Maurer-Cartan equation for gentle algebras

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