The solenoidal Virasoro algebra and its simple weight modules

Abstract

Let $A_n=\mathbb{C}[t_i^{\pm1},~1\leq i\leq n]$ be the algebra of Laurent polynomials in $n$-variables. Let $\mu=(\mu_1,\ldots,\mu_n)$ be a generic vector in $\mathbb{C}^n$ and $\Gamma_{\mu}=\{\mu\cdot\alpha,\alpha\in \mathbb{Z}^n\}$ where $\mu\cdot\alpha=\displaystyle\sum_{i=1}^n\mu_i\alpha_i$ for $\alpha=(\alpha_1,\ldots,\alpha_n)\in \mathbb{Z}^n$. Denote by $d_\mu$ the vector field: $$d_\mu=\displaystyle\sum_{i=1}^n\mu_it_i\frac{d}{dt_i}.$$ In \cite{BiFu}, Y. Billig and V. Futorny introduce the solenoidal Lie algebra $\mathbf{W}(n)_{\mu}:=A_nd_\mu$, 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 $\mathbf{W}(n)_{\mu}$. We obtain a rank $n$ Virasoro algebra called the solenoidal Virasoro algebra $\mathbf{Vir}(n)_\mu$. In the second part, we recall in the case of $\mathbf{Vir}(n)_\mu$, 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 $\mathbf{Vir}(n)_\mu$-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.