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

TL;DR

This paper explores the structure of super cluster algebras derived from super Plücker and super Ptolemy relations, extending known results to super Grassmannians and revealing new forms and properties of these relations.

Contribution

It develops the super cluster structure of super Grassmannians and simplifies super Plücker relations for specific cases, advancing the understanding of super cluster algebras.

Findings

Super cluster structure of $ ext{Gr}_{2|0}(n|1)$ established for arbitrary n.

Super Ptolemy relation can be transformed into classical form with decoupled variables.

New simple form of super Plücker relations for $ ext{Gr}_{r|1}(n|1)$ obtained.

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'').