Classification of U(h)-Free Modules over the Heisenberg-Virasoro type Lie superalgebra via Polynomial Shift Operators
Abstract
Let 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 ℤ2-graded ℒ-modules that are free of super-rank 1|1 over U(h)=ℂ[L0,F0]. 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 2×2 matrices with entries associated with polynomial shift-differential operators.
Related
- Local and Quasi Derivations of the Mirror Heisenberg-Virasoro Algebra
- Classification of finite energy positive solutions to Yamabe-type equation on the fifteen dimensional octonionic Heisenberg group
- Polynomial Spectral Gap Estimates for Finitely Differentiable Quasi-Periodic Schrödinger Operators
- The spectral structure of Toeplitz operators with polynomial symbols on the harmonic Bergman space
- Arithmetic k-Polynomial Dynamical Localization for k Power-Law Quasi-Periodic Long-Range Operators on ℓ2(ℤd)
BibTeX
@misc{kong2026classification,
title = {{Classification of $U(\mathfrak h)$-Free Modules over the Heisenberg-Virasoro type Lie superalgebra via Polynomial Shift Operators}},
author = {Yan Kong and Haibo Chen and Yucai Su},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.32105},
eprint = {2609.32105},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32105},
}
RIS
TY - UNPB
TI - Classification of U(h)-Free Modules over the Heisenberg-Virasoro type Lie superalgebra via Polynomial Shift Operators
AU - Yan Kong
AU - Haibo Chen
AU - Yucai Su
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2609.32105
UR - https://arxiv.org/abs/2609.32105
AB - Let 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 ℤ_2-graded ℒ-modules that are free of super-rank 1|1 over U(h)=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 2×2 matrices with entries associated with polynomial shift-differential operators.
ER -