# Unified products of Leibniz conformal algebras

## TL;DR

This paper introduces a unified product framework to classify Leibniz conformal algebra extensions, providing new cohomological tools and applying them to specific free module cases with illustrative examples.

## Contribution

It develops a unified product approach and cohomological classification for Leibniz conformal algebra extensions, including special product types and explicit examples.

## Key findings

- Classified Leibniz conformal algebra extensions using cohomological objects.
- Introduced twisted, crossed, and bicrossed products as special cases.
- Provided explicit examples illustrating the theory.

## Abstract

The aim of this paper is to provide an answer to the $\mathbb{C}[\partial]$-split extending structures problem for Leibniz conformal algebras, which asks that how to describe all Leibniz conformal algebra structures on $E=R\oplus Q$ up to an isomorphism such that $R$ is a Leibniz conformal subalgebra. For this purpose, an unified product of Leibniz conformal algebras is introduced. Using this tool, two cohomological type objects are constructed to classify all such extending structures up to an isomorphism. Then this general theory is applied to the special case when $R$ is a free $\mathbb{C}[\partial]$-module and $Q$ is a free $\mathbb{C}[\partial]$-module of rank one. Finally, the twisted product, crossed product and bicrossed product between two Leibniz conformal algebras are introduced as special cases of the unified product, and some examples are given.