Maurer-Cartan equation for gentle algebras

TL;DR

This paper investigates the conditions under which the $L_$-structure on the shifted Bardzell's complex of a gentle algebra becomes nilpotent and describes Maurer-Cartan elements in these cases.

Contribution

It provides new criteria for nilpotency of the $L_$-structure and characterizes Maurer-Cartan elements for gentle algebras under certain conditions.

Findings

Identifies conditions for nilpotency of the $L_$-structure.

Describes Maurer-Cartan elements in nilpotent cases.

Focuses on gentle algebras with specific quiver hypotheses.

Abstract

Let $A= \Bbbk Q/I$ be a finite-dimensional gentle algebra. In this article, under some hypothesis on the quiver $Q$, we give conditions for nilpotency of the $L_\infty$-structure on the shifted Bardzell's complex $B(A)[1]$. For nilpotent cases, we describe Maurer-Cartan elements.