Regular actions and semisimplicity of conformal modules over the general conformal algebra

TL;DR

This paper defines regular actions in conformal modules over Lie conformal algebras and establishes semisimplicity criteria for modules over the general conformal algebra, also classifying Virasoro elements and constructing new modules.

Contribution

It introduces the concept of regular actions and provides semisimplicity criteria for modules over $\, ext{gc}_N$, along with classifying Virasoro elements and constructing new modules.

Findings

Semisimplicity of modules characterized by regular actions and Virasoro elements.

Classification of Virasoro elements in $\, ext{gc}_N$.

Construction of numerous new Virasoro conformal modules.

Abstract

We introduce the notion of a regular action in the category of conformal modules over Lie conformal algebras with Virasoro elements. We show that a finite conformal module over the general conformal algebra $\mathfrak{gc}_1$ (resp., $\mathfrak{gc}_N$ with $N\ge2$) is semisimple if and only if there exists a pair of different Virasoro elements (resp., canonical Virasoro elements) with regular actions. Along the way to finding a semisimplicity criteria, we also discuss the classification of Virasoro elements of $\mathfrak{gc}_N$ in-depth, leading us to construct a huge number of new Virasoro conformal modules.