Deformation maps on quasi-twilled Lie conformal algebras

In this paper, we develop a unified approach for various operators on Lie conformal algebras. Given a quasi-twilled Lie conformal algebra $(\text{Ep},\text{Vs},\text{Ws})$, we introduce two dual families of operators: \emph{right deformation maps} $D:\text{Vs}\to \text{Ws}$ and \emph{left deformation maps} $B:\text{Ws}\to \text{Vs}$. Each family simultaneously subsumes several classical structures: modified $r$-matrices, crossed homomorphisms, derivations, and Lie conformal algebra homomorphisms in the right case, relative Rota-Baxter operators, twisted Rota-Baxter operators, Reynolds operators, and deformation maps of matched pairs in the left case. Using Voronov’s derived bracket method, we construct the controlling homotopy algebras: a curved $L_{\mathrm{\infty }}$-algebra governing right deformation maps and an $L_{\mathrm{\infty }}$-algebra governing left deformation maps, with Maurer-Cartan elements precisely characterizing each type. We further develop the associated deformation theories via twisted $L_{\mathrm{\infty }}$-algebras and define cohomology complexes for both types of deformation maps, recovering and extending the cohomologies of all classical and conformal operators already developed in the literature.