The solenoidal Virasoro algebra and its simple weight modules

Abstract

Let $A_n=\mathbb{C}[t_i^{\pm 1},1\le i\le n]$ be the algebra of Laurent polynomials in $n$-variables. Let $\mu =({\mu }_1,\dots ,{\mu }_n)$ be a generic vector in ${\mathbb{C}}^n$ and ${\Gamma }_{\mu }=\{\mu \cdot \alpha ,\alpha \in {\mathbb{Z}}^n\}$ where $\mu \cdot \alpha =\sum _{i=1}^n{\mu }_i{\alpha }_i$ for $\alpha =({\alpha }_1,\dots ,{\alpha }_n)\in {\mathbb{Z}}^n$. Denote by $d_{\mu }$ the vector field: $$d_{\mu }=\sum _{i=1}^n{\mu }_i{t}_i\frac{d}{d{t}_i}.$$ In \cite{BiFu}, Y. Billig and V. Futorny introduce the solenoidal Lie algebra $\mathbf{W}(n{)}_{\mu }:=A_n{d}_{\mu }$, 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.

14 pages. arXiv admin note: text overlap with arXiv:math/0308133, arXiv:math/0607614 by other authors