Classifying complements for conformal algebras

Let R subset of E be two Lie conformal algebras and Q be a given complement of R in E. Classifying complements problem asks for describing and classifying all complements of R in E up to an isomorphism. It is known that E is isomorphic to a bicrossed product of R and Q. We show that any complement of R in E is isomorphic to a deformation of Q associated to the bicrossed product. A classifying object is constructed to parameterize all R-complements of E. Several explicit examples are provided. Similarly, we also develop a classifying complements theory of associative conformal algebras.