Data

1. Title: Algebraic properties of zig-zag algebras
2. Authors: Michael Ehrig and Daniel Tubbenhauer
3. Status: Preprint. Last update: Mon, 30 Jul 2018 05:03:56 GMT

Abstract

We give necessary and sufficient conditions for zig-zag 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 zig-zag 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 $0$.