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Simple restricted modules over a new Lie superalgebra extended by the Ovsienko--Roger algebra

Jinrong Wang, Xiaoqing Yue

TL;DR

This work introduces the infinite-dimensional Lie superalgebra $\mathcal{S}$, a super extension of the Ovsienko--Roger framework, obtained as a central extension of the annihilation algebra of a rank $(2+1)$ Lie conformal superalgebra. It develops a systematic construction of simple restricted $\mathcal{S}$-modules by inducing from simple modules over finite-dimensional solvable subquotients and achieves a classification of simple generalized Verma modules, showing that Verma modules for $\mathcal{S}$ are always reducible. The results tie the restricted representation theory of $\mathcal{S}$ to well-studied structures in the Ramond/Neveu–Schwarz family and to Block-type modules, while demonstrating a precise equivalence among several natural finiteness and nilpotence conditions for simple restricted modules. Overall, the paper extends the landscape of restricted representations in Lie superalgebras and provides concrete constructions and classifications that may inform further study of related vertex superalgebras and deformations.

Abstract

In this paper, we introduce a new infinite-dimensional Lie superalgebra $\mathcal{S}$ called the super extended Ovsienko--Roger algebra. This algebra is obtained by determining the annihilation superalgebra of the Lie conformal superalgebra $S=S_{\bar0}\oplus S_{\bar{1}}$ with $S_{\bar{0}}=\mathbb{C}[\partial]L\oplus\mathbb{C}[\partial]W$, $S_{\bar{1}}=\mathbb{C}[\partial]G$ and non-trivial $λ$-brackets $[L_λL]=(\partial+2λ)L$, $[L_λG]=(\partial+λ)G$, $[L_λW]=[G_λG]=\partial W$. Then we construct a class of simple restricted $\mathcal{S}$-modules, which are induced from simple modules of some finite dimensional solvable Lie superalgebras under certain conditions. Moreover, we obtain the classification of simple generalized Verma modules over $\mathcal{S}$ and we show that the Verma module of $\mathcal{S}$ is always reducible.

Simple restricted modules over a new Lie superalgebra extended by the Ovsienko--Roger algebra

TL;DR

This work introduces the infinite-dimensional Lie superalgebra , a super extension of the Ovsienko--Roger framework, obtained as a central extension of the annihilation algebra of a rank Lie conformal superalgebra. It develops a systematic construction of simple restricted -modules by inducing from simple modules over finite-dimensional solvable subquotients and achieves a classification of simple generalized Verma modules, showing that Verma modules for are always reducible. The results tie the restricted representation theory of to well-studied structures in the Ramond/Neveu–Schwarz family and to Block-type modules, while demonstrating a precise equivalence among several natural finiteness and nilpotence conditions for simple restricted modules. Overall, the paper extends the landscape of restricted representations in Lie superalgebras and provides concrete constructions and classifications that may inform further study of related vertex superalgebras and deformations.

Abstract

In this paper, we introduce a new infinite-dimensional Lie superalgebra called the super extended Ovsienko--Roger algebra. This algebra is obtained by determining the annihilation superalgebra of the Lie conformal superalgebra with , and non-trivial -brackets , , . Then we construct a class of simple restricted -modules, which are induced from simple modules of some finite dimensional solvable Lie superalgebras under certain conditions. Moreover, we obtain the classification of simple generalized Verma modules over and we show that the Verma module of is always reducible.
Paper Structure (13 sections, 13 theorems, 101 equations)

This paper contains 13 sections, 13 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lie superalgebra
  4. Lie conformal superalgebra
  5. New Lie superalgebras derived from $S$
  6. Annihilation superalgebra $\mathrm{Lie }(S)$
  7. Central extension
  8. Lie superalgebra $\mathcal{S}$
  9. Construction of simple restricted $\mathcal{S}$-modules
  10. Characterization of simple restricted modules
  11. Examples of restricted $\mathcal{S}$-modules
  12. Verma module
  13. Whittaker module

Key Result

Lemma 3.1

Suppose that the Lie superalgebra $\bar{\mathcal{S}}^0$ is the annihilation superalgebra of $S$. Then $\bar{\mathcal{S}}^0=\mathrm{Lie }(S)=\bigoplus_{m\in\mathbb{Z}}(\mathbb{C} L_m\oplus\mathbb{C} W_m\oplus\mathbb{C} G_m)$ satisfies the following relations: where $|L_m|=|W_m|={\bar{0}}$ and $|G_m|={\bar{1}}$, $m,n\in\mathbb{Z}$.

Theorems & Definitions (38)

  • Definition 1.1
  • Definition 1.2
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Lemma 3.1
  • proof
  • Definition 3.2
  • ...and 28 more