Verma modules for rank two Heisenberg-Virasoro algebra

Abstract

Let $⪯$ be a compatible total order on the additive group ${\mathbb{Z}}^2$, and $L$ be the rank two Heisenberg-Virasoro algebra. For any $\mathbf{c}=(c_1,c_2,c_3,c_4)\in {\mathbb{C}}^4$, we define ${\mathbb{Z}}^2$-graded Verma module $M(\mathbf{c},⪯)$ for the Lie algebra $L$. A necessary and sufficient condition for the Verma module $M(\mathbf{c},⪯)$ to be irreducible is provided. Moreover, the maximal ${\mathbb{Z}}^2$-graded submodules of the Verma module $M(\mathbf{c},⪯)$ are characterized when $M(\mathbf{c},⪯)$ is reducible.