arXiv subjects

Walid Mhiri

Publications and source records attributed to Walid Mhiri.

2 recordsLinked to original sources

arXiv2024

The solenoidal Heisenberg Virasoro algebra and its simple weight modules

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.

math.RT↗

arXiv2024

The solenoidal Virasoro algebra and its simple weight modules

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.

math.RT↗