On super cluster algebras based on super Pl\"ucker and super Ptolemy relations

Abstract

We study super cluster algebra structure arising in examples provided by super Pl\"{u}cker and super Ptolemy relations. We develop the super cluster structure of the super Grassmannians $\Gr_{2|0}(n|1)$ for arbitrary $n$, which was indicated earlier in our joint work with Th. Voronov. For the super Ptolemy relation for the decorated super Teichm\"{u}ller space of Penner-Zeitlin, we show how by a change of variables it can be transformed into the classical Ptolemy relation with the new even variables decoupled from odd variables. We also analyze super Pl\"{u}cker relations for general super Grassmannians and obtain a new simple form of the relations for $\Gr_{r|1}(n|1)$. To this end, we establish properties of Berezinians of certain type matrices (which we call ``wrong'').