Representations of the C-series related to the q-analog Virasoro-like Lie algebra

Abstract

In this paper, we study the representation of an infinite-dimensional Lie algebra $C$ related to the q-analog Virasoro-like Lie algebra. We give the necessary and sufficient conditions for the highest weight irreducible module $V\left(\varphi \right)$ of $C$ to be a Harish-Chandra module. We prove that the Verma $C$-module $\overline{V}\left(\varphi \right)$ is either irreducible or has the corresponding irreducible highest weight $C$-module $V\left(\varphi \right)$ that is a Harish-Chandra module. We also give the maximal proper submodule of the Verma module $\overline{V}\left(\varphi \right)$ and the e-character of the irreducible highest weight $C$-module $V\left(\varphi \right)$ when the highest weight ϕ satisfies some natural conditions. Furthermore, we give the classification of the Harish-Chandra $C$-modules with nontrivial central charge.