Another class of simple graded Lie conformal algebras that cannot be embedded into general Lie conformal algebras

TL;DR

This paper classifies a new class of simple graded Lie conformal algebras that cannot be embedded into general Lie conformal algebras, expanding understanding of their structure and examples, including some of Block type.

Contribution

It introduces a new class of simple graded Lie conformal algebras with specific properties, not embeddable into general Lie conformal algebras, and includes examples of Block type.

Findings

Classified a new class of simple graded Lie conformal algebras.

Identified these algebras include some of Block type.

Proved these algebras cannot be embedded into general Lie conformal algebras.

Abstract

In a previous paper by the authors, we obtain the first example of a finitely freely generated simple $\mathbb Z$-graded Lie conformal algebra of linear growth that cannot be embedded into any general Lie conformal algebra. In this paper, we obtain, as a byproduct, another class of such Lie conformal algebras by classifying $\mathbb Z$-graded simple Lie conformal algebras ${\cal G}=\oplus_{i=-1}^\infty{\cal G}_i$ satisfying the following, (1) ${\cal G}_0\cong{\rm Vir}$, the Virasoro conformal algebra; (2) Each ${\cal G}_i$ for $i\ge-1$ is a ${\rm Vir}$-module of rank one. These algebras include some Lie conformal algebras of Block type.