Tensor product modules over the twisted Heisenberg–Virasoro algebra

Abstract

Let $\mathcal{H}$ be the twisted Heisenberg–Virasoro algebra. In this paper, we first present a method to produce a class of tensor product $\mathcal{H}$-modules, by which we can obtain the known weight $\mathcal{H}$-modules $\mathcal{M}(V,A_{\alpha ,b})$ and non-weight $\mathcal{H}$-modules $\mathcal{M}(V,\mathrm{\Omega }(\lambda ))$. Then we study a class of linear tensor product non-weight modules $\mathcal{M}(V,\mathrm{\Omega }({\lambda }_0))\otimes \underset{i=1}{\overset{m}{⨂}}\mathrm{\Omega }({\lambda }_i,{\alpha }_i,{\beta }_i)$ over $\mathcal{H}$. The necessary and sufficient conditions for $\mathcal{M}(V,\mathrm{\Omega }({\lambda }_0))\otimes \underset{i=1}{\overset{m}{⨂}}\mathrm{\Omega }({\lambda }_i,{\alpha }_i,{\beta }_i)$ to be irreducible are obtained. We also determine the necessary and sufficient conditions for two such irreducible $\mathcal{H}$-modules to be isomorphic.

Keywords:

• Twisted Heisenberg–Virasoro algebra
• tensor product module
• irreducible module

Mathematics Subject Classifications (2010):

• 17B10
• 17B65
• 17B68

Acknowledgments

This work was partially supported by the NSFC (11801369, 11871421, 11431010, 11971350), the Zhejiang Provincial Natural Science Foundation (No. LY20A010022) and the Scientific Research Foundation of Hangzhou Normal University (No. 2019QDL012). The authors thank the referees for nice suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Natural Science Foundation of China [11431010,11801369,11871421,11971350] and Zhejiang Provincial Natural Science Foundation [LY20A010022] and Scientific Research Foundation of Hangzhou Normal University [2019QDL012].