Data Abstract A few extra words
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 $0$.