Poisson structures and star products on quasimodular forms

TL;DR

This paper classifies all Poisson structures on quasimodular forms extending classical brackets and uses them to develop formal deformations of their algebraic structure.

Contribution

It provides a complete classification of Poisson structures on quasimodular forms and constructs associated formal deformations, extending known modular form brackets.

Findings

Complete classification of Poisson structures on quasimodular forms.

Construction of formal deformations based on these structures.

Extension of the first Rankin-Cohen bracket to quasimodular forms.

Abstract

We construct and classify all Poisson structures on quasimodular forms that extend the one coming from the first Rankin-Cohen bracket on the modular forms. We use them to build formal deformations on the algebra of quasimodular forms.