Cluster algebras and Weil-Petersson forms

TL;DR

This paper extends the study of cluster algebras by exploring their geometric structures through closed 2-forms, generalizing previous work on Poisson properties and connecting to Weil-Petersson forms in Teichmüller theory.

Contribution

It introduces a new approach using closed 2-forms compatible with cluster algebras for general transition matrices, linking algebraic and geometric structures.

Findings

Generalized the geometric interpretation of cluster algebras

Connected cluster algebra structures to Weil-Petersson forms

Provided a framework for understanding Poisson properties in broader cases

Abstract

In our previous paper we have discussed Poisson properties of cluster algebras of geometric type for the case of a nondegenerate matrix of transition exponents. In this paper we consider the case of a general matrix of transition exponents. Our leading idea that a relevant geometric object in this case is a certain closed 2-form compatible with the cluster algebra structure. The main example is provided by Penner coordinates on the decorated Teichmueller space, in which case the above form coincides with the classic Weil-Petersson symplectic form.