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

Abstract

We construct a transferred $L_{\infty }$-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 $L_{\infty }$-formal, and contrast this with Wong's non-formality result for the dual numbers $\mathbb{K}[\theta ]/⟨{\theta }^2⟩$, leading to a general formality classification problem.

60 pages, any comments are welcome