Classification of uniformly bounded simple Lie conformal algebras with upper bound one

TL;DR

The paper resolves the classification problem for uniformly bounded simple Lie conformal algebras with upper bound one by proving finite generation and analyzing the simple graded case. It develops a two-track approach: (i) a preliminary framework for rank-one graded algebras and (ii) a detailed split into integral versus non-integral cases within class $\mathcal{V}$, yielding explicit isomorphism types such as $SCL_2(b,s)$, $Cur\mathfrak{g}$, $\mathcal{V}(s)$, $CL_2(b_1,s)$, $Vir$, and $CL_1(s)$, with $\mathfrak{g}$ uniformly bounded. The integral scenario collapses to current algebras $Cur\mathfrak{g}$ for simple graded $\mathfrak{g}$, while the non-integral branch recovers loop-like Virasoro/Witt structures, culminating in a complete classification of simple graded Lie conformal algebras with bound one. These results bridge finite and infinite families in Lie conformal theory and have implications for related vertex-algebra constructions.

Abstract

In this paper, we prove that uniformly bounded simple Lie conformal algebra must be finitely generated. Furthermore, we give a completely classification of simple uniformly bounded Lie conformal algebras with upper bound one.

Theorems & Definitions (72)

• Proposition 1.1
• Theorem 1.2
• Definition 2.1
• Example 2.2
• Example 2.3
• Definition 2.4
• Definition 2.5
• Proposition 2.6
• Definition 3.1
• Proposition 3.2