Algebraic constructions for left-symmetric conformal algebras

Let $R$ be a left-symmetric conformal algebra and $Q$ be a $\mathbb{C}[\mathrm{\partial }]$-module. We introduce the notion of a unified product for left-symmetric conformal algebras and apply it to construct an object ${\mathcal{H}}_R^2(Q,R)$ to describe and classify all left-symmetric conformal algebra structures on the direct sum $E=R\oplus Q$ as a $\mathbb{C}[\mathrm{\partial }]$-module such that $R$ is a subalgebra of $E$ up to isomorphism whose restriction on $R$ is the identity map. Moreover, we study ${\mathcal{H}}_R^2(Q,R)$ in detail when $Q$, $R$ are free as $\mathbb{C}[\mathrm{\partial }]$-modules and $\text{rank}Q=1$. Some special products such as crossed product and bicrossed product are also investigated.