AI 中文总结
本文利用多项式移位算子分类了Heisenberg-Virasoro型李超代数上超秩$1|1$的$U(\mathfrak h)$-自由模，确定了三角族$\Omega^+(\lambda,\beta,d)$及平凡模结构，并给出该代数的$2\times2$矩阵新实现。
AI 中文摘要
设$\mathcal L$为Heisenberg-Virasoro型李超代数，其偶部为扭曲的Heisenberg-Virasoro代数，奇部为阿贝尔理想。我们分类了在$U(\mathfrak h)=\mathbb C[L_0,F_0]$上自由且超秩为$1|1$的$\mathbb Z_2$-分次$\mathcal L$-模。利用多项式移位算子，我们将定义关系表示为生成元矩阵元的函数方程。假设偶子代数的秩一分类已知，我们证明：在奇作用非平凡的情况下，每个模（在奇偶翻转意义下）属于三角族$\Omega^+(\lambda,\beta,d)$；而在奇作用平凡的情况下，每个模是偶子代数两个秩一模的直和。我们确定了分次同构类：在三角族内，耦合多项式$d$在相差一个非零标量乘法的意义下不变。我们还描述了典范真子模，并证明分类中的每个模都是可约的。特别地，我们利用与多项式移位微分算子相关的$2\times2$矩阵元给出了Heisenberg-Virasoro型李超代数的一个新实现。
英文摘要
Let \(\mathcal L\) be the Heisenberg-Virasoro type Lie superalgebra whose even part is the twisted Heisenberg-Virasoro algebra and whose odd part is an abelian ideal. We classify \(\mathbb Z_2\)-graded \(\mathcal L\)-modules that are free of super-rank \(1|1\) over \(U(\mathfrak h)=\mathbb C[L_0,F_0]\). Using polynomial shift operators, we express the defining relations as functional equations for the matrix entries of the generators. Assuming the rank one classification for the even subalgebra, we show that, up to parity reversal, every module with nontrivial odd action belongs to a triangular family \(Ω^+(λ,β,d)\), whereas every module with trivial odd action is a direct sum of two rank-one modules for the even subalgebra. We determine the graded isomorphism classes: within the triangular family, the coupling polynomial \(d\) is invariant up to multiplication by a non-zero scalar. We also describe canonical proper submodules and show that every module in the classification is reducible. In particular, we provide a new realization of the Heisenberg-Virasoro type Lie superalgebra using $2\times2$ matrices with entries associated with polynomial shift-differential operators.
