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:
