Data

  • Title: Algebraic properties of zigzag algebras
  • Authors: Michael Ehrig and Daniel Tubbenhauer
  • Status: Comm. Algebra 48 (2020), no.1, 11-36. Last update: Tue, 2 Jul 2019 11:28:16 UTC
  • ArXiv link: https://arxiv.org/abs/1807.11173
  • ArXiv version = 0.99 published version
  • LaTex Beamer presentation: Slides1

Abstract

We give necessary and sufficient conditions for zigzag algebras and certain generalizations of them to be (relative) cellular, quasi-hereditary or Koszul.

A few extra words

Let \(\mathrm{Z}_{\rightleftarrows}=\mathrm{Z}_{\rightleftarrows}(\Gamma)\) be the zigzag algebra associated to a finite, connected, simple graph \(\Gamma\). The purpose of this note is to show the following.

Theorem A
\(\mathrm{Z}_{\rightleftarrows}\) is cellular if and only if \(\Gamma\) is a finite type \(\mathsf{A}\) graph. \(\mathrm{Z}_{\rightleftarrows}\) is relative cellular if and only if \(\Gamma\) is a finite or affine type \(\mathsf{A}\) graph.
Further, in all cases where \(\mathrm{Z}_{\rightleftarrows}\) is (relative) cellular, the path length endows it with the structure of a graded (relative) cellular algebra.
Theorem B
\(\mathrm{Z}_{\rightleftarrows}\) is never quasi-hereditary.
Theorem C
\(\mathrm{Z}_{\rightleftarrows}\) is Koszul if and only if \(\Gamma\) is not a type \(\mathsf{ADE}\) graph.

Moreover, in all cases we construct the corresponding data explicitly.

Let further \(\mathrm{Z}_{\rightleftarrows}^{\mathtt{B}}=\mathrm{Z}_{\rightleftarrows}^{\mathtt{B}}(\Gamma)\) be the zigzag algebra with a vertex-loop condition (vertex condition for short) set of vertices \(\mathtt{B}\neq\emptyset\). Using the same ideas as for \(\mathrm{Z}_{\rightleftarrows}\) we can also prove:

Theorem A\(\mathtt{B}\)
\(\mathrm{Z}_{\rightleftarrows}^{\mathtt{B}}\) is cellular if and only if \(\Gamma\) is a finite type \(\mathsf{A}\) graph and the vertex condition is imposed on one leaf. \(\mathrm{Z}_{\rightleftarrows}^{\mathtt{B}}\) is relative cellular in exactly the same cases.
Theorem B\(\mathtt{B}\)
\(\mathrm{Z}_{\rightleftarrows}^{\mathtt{B}}\) is quasi-hereditary if and only if \(\Gamma\) is a finite type \(\mathsf{A}\) graph and the vertex condition is imposed on one leaf.
Theorem C\(\mathtt{B}\)
\(\mathrm{Z}_{\rightleftarrows}^{\mathtt{B}}\) is always Koszul.

Here is an example of how a linear projective resolution might look like:

which gives a linear projective resolution of the simple corroding to the vertex 0.