AI 中文总结
该文分类了环镜像海森堡-维拉索罗代数上的简单哈里什-钱德拉模，其分为三类，还分类了n≥2时截断镜像海森堡-维拉索罗代数ℒ(n)的此类模，简化证明了相关定理，方法可推广至其他李（超）代数。
AI 中文摘要
环镜像海森堡-维拉索罗代数是环海森堡-维拉索罗代数的一个嵌入子代数，它包含一类有趣的截断子代数，包括高夫型（Takiff type）和𝔟𝔪𝔰₃型的截断子代数。我们对环镜像海森堡-维拉索罗代数上的简单哈里什-钱德拉模给出了完整分类，其简单模分为三类：最高权模、最低权模和中间系列的赋值模。作为副产品，我们对n≥2时的截断镜像海森堡-维拉索罗代数ℒ(n)上的所有简单哈里什-钱德拉模进行了分类。借助d参数族中的移位算子，我们给出了Y. Billig和V. Futorny在论文《具有有限维权空间的简单Wₙ模的分类》（载于《纯粹与应用数学杂志》第720卷，2016年，第199-216页）中定理3.3的更简洁证明，该定理阐述了关键的Billig-Futorny恒等式。此外，我们的方法可推广到计算其他一些李（超）代数上一致有界模的零化子。
英文摘要
The loop mirror Heisenberg-Virasoro algebra, an embedded subalgebra of the loop Heisenberg-Virasoro algebra, admits a family of interesting truncated subalgebras including those of Takiff type and \(\mathfrak{bms}_3\) type. We give a complete classification of simple Harish-Chandra modules over the loop mirror Heisenberg-Virasoro algebra, whose simple modules fall into three categories, highest weight modules, lowest weight modules, and evaluation modules of the intermediate series. As a by-product, we classify all simple Harish-Chandra modules over the truncated mirror Heisenberg-Virasoro algebras \(\mathcal{L}(n)\) for \(n\geq2\). By virtue of shift operators in the \(d\)-parameter family, we give a more streamlined proof of Theorem 3.3 from the work [Classification of simple $W_n$-modules with finite-dimensional weight spaces, {\it J. Reine Angew. Math.}, {\bf 720} (2016), 199-216] by Y. Billig and V. Futorny, which states the key Billig-Futorny identity. Furthermore, our approach can be extended to the computation of annihilators for uniformly bounded modules over some other Lie (super)algebras.
