The solenoidal Heisenberg Virasoro algebra and its simple weight modules

Abstract

Let $A_n=\mathbb{C}[t_i^{\pm1},~1\leq i\leq n]$ and $\mathbf{W}(n)_\mu=A_nd_\mu$ the solenoidal Lie algebra introduced by Y.Billig and V.Futorny in \cite{BiFu2}, where $\mu=(\mu_1,\ldots,\mu_n)\in\mathbb{C}^n$ is a generic vector and $$d_\mu=\sum_{i=1}^n\mu_it_i\frac{\partial}{\partial t_i}.$$ We consider the semi-direct product Lie algebra $\mathbf{WA}(n)_\mu:=\mathbf{W}(n)_\mu\ltimes A_n$. In the first part, We prove that $\mathbf{WA}(n)_\mu$ 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 $\mathbf{HVir}(n)_\mu$ and it will be called the solenoidal Heisenberg-Virasoro algebra. Then we will study Harish-Chandra modules of $\mathbf{HVir}(n)_\mu$ 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 $\mathbf{HVir}(n)_\mu$ 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.