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A class of uniformly bounded simple $\mathbb{Z}$-graded Lie conformal algebras

Maosen Xu

TL;DR

The paper addresses the classification of uniformly bounded simple Z-graded Lie conformal algebras with L_0 = Vir and rank L_i ≤ 1. By analyzing the structure constants under Jacobi identities and exploiting the Gel'fand-Dorfman/Novikov algebra framework, it derives the possible forms and supports, culminating in a complete list of isomorphism classes. The main result identifies four families, namely Vir·V(s), CL_1(s), CL_2(b,s), and the simple graded ideal SCL_2(b,s) (the latter arising when 2b ∈ Z), with explicit bracket formulas for each. This work advances the understanding of infinite simple Lie conformal algebras of finite growth and links their structure to Gel'fand-Dorfman and Novikov algebras.

Abstract

In this paper, we classify the following simple $\mathbb{Z}$-graded Lie conformal algebras $\mathcal{L}=\bigoplus_{i\in \mathbb{Z}}\mathcal{L}_i$ such that (1)$rank\mathcal{L}_i\leq 1$, (2)$\mathcal{L}_0$ is the Virasoro Lie conformal algebra.

A class of uniformly bounded simple $\mathbb{Z}$-graded Lie conformal algebras

TL;DR

The paper addresses the classification of uniformly bounded simple Z-graded Lie conformal algebras with L_0 = Vir and rank L_i ≤ 1. By analyzing the structure constants under Jacobi identities and exploiting the Gel'fand-Dorfman/Novikov algebra framework, it derives the possible forms and supports, culminating in a complete list of isomorphism classes. The main result identifies four families, namely Vir·V(s), CL_1(s), CL_2(b,s), and the simple graded ideal SCL_2(b,s) (the latter arising when 2b ∈ Z), with explicit bracket formulas for each. This work advances the understanding of infinite simple Lie conformal algebras of finite growth and links their structure to Gel'fand-Dorfman and Novikov algebras.

Abstract

In this paper, we classify the following simple -graded Lie conformal algebras such that (1), (2) is the Virasoro Lie conformal algebra.
Paper Structure (3 sections, 15 theorems, 110 equations, 1 table)

This paper contains 3 sections, 15 theorems, 110 equations, 1 table.

Table of Contents

  1. introduction
  2. Preliminary
  3. Classification of simple $\mathbb{Z}$-graded Lie conformal algebras $\mathcal{L}$

Key Result

Theorem 1.1

Suppose that $\mathcal{L}$ is a $\mathbb{Z}$-graded simple Lie conformal algebra satisfying $(C1)$ and $(C2)$. Then $\mathcal{L}$ must be isomorphic to:

Theorems & Definitions (35)

  • Theorem 1.1
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.1
  • Remark 2.7
  • Definition 2.8
  • ...and 25 more