Representations of the N=1 Heisenberg-Virasoro superalgebra

TL;DR

This paper classifies non-weight modules over the N=1 Heisenberg-Virasoro superalgebra, determines their submodules, and explores related modules over subalgebras, advancing understanding of their structure and representations.

Contribution

It provides a complete classification of certain non-weight modules over the N=1 Heisenberg-Virasoro superalgebra and clarifies their submodule structures.

Findings

Classified all submodules of reducible non-weight modules.

Constructed irreducible quotient modules explicitly.

Extended analysis to modules over subalgebras such as the Neveu-Schwarz algebra.

Abstract

We first define a class of non-weight modules over the N=1 Heisenberg-Virasoro superalgebra $\mathfrak{g}$, which are reducible modules. Then we give all submodules of such modules, and present the corresponding irreducible quotient modules which were exactly studied in \cite{DL}. Also, we prove that those modules constitute a complete classification of $U(\mathfrak{h})$-free modules of rank $2$ over $\mathfrak{g}$, where $\mathfrak{h}=\C L_{0}\oplus\C H_{0}$ is the degree-0 part of $\mathfrak{g}$. As an application, we obtain a class of weight $\mathfrak{g}$-modules from those non-weight $\mathfrak{g}$-modules by weighting functor. Furthermore, we study the non-weight modules over the four subalgebras of $\mathfrak{g}$: (i) the Heisenberg-Virasoro algebra; (ii) the Neveu-Schwarz algebra; (iii) the Fermion-Virasoro algebra; (iv) the Heisenberg-Clifford superalgebra. As far as we know, those non-weight Heisenberg-Virasoro modules were constructed in \cite{HCS}, but the structure of submodules was not clear. In this paper, we determine all submodules of them, and we show the corresponding irreducible quotient modules which were exactly defined in \cite{CG}.