $\mu^{\mathcal{H}}$ given by $\mu^{\mathcal{H}}(u_i)=-\beta_{i-1}^{-1} u_i$ and $\mu^{\mathcal{H}}(d_i)=-\beta_{i-1}^{-1} d_

Theorems & Definitions (35)

  • Definition 1
  • Theorem A: Theorem \ref{['thm.CY']}
  • Theorem B: Theorem \ref{['thm.main']}
  • Theorem C: Theorem \ref{['thm.iso']}
  • Proposition 1.1
  • proof
  • Example 1.2
  • Corollary 1.3
  • proof
  • Proposition 2.1
  • ...and 25 more