# Extending structures for associative conformal algebras

## TL;DR

This paper develops a cohomological framework for classifying extending structures of associative conformal algebras, especially those free as modules, using unified products and flag datums.

## Contribution

It introduces a cohomological approach to classify $C[partial]$-split extensions of associative conformal algebras, including explicit classifications for rank 1 cases.

## Key findings

- Constructed a cohomological object for classifying extensions.
- Described extensions of free associative conformal algebras via flag datums.
- Provided detailed classification up to equivalence for rank 1 cases.

## Abstract

In this paper, we give a study of the $\mathbb{C}[\partial]$-split extending structures problem for associative conformal algebras. Using the unified product as a tool, which includes interesting products such as bicrossed product, cocycle semi-direct product and so on, a cohomological type object is constructed to characterize the $\mathbb{C}[\partial]$-split extending structures for associative conformal algebras. Moreover, using this theory, the extending structures of an associative conformal algebra $A$ which is free as a $\mathbb{C}[\partial]$-module by the $\mathbb{C}[\partial]$-module $Q=\mathbb{C}[\partial]x$ are described using flag datums of $A$. Furthermore, we give a classification of the extending structures of $A$ by $Q=\mathbb{C}[\partial]x$ in detail up to equivalence when $A$ is a free associative conformal algebra of rank 1.