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Lie conformal superalgebras of rank (2 + 1)

Jinrong Wang, Xiaoqing Yue

TL;DR

This work delivers a complete classification of Lie conformal superalgebras of rank (2+1) up to isomorphism, including a determination of their automorphism groups. Building on the rank-two BCH classification and conformal module theory, the authors partition the algebras into four types (A–D) with 18 nontrivial cases plus 5 trivial ones, and they analyze the action of the odd part as a module over the even part. They then classify all finite nontrivial irreducible conformal modules (FNICMs) for these algebras, showing that FNICMs are restricted to rank one or rank (1+1) with explicit module families and actions. The results illuminate the representation theory of rank (2+1) Lie conformal superalgebras, reveal connections to classical objects like the Neveu–Schwarz and Heisenberg–Virasoro algebras, and yield corollaries about single super deformations and reductions to even parts that inform future applications in conformal field theory and related algebraic structures.

Abstract

In this paper, Lie conformal superalgebras of rank (2 + 1) are completely classified (up to isomorphism) and their automorphism groups are determined. Furthermore, we give the classification of the finite irreducible conformal modules over them and the actions are explicitly described.

Lie conformal superalgebras of rank (2 + 1)

TL;DR

This work delivers a complete classification of Lie conformal superalgebras of rank (2+1) up to isomorphism, including a determination of their automorphism groups. Building on the rank-two BCH classification and conformal module theory, the authors partition the algebras into four types (A–D) with 18 nontrivial cases plus 5 trivial ones, and they analyze the action of the odd part as a module over the even part. They then classify all finite nontrivial irreducible conformal modules (FNICMs) for these algebras, showing that FNICMs are restricted to rank one or rank (1+1) with explicit module families and actions. The results illuminate the representation theory of rank (2+1) Lie conformal superalgebras, reveal connections to classical objects like the Neveu–Schwarz and Heisenberg–Virasoro algebras, and yield corollaries about single super deformations and reductions to even parts that inform future applications in conformal field theory and related algebraic structures.

Abstract

In this paper, Lie conformal superalgebras of rank (2 + 1) are completely classified (up to isomorphism) and their automorphism groups are determined. Furthermore, we give the classification of the finite irreducible conformal modules over them and the actions are explicitly described.
Paper Structure (19 sections, 46 theorems, 110 equations, 4 tables)

This paper contains 19 sections, 46 theorems, 110 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lie conformal superalgebra
  4. Conformal module
  5. Lie conformal superalgebra of rank $(2+1)$
  6. Type $\mathcal{A}$
  7. Type $\mathcal{B}$
  8. Type $\mathcal{C}$
  9. Type $\mathcal{D}$
  10. Brief summary
  11. Automorphism groups
  12. Representations of Lie conformal superalgebras of rank $(2+1)$
  13. Classifications of finite nontrivial irreducible conformal modules (FNICMs)
  14. FNICMs of type $\mathcal{A}$
  15. FNICMs of type $\mathcal{B},\mathcal{C}$
  16. ...and 4 more sections

Key Result

Lemma 2.7

Let $R$ be a Lie conformal algebra and $M$ be a finite nontrivial irreducible conformal $R$-module. Then $M$ has no nonzero torsion elements and is free of finite rank as a $\mathbb{C}[\partial]$-module.

Theorems & Definitions (71)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • Proposition 2.8
  • Proposition 2.9
  • Definition 2.10
  • ...and 61 more