Verma modules for rank two Heisenberg-Virasoro algebra

TL;DR

This paper studies the structure of Verma modules over the rank two Heisenberg-Virasoro algebra, providing conditions for their irreducibility and describing their maximal submodules.

Contribution

It introduces a classification of Verma modules for the algebra and characterizes their reducibility and submodule structure.

Findings

Irreducibility conditions for Verma modules are established.

Maximal submodules are characterized when modules are reducible.

Provides a complete description of module structure for the algebra.

Abstract

Let $\preceq$ 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 $M(\mathbf{c}, \preceq)$ for the Lie algebra $L$. A necessary and sufficient condition for the Verma module $M(\mathbf{c}, \preceq)$ to be irreducible is provided. Moreover, the maximal $\mathbb{Z}^2$-graded submodules of the Verma module $M(\mathbf{c}, \preceq)$ are characterized when $M(\mathbf{c}, \preceq)$ is reducible.