Generalized Oscillator Representations of the Twisted Heisenberg-Virasoro Algebra

Abstract

In this paper, we first obtain a general result on sufficient conditions for tensor product modules to be simple over an arbitrary Lie algebra. We classify simple smooth modules over the infinite-dimensional Heisenberg algebra $\mathfrak {H}$, and then obtain a lot of simple modules over the twisted Heisenberg-Virasoro algebra $\widetilde {\mathcal {V}}$ from generalized oscillator representations of $\widetilde {\mathcal {V}}$ by extending these $\mathfrak {H}$-modules. Using generalized oscillator representations we give the necessary and sufficient conditions for Whittaker modules over $\widetilde {\mathcal {V}}$ (in the more general setting) to be simple. We use the “shifting technique” to determine the necessary and sufficient conditions for the tensor products of highest weight modules and modules of intermediate series over $\widetilde {\mathcal {V}}$ to be simple. At last we establish the “embedding trick” to obtain a lot more simple $\widetilde {\mathcal {V}}$-modules.