Cohomology and deformation theory of 𝒪-operators on Hom-Lie conformal algebras

In this paper, we aim to introduce the cohomology of [Formula: see text]-operators defined on the Hom-Lie conformal algebra with the coefficients from its representation. To obtain the desired results, we constructed a differential graded Lie algebra. We show that the differential maps on the cochain complex of the graded Lie algebra can also be described by using the Maurer–Cartan element. The [Formula: see text]-operator on the given Hom-Lie conformal algebra serves as a Maurer–Cartan element, which yields the differential map associated with an [Formula: see text]-operator. Next, we provide the notion of Hom-pre-Lie conformal algebra, which induces a sub-adjacent Hom-Lie conformal algebra structure. The differential of this sub-adjacent Hom-Lie conformal algebra is related to the differential map in terms of the [Formula: see text]-operator. Finally, as an application to the cohomology, we provide the deformation of [Formula: see text]-operators on the Hom-Lie conformal algebras, where we discuss linear and formal deformations in detail.