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The solenoidal Heisenberg Virasoro algebra and its simple weight modules

Boujemaa Agrebaoui, Walid Mhiri

Abstract

Let $A_n=\mathbb{C}[t_i^{\pm1},~1\leq i\leq n]$ and $\mathbf{W}(n)_μ=A_nd_μ$ the solenoidal Lie algebra introduced by Y.Billig and V.Futorny in \cite{BiFu2}, where $μ=(μ_1,\ldots,μ_n)\in\mathbb{C}^n$ is a generic vector and $$d_μ=\sum_{i=1}^nμ_it_i\frac{\partial}{\partial t_i}.$$ We consider the semi-direct product Lie algebra $\mathbf{WA}(n)_μ:=\mathbf{W}(n)_μ\ltimes A_n$. In the first part, We prove that $\mathbf{WA}(n)_μ$ has a unique three-dimensional universal central extension. In fact we construct a higher rank Heisenberg-Virasoro algebra (see \cite{LiuGuo, LdZ}). It will be denoted by $\mathbf{HVir}(n)_μ$ and it will be called the solenoidal Heisenberg-Virasoro algebra. Then we will study Harish-Chandra modules of $\mathbf{HVir}(n)_μ$ following \cite{LiuGuo}. We will obtain two classes of Harich-Chandra modules: generalized highest weight modules(\textbf{GHW} modules) and intermediate series modules. Our results are particular cases of \cite{LiuGuo}. In the end, we will construct $\mathbf{HVir}(n)_μ$ Verma modules using the lexicographic order on $\mathbb{Z}^{n}$. In particular we give examples of irreducible weight modules which have infinite dimensional weight spaces.

The solenoidal Heisenberg Virasoro algebra and its simple weight modules

Abstract

Let and the solenoidal Lie algebra introduced by Y.Billig and V.Futorny in \cite{BiFu2}, where is a generic vector and We consider the semi-direct product Lie algebra . In the first part, We prove that has a unique three-dimensional universal central extension. In fact we construct a higher rank Heisenberg-Virasoro algebra (see \cite{LiuGuo, LdZ}). It will be denoted by and it will be called the solenoidal Heisenberg-Virasoro algebra. Then we will study Harish-Chandra modules of following \cite{LiuGuo}. We will obtain two classes of Harich-Chandra modules: generalized highest weight modules(\textbf{GHW} modules) and intermediate series modules. Our results are particular cases of \cite{LiuGuo}. In the end, we will construct Verma modules using the lexicographic order on . In particular we give examples of irreducible weight modules which have infinite dimensional weight spaces.
Paper Structure (7 sections, 5 theorems, 87 equations)

This paper contains 7 sections, 5 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. The solenoidal Heisenberg-Virasoro algebra $\mathbf{HVir}(n)_{\mu}$
  3. Harish Chandra modules for ${\mathbf{HVir}(n)_\mu}$
  4. Generalities on Harish-Chandra modules
  5. Intermediate series of $\mathbf{HVir}(n)_\mu$
  6. Generalized highest weight modules
  7. Simple Weight ${\mathbf{HVir}(n)_\mu}$-modules having infinite dimensional weight spaces

Key Result

Theorem 2.1

The second cohomology space $H^2(\mathbf{WA}(n)_{\mu},\mathbb{C})$ is three dimensional and it is generated by the following $2$-cocycles $C_{\mu,1},~C_{\mu,2},~C_{\mu,3}:\mathbf{WA}(n)_{\mu}\times\mathbf{WA}(n)_{\mu} \longrightarrow \mathbb{C}$ defined by:

Theorems & Definitions (15)

  • Theorem 2.1
  • proof
  • Definition 2.2
  • Remark 2.3
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • Remark 3.4
  • Proposition 3.5
  • proof
  • ...and 5 more