Abstract

The aim of this article 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, a 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.

摘要

本文的目的是为 $\mathbb{C}[\partial ]$莱布尼兹保形代数的分拆扩展结构问题，该问题要求如何描述所有莱布尼兹保形代数结构 $E=\mathrm{[R}\oplus 问$直到R是Leibniz保形子代数的同构。为此，引入了莱布尼兹保形代数的统一乘积。使用该工具，构造了两个同调类型对象，以将所有此类扩展结构分类为同构。然后将此通用理论应用于当R为自由时的特殊情况$\mathbb{C}[\partial ]$-module和Q是免费的$\mathbb{C}[\partial ]$-排名第一的模块。最后，介绍了两个莱布尼兹共形代数之间的扭曲积，交叉积和双叉积，作为统一积的特例，并给出了一些例子。