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Non-weight modules over gap-$p$ Virasoro algebras

Chengkang Xu, Fulin Chen, Shaobin Tan

TL;DR

The article advances the theory of non-weight representations for the gap-$p$ Virasoro algebra $\mathfrak{g}$ by (i) constructing and classifying irreducible Whittaker modules through free-field realizations, (ii) classifying rank-1 $\mathcal{U}(\mathbb{C}L_0)$-free modules and establishing their irreducibility criteria, and (iii) analyzing the irreducibility of tensor products between these free modules and irreducible restricted modules. The authors provide precise irreducibility criteria for universal Whittaker modules at zero and nonzero levels, and they give a structural decomposition of irreducible Whittaker $\mathfrak{g}$-modules in terms of tensor products of subalgebra modules and quotients. A key methodological contribution is the free-field realization on polynomial rings and the tensor-product framework that yields clean irreducibility and isomorphism criteria. The appendix situates these results in a vertex-algebra context, presenting a twisted vertex Lie algebra perspective that underpins the realizations and constructions and connects $\mathfrak{g}$ to twisted modules over certain vertex algebras. Overall, the work provides systematic construction, classification, and interrelations of non-weight modules for gap-$p$ Virasoro algebras with potential applications to related vertex-algebra structures and representation theory.

Abstract

In this paper, we study non-weight modules over gap-$p$ Virasoro algebras, including Whittaker modules, $\mathcal{U}(\mathbb{C} L_0)$-free modules and their tensor products. We establish necessary and sufficient conditions for universal Whittaker modules to be irreducible and study the structure of irreducible Whittaker modules. The $\mathcal{U}(\mathbb{C} L_0)$-free modules of rank 1 are classified and the irreducibility of such modules are determined. Moreover, the irreducibility of tensor products of $\mathcal{U}(\mathbb{C} L_0)$-free modules of rank 1 and irreducible restricted modules is also determined.

Non-weight modules over gap-$p$ Virasoro algebras

TL;DR

The article advances the theory of non-weight representations for the gap- Virasoro algebra by (i) constructing and classifying irreducible Whittaker modules through free-field realizations, (ii) classifying rank-1 -free modules and establishing their irreducibility criteria, and (iii) analyzing the irreducibility of tensor products between these free modules and irreducible restricted modules. The authors provide precise irreducibility criteria for universal Whittaker modules at zero and nonzero levels, and they give a structural decomposition of irreducible Whittaker -modules in terms of tensor products of subalgebra modules and quotients. A key methodological contribution is the free-field realization on polynomial rings and the tensor-product framework that yields clean irreducibility and isomorphism criteria. The appendix situates these results in a vertex-algebra context, presenting a twisted vertex Lie algebra perspective that underpins the realizations and constructions and connects to twisted modules over certain vertex algebras. Overall, the work provides systematic construction, classification, and interrelations of non-weight modules for gap- Virasoro algebras with potential applications to related vertex-algebra structures and representation theory.

Abstract

In this paper, we study non-weight modules over gap- Virasoro algebras, including Whittaker modules, -free modules and their tensor products. We establish necessary and sufficient conditions for universal Whittaker modules to be irreducible and study the structure of irreducible Whittaker modules. The -free modules of rank 1 are classified and the irreducibility of such modules are determined. Moreover, the irreducibility of tensor products of -free modules of rank 1 and irreducible restricted modules is also determined.

Paper Structure

This paper contains 13 sections, 27 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Whittaker $\mathfrak{g}$-modules
  4. Free field realizations of irreducible Whittaker $\mathfrak{g}$-modules
  5. Certain tensor product $\mathfrak{g}$-modules
  6. Whittaker $\mathfrak{g}$-modules of zero level
  7. Structure of Whittaker $\mathfrak{g}$-modules
  8. $\mathcal{U}(\mathbb C L_0)$-free $\mathfrak{g}$-modules
  9. Tensor product of $\mathcal{U}(\mathbb C L_0)$-free modules and restricted modules
  10. Appendix
  11. Vertex Lie algebras $\mathcal{N}_J$
  12. $\sigma_J$-twist of $\mathcal{N}_J$
  13. A generalization of Proposition \ref{['proprealize']}

Key Result

Proposition 2.6

LGZ Let $m\in\mathbb{Z}_+$ and let $\phi$ be a Whittaker function on the subalgebra $\mathcal{V}_{\geq m}=\oplus_{k\geq m}\mathbb C L_k$ of $\mathcal{V}$. Then the universal Whittaker $\mathcal{V}$-module $M_{\mathcal{V},\phi}$ is irreducible if and only if $\left(\phi(L_{2m-1}),\phi(L_{2m})\right)\

Theorems & Definitions (49)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Lemma 2.8
  • Proposition 3.1
  • proof
  • ...and 39 more