Super Caldero--Chapoton map for type $A$

Abstract

One can explicitly compute the generators of a surface cluster algebra either combinatorially, through dimer covers of snake graphs, or homologically, through the CC-map applied to indecomposable modules over the appropriate algebra. Recent work by Musiker, Ovenhouse and Zhang used Penner and Zeitlin's decorated super Teichm{\"u}ller theory to define a super version of the cluster algebra of type $A$ and gave a combinatorial formula to compute the even generators. We extend this theory by giving a homological way of explicitly computing these generators by defining a super CC-map for type $A$.