Transferred $L_{\mathrm{\infty }}$-structures and a classification of binary brackets for finite-dimensional skew-gentle algebras

We construct a transferred $L_{\mathrm{\infty }}$-structure on the shifted Chouhy–Solotar cochain complex of a finite-dimensional skew-gentle algebra and give explicit basis-level formulas for its binary operation $l_2$ in all shifted degrees. We further prove that the Hochschild dg Lie algebra of an $A_n$-type skew-gentle algebra is homotopy abelian, and hence $L_{\mathrm{\infty }}$-formal, and contrast this with Wong’s non-formality result for the dual numbers $\mathbb{K}[\theta ]\mathrm{/}⟨{\theta }^2⟩$, leading to a general formality classification problem.