Regular actions and semisimplicity of conformal modules over the general conformal algebra
Yucai Su, Chunguang Xia
TL;DR
This work introduces the notion of regular actions for conformal modules over Lie conformal algebras with Virasoro elements and derives semisimplicity criteria for finite modules over the general conformal algebras ${\mathfrak{gc}}_1$ and ${\mathfrak{gc}}_N$. The main results show that a finite ${\mathfrak{gc}}_1$-module $V$ is semisimple iff there exist two distinct Virasoro elements whose actions on a ${\mathbb{C}}[\partial]$-basis are regular, while for $N\ge2$ the criterion uses two distinct canonical Virasoro elements with regular actions. Along the way, the paper classifies Virasoro elements of ${\mathfrak{gc}}_1$ completely, develops a detailed framework for canonical and standard Virasoro elements in ${\mathfrak{gc}}_N$, and constructs many new Virasoro conformal modules. A key part of the proofs leverages the Heisenberg–Virasoro subalgebra for $N=1$ and Skolem–Noether rigidity for $N\ge2$, linking regularity to semisimplicity via decomposition into irreducible constituents. These results provide Weyl-type semisimplicity criteria in conformal representation theory and yield a rich supply of new Virasoro modules with potential applications to conformal and vertex algebra contexts.
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.