Classifying complements for conformal algebras

TL;DR

This paper develops a classification framework for complements in Lie and associative conformal algebras, showing that all complements are deformations of a given subalgebra and constructing a parameterizing classifying object.

Contribution

It introduces a novel approach to classify all complements of a subalgebra in conformal algebras via deformations and bicrossed products, extending to associative conformal algebras.

Findings

All complements are isomorphic to deformations of a fixed complement.

A classifying object parameterizes all R-complements of E.

Explicit examples illustrate the classification method.

Abstract

Let $R\subseteq E$ 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.