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Extensions of finite irreducible modules over rank two Lie conformal algebra

Lipeng Luo, Yucai Su, Mengjun Wang

TL;DR

This work delivers a complete classification of extensions of finite irreducible conformal modules over rank-two Lie conformal algebras, covering semisimple, solvable, and the two non-semisimple non-solvable types (Type I and Type II). The authors reduce the extension problem to solving cocycle equations, yielding explicit polynomial cocycles that describe nontrivial extensions between modules such as $V_{\\delta,\\alpha,\\beta}$, $V_{\\phi_A,\\phi_B}$, and $V_{\\alpha,\\beta}$ across the four algebra families. The results reveal a spectrum of extension dimensions, including 0, 1, 2, and, in many solvable cases, infinite-dimensional Ext spaces, with precise conditions under which each occurs and complete formulas for the cocycles. The findings advance the understanding of indecomposable finite conformal modules and provide a comprehensive foundation for further study in conformal representation theory and related algebraic structures.

Abstract

In this paper, we give a complete classification of extensions of finite irreducible conformal modules over rank two Lie conformal algebras.

Extensions of finite irreducible modules over rank two Lie conformal algebra

TL;DR

This work delivers a complete classification of extensions of finite irreducible conformal modules over rank-two Lie conformal algebras, covering semisimple, solvable, and the two non-semisimple non-solvable types (Type I and Type II). The authors reduce the extension problem to solving cocycle equations, yielding explicit polynomial cocycles that describe nontrivial extensions between modules such as , , and across the four algebra families. The results reveal a spectrum of extension dimensions, including 0, 1, 2, and, in many solvable cases, infinite-dimensional Ext spaces, with precise conditions under which each occurs and complete formulas for the cocycles. The findings advance the understanding of indecomposable finite conformal modules and provide a comprehensive foundation for further study in conformal representation theory and related algebraic structures.

Abstract

In this paper, we give a complete classification of extensions of finite irreducible conformal modules over rank two Lie conformal algebras.
Paper Structure (18 sections, 31 theorems, 115 equations)

This paper contains 18 sections, 31 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. For semisimple rank two Lie conformal algebras
  4. $0\longrightarrow \mathbb{C}c_{\eta} \longrightarrow E\longrightarrow V_{\delta,\alpha,\beta} \longrightarrow 0$
  5. $0\longrightarrow V_{\delta,\alpha,\beta} \longrightarrow E\longrightarrow\mathbb{C}c_{\eta} \longrightarrow 0$
  6. $0\longrightarrow V_{\delta,\alpha,\beta} \longrightarrow E\longrightarrow V_{\bar{\delta},\bar{\alpha},\bar{\beta}} \longrightarrow 0$
  7. For solvable rank two Lie conformal algebras
  8. $0\longrightarrow \mathbb{C}c_{\eta} \longrightarrow E\longrightarrow V_{\phi_A,\phi_B} \longrightarrow 0$
  9. $0\longrightarrow V_{\phi_A,\phi_B} \longrightarrow E\longrightarrow\mathbb{C}c_{\eta} \longrightarrow 0$
  10. $0\longrightarrow V_{\phi_A,\phi_B} \longrightarrow E\longrightarrow V_{\bar{\phi}_A,\bar{\phi}_B} \longrightarrow 0$
  11. For rank two Lie conformal algebras that are of Type I
  12. $0\longrightarrow \mathbb{C}c_{\eta} \longrightarrow E\longrightarrow V_{\alpha,\beta,\phi} \longrightarrow 0$
  13. $0\longrightarrow V_{\alpha,\beta,\phi} \longrightarrow E\longrightarrow\mathbb{C}c_{\eta} \longrightarrow 0$
  14. $0\longrightarrow V_{\alpha,\beta,\phi} \longrightarrow E\longrightarrow V_{\bar{\alpha},\bar{\beta},\bar{\phi}} \longrightarrow 0$
  15. For rank two Lie conformal algebras that are of Type II
  16. ...and 3 more sections

Key Result

Proposition 2.5

[bch, Theorem 3.21] Let $\mathcal{R}$ be a rank two Lie conformal algebra that is not semisimple.

Theorems & Definitions (65)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Proposition 2.5
  • Remark 2.6
  • Proposition 2.7: xhw, Theorem 3.2
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 55 more