Classification of Simple Harish-Chandra Modules over the Loop Mirror HeisenbergVirasoro Algebra
Abstract
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 bms3 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 ℒ(n) for n≥2. 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 Wn-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.
Related
- Representation theory of the real Gelfand order and real Harish-Chandra modules for SL2(ℝ)
- Simple Harish-Chandra bimodules in positive characteristics
- simple cartan-free modules of finite rank over the super-virasoro algebras
- Geometry of holonomic modules over the first Weyl algebra
- Alperin weight bijections and dimensions of simple modules
BibTeX
@misc{chen2026classification,
title = {{Classification of Simple Harish-Chandra Modules over the Loop Mirror HeisenbergVirasoro Algebra}},
author = {Haibo Chen and Xiansheng Dai and Yucai Su},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30187},
eprint = {2608.30187},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30187},
}
RIS
TY - UNPB
TI - Classification of Simple Harish-Chandra Modules over the Loop Mirror HeisenbergVirasoro Algebra
AU - Haibo Chen
AU - Xiansheng Dai
AU - Yucai Su
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30187
UR - https://arxiv.org/abs/2608.30187
AB - 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 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 ℒ(n) for n≥2. 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.
ER -