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The solenoidal Virasoro algebra and its simple weight modules

Boujemaa Agrebaoui, Walid Mhiri

Abstract

Let $A_n=\mathbb{C}[t_i^{\pm1},~1\leq i\leq n]$ be the algebra of Laurent polynomials in $n$-variables. Let $μ=(μ_1,\ldots,μ_n)$ be a generic vector in $\mathbb{C}^n$ and $Γ_μ=\{μ\cdotα,α\in \mathbb{Z}^n\}$ where $μ\cdotα=\displaystyle\sum_{i=1}^nμ_iα_i$ for $α=(α_1,\ldots,α_n)\in \mathbb{Z}^n$. Denote by $d_μ$ the vector field: $$d_μ=\displaystyle\sum_{i=1}^nμ_it_i\frac{d}{dt_i}.$$ In \cite{BiFu}, Y. Billig and V. Futorny introduce the solenoidal Lie algebra $\mathbf{W}(n)_μ:=A_nd_μ$, where the Lie structure is given by the commutators of vector fields. In the first part of this paper, we study the universal central extension of $\mathbf{W}(n)_μ$. We obtain a rank $n$ Virasoro algebra called the solenoidal Virasoro algebra $\mathbf{Vir}(n)_μ$. In the second part, we recall in the case of $\mathbf{Vir}(n)_μ$, the well know Harich-Chandra modules for generalized Virasoro algebra studied in \cite{Su,Su1,LuZhao}. In the third part, we construct irreducible highest and lowest $\mathbf{Vir}(n)_μ$-modules using triangular decomposition given by lexicographic order on $\mathbb{Z}^{n}$. We prove that these modules are weight modules which have infinite dimensional weight spaces.

The solenoidal Virasoro algebra and its simple weight modules

Abstract

Let be the algebra of Laurent polynomials in -variables. Let be a generic vector in and where for . Denote by the vector field: In \cite{BiFu}, Y. Billig and V. Futorny introduce the solenoidal Lie algebra , where the Lie structure is given by the commutators of vector fields. In the first part of this paper, we study the universal central extension of . We obtain a rank Virasoro algebra called the solenoidal Virasoro algebra . In the second part, we recall in the case of , the well know Harich-Chandra modules for generalized Virasoro algebra studied in \cite{Su,Su1,LuZhao}. In the third part, we construct irreducible highest and lowest -modules using triangular decomposition given by lexicographic order on . We prove that these modules are weight modules which have infinite dimensional weight spaces.
Paper Structure (7 sections, 13 theorems, 63 equations)

This paper contains 7 sections, 13 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. The solenoidal Virasoro algebra
  3. Harish-Chandra modules over $\mathbf{Vir}(n)_\mu$
  4. Generalities on Harish-Chandra modules
  5. Classification of cuspidal modules over $\mathbf{Vir}(n)_\mu$
  6. Generalized highest weight modules over $\mathbf{Vir}(n)_{\mu}$
  7. $\mathbf{Vir}(n)_\mu$-modules with infinite dimensional weight spaces

Key Result

Theorem 2.1

The solenoidal-Witt algebra $\mathbf{W}(n)_{\mu}$ has a one-dimensional universal central extension. It is generated by the $2$-cocycle: The central extension $\mathbf{Vir}(n)_{\mu}:=\mathbf{W}(n)_{\mu}\oplus \mathbb{C}c_{\mu}$, is called the solenoidal Virasoro algebra. Its Lie structure in the basis $\{e_{\mu\cdot\alpha}=t^{\alpha}d_{\mu},c_{\mu}|\mu\cdot\alpha\in\Gamma_{\mu}\}$, is generated b

Theorems & Definitions (25)

  • Theorem 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Remark 2.4
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Proposition 3.4
  • ...and 15 more