Abstract
We study two classes of non-weight modules \(\Omega (\lambda ,\alpha ,\beta )\otimes \mathrm {Ind}(M)\) and \({\mathcal {M}}\big (V,\Omega (\lambda ,\alpha ,\beta )\big )\) over the twisted Heisenberg–Virasoro algebra in this paper. We present conditions under which these modules are irreducible, the necessary and sufficient conditions under which modules in each class are isomorphic, and also show that the irreducible modules in these two classes are new. Finally, we construct non-weight modules \(\mathrm {Ind}_{{\underline{y}},\lambda }({\mathbb {C}}_{RS})\) and \(\mathrm {Ind}_{{\underline{z}},\lambda }({\mathbb {C}}_{PQ})\) over the twisted Heisenberg–Virasoro algebra and then apply the established results to give irreducible conditions for these modules.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Arbarello, E., DeConcini, C., Kac, V.G., Procesi, C.: Moduli spaces of curves and representation theory. Commun. Math. Phys. 117, 1–36 (1988)
Billig, Y.: Representations of the twisted Heisenberg–Virasoro algebra at level zero. Can. Math. Bull. 46, 529–537 (2003)
Chen, H., Guo, X.: New simple modules for the Heisenberg–Virasoro algebra. J. Algebra 390, 77–86 (2013)
Chen, H., Guo, X.: Non-weight modules over the Heisenberg-Virasoro algebra and the \(W\) algebra \(W(2,2)\). J. Algebra Appl. 16, 1750097 (2017)
Chen, H., Guo, X., Zhao, K.: Tensor product weight modules over the Virasoro algebra. J. Lond. Math. Soc. 88, 829–844 (2013)
Chen, H., Han, J., Su, Y.: A class of simple weight modules over the twisted Heisenberg–Virasoro algebra. J. Math. Phys. 57, 101705 (2016). 7 pp
Chen, H., Li, J.: Left-symmetric algebra structures on the twisted Heisenberg–Virasoro algebra. Sci. China Math. 57, 469–476 (2014)
Han, J., Chen, Q., Su, Y.: Modules over the algebra \({\cal{V}} ir(a, b)\). Linear Algebra Appl. 515, 11–23 (2017)
Kaplansky, I., Santharoubane, L.J.: Harish–Chandra Modules Over the Virasoro Algebra, Math. Sci. Res. Inst. Publ., vol. 4, pp. 217–231. Springer, New York (1985)
Liu, D., Pei, Y., Zhu, L.: Lie bialgebra structures on the twisted Heisenberg–Virasoro algebra. J. Algebra 359, 35–48 (2012)
Liu, D., Jiang, C.: Harish-Chandra modules over the twisted Heisenberg–Virasoro algebra. J. Math. Phys. 49, 012901 (2008). 13 pp
Liu, D., Wu, Y., Zhu, L.: Whittaker modules for the twisted Heisenberg–Virasoro algebra. J. Math. Phys. 51, 023524 (2010). 12 pp
Lü, R., Zhao, K.: Irreducible Virasoro modules from irreducible Weyl modules. J. Algebra 414, 271–287 (2014)
Liu, G., Zhao, Y.: Generalized polynomial modules over the Virasoro algebra. Proc. Am. Math. Soc. 144, 5103–5112 (2016)
Lü, R., Zhao, K.: Classification of irreducible weight modules over the twisted Heisenberg–Virasoro algebra. Commun. Contemp. Math. 12, 183–205 (2010)
Mathieu, O.: Classification of Harish-Chandra modules over the Virasoro Lie algebra. Invent. Math. 107, 225–234 (1992)
Mazorchuk, V., Weisner, E.: Simple Virasoro modules induced from codimension one subalgebras of the positive part. Proc. Am. Math. Soc. 142, 3695–3703 (2014)
Nilsson, J.: \(U({\mathfrak{h}})\)-free modules and coherent families. J. Pure Appl. Algebra 220, 1475–1488 (2016)
Radobolja, G.: Subsingular vectors in Verma modules, and tensor product of weight modules over the twisted Heisenberg–Virasoro algebra and \(W(2,2)\) algebra. J. Math. Phys. 54, 071701 (2013). 24 pp
Shen, R., Jiang, C.: The derivation algebra and automorphism group of the twisted Heisenberg–Virasoro algebra. Commun. Algebra 34, 2547–2558 (2006)
Shen, R., Jiang, Q., Su, Y.: Verma modules over the generalized Heisenberg–Virasoro algebra. Commun. Algebra 04, 1464–1473 (2008)
Shen, R., Su, Y.: Classification of irreducible weight modules with a finite-dimensional weight space over twisted Heisenberg–Virasoro algebra. Acta Math. Sin. (E. S.) 23, 189–192 (2007)
Tan, H., Zhao, K.: Irreducible Virasoro modules from tensor products. Ark. Mat. 54, 181–200 (2016)
Tan, H., Zhao, K.: Irreducible Virasoro modules from tensor products \((\rm II)\). J. Algebra 394, 357–373 (2013)
Zhang, H.: A class of representations over the Virasoro algebra. J. Algebra 190, 1–10 (1997)
Acknowledgements
The authors would like to thank the referee very much for valuable suggestions and comments to put this paper in better shape. Jianzhi Han was partially supported by the NSFC (11501417, 11671247). Yucai Su and Xiaoqing Yue were partially supported by the NSFC (11431010).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, H., Han, J., Su, Y. et al. Two classes of non-weight modules over the twisted Heisenberg–Virasoro algebra. manuscripta math. 160, 265–284 (2019). https://doi.org/10.1007/s00229-018-1059-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-018-1059-3