Explicit Hochschild Cocycles for Gentle Algebras

TL;DR

This paper provides an explicit construction of Hochschild cocycles for gentle algebras, extending previous methods to cases with a single puncture, thereby enhancing understanding of their deformation theory.

Contribution

It introduces a new explicit method for constructing Hochschild cocycles for gentle algebras, applicable even when the punctured surface has only one puncture.

Findings

Explicit Hochschild cocycles constructed for all punctured surfaces.

Method extends previous implicit constructions to single puncture cases.

Enhances deformation theory analysis for gentle algebras.

Abstract

Hochschild cohomology is crucial for understanding deformation theory. In arXiv:2304.10223, we have computed the Hochschild cohomology for gentle algebras of punctured surfaces. The construction of that paper is rather implicit and fails if the punctured surface has only a single puncture. In the present note, we supplement the earlier method by providing an explicit construction of Hochschild cocycles which also succeeds in the case of a single puncture.