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Simple weight modules over the $N=1$ Heisenberg–Virasoro superalgebra

Part of: Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  16 December 2024

Chao Lan  and
Dong Liu
Chao Lan
Affiliation:
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, Xinjiang, China e-mail: [email protected]
Dong Liu*
Affiliation:
Department of Mathematics, Huzhou University, Zhejiang Huzhou, 313000, China
*
e-mail: [email protected]
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Abstract

This paper presents a classification of all simple Harish-Chandra modules for the $N=1$ Heisenberg–Virasoro superalgebra, which turn out to be highest weight modules, lowest weight modules, and evaluation modules of the intermediate series (all weight spaces are one dimensional). Moreover, a characterization of the tensor product of highest weight modules with intermediate series modules is obtained.

Keywords

Heisenberg–Virasoro superalgebraHarish-Chandra moduleVerma moduletensor productintermediate series

MSC classification

Primary: 17B65: Infinite-dimensional Lie (super)algebras 17B68: Virasoro and related algebras 17B70: Graded Lie (super)algebras
Type
Article
Information
Canadian Mathematical Bulletin , Volume 68 , Issue 1 , March 2025 , pp. 198 - 212
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

This work was supported by the National Natural Science Foundation of China (Grant No. 12071405).

References

Adamović, D., Jandrić, B., and Radobolja, G., On the N = 1 super Heisenberg-Virasoro vertex algebra, Lie Groups, Number Theory, and Vertex Aglebras . In: Proceedings of the Conference “Representation Theory XVI,” June 24–29, IUC-Dubrovnik, Contemporary Mathematics, Croatia, 2019.Google Scholar
Adamovic, D., Jandric, B., and Radobolja, G., The N = 1 super Heisenberg-Virasoro vertex algebra at level zero . J. Algebra Appl. 21(2022), no. 12, 2350003.CrossRefGoogle Scholar
Arbarello, E., De Concini, C., Kac, V. G., and Procesi, C., Moduli spaces of curves and representation theory . Commun. Math. Phys. 117(1988), 1–36.CrossRefGoogle Scholar
Astashkevich, A., On the structure of Verma modules over Virasoro and Neveu–Schwarz algebras . Commun. Math. Phys. 186(1997), no. 3, 531–562.CrossRefGoogle Scholar
Billig, Y. and Futorny, V., Classification of simple representations of Lie algebra of vector fields on a torus . J. Reine Angew. Math. 720(2016), 199–216.CrossRefGoogle Scholar
Billig, Y., Futorny, V., Iohara, K., and Kashuba, I., Classification of simple strong Harish-Chandra W(m,n)-modules.Google Scholar
Cai, Y., Liu, D., and Lü, R., Classification of simple Harish-Chandra modules over the N = 1 Ramond algebra . J. Algebra 567(2021), 114–127.CrossRefGoogle Scholar
Cai, Y. and Lü, R., Classification of simple Harish-Chandra modules over the super Virasoro algebra and its contact subalgebra . J. Pure Appl. Algebra 226(2022), 106866.CrossRefGoogle Scholar
Chen, H., Guo, X., and Zhao, K., Tensor product weight modules over the Virasoro algebra . J. Lond. Math. Soc. (2). 88(2013), 829–834.CrossRefGoogle Scholar
Dilxat, M., Chen, L., and Liu, D., Classification of simple Harish-Chandra modules over the Ovsienko–Roger superalgebra . Proc. R. Soc. Edinb. A: Math. 154(2024), no. 2, 483–493.CrossRefGoogle Scholar
He, Y., Liu, D., and Wang, Y., Simple Harish-Chandra modules over the superconformal current algebras. J. Algebra 658(2024), no. 15, 227–246.CrossRefGoogle Scholar
Kac, V. G. and Todorov, I. T., Superconformal current algebras and their unitary representations . Commun. Math. Phys. 102(1985), 337–347.CrossRefGoogle Scholar
Kaplansky, I. and Santharoubane, L. J., Harish-Chandra modules over the Virasoro algebra . In: Infinite-dimensional groups with applications (Berkeley, CA, 1984), Mathematical Sciences Research Institute Publications, 4, Springer, New York, 1985, pp. 217–231.CrossRefGoogle Scholar
Liu, D., Pei, Y., and Xia, L., Classification of simple modules with finite-dimensional weight spaces for the N = 2 Ramond algebra (in Chinese) . Sci. Sin. Math. 53(2023), 1181–1194.Google Scholar
Liu, D., Pei, Y., Xia, L., and Zhao, K., Simple smooth modules over the superconformal current algebra.Google Scholar
Liu, D. and Zhang, X., Tensor product weight modules of Schrödinger–Virasoro algebras . Front. Math. China 14(2019), no. 2, 381–393.CrossRefGoogle Scholar
Lü, R. and Zhao, K., Classification of irreducible weight modules over the twisted Heisenberg–Virasoro algebra . Commun. Contemp. Math. 12(2010), 183–205.CrossRefGoogle Scholar
Lü, R. and Zhao, K., Generalized oscillator representations of the twisted Heisenberg–Virasoro algebra . Algebra Represent. Theory 23(2020), 1417–1442.CrossRefGoogle Scholar
Mathieu, O., Classification of Harish-Chandra modules over the Virasoro Lie algebra . Invent. Math. 107(1992), 225–234.CrossRefGoogle Scholar
Pei, Y. and Bai, C., Balinsky-Novikov superalgebras and infinite-dimensional Lie superalgebras . J. Algebra Appl. 11(2012), no. 6, 1250119.CrossRefGoogle Scholar
Radobolja, G., Subsingular vectors in Verma modules, and tensor product modules over the twisted Heisenberg–Virasoro algebra and W(2,2) algebra . J. Math. Phys. 54(2013), 071701.CrossRefGoogle Scholar
Su, Y., Classification of Harish-Chandra modules over the super Virasoro algebras . Commun. Algebra 23(1995), 3653–3675.CrossRefGoogle Scholar
Xue, Y. and Lü, R., Simple weight modules with finite dimensional weight spaces over Witt superalgebras . J. Algebra 574(2021), 92–116.CrossRefGoogle Scholar
Xue, Y. and Zhao, K., Simple representations of the Fermion-Virasoro algebras.Google Scholar
Zhang, H., A class of representations over the Virasoro algebra . J Algebra 190(1997), no. 1, 1–10. CrossRefGoogle Scholar
Zhang, X., Tensor product weight representations of the Neveu–Schwarz algebra . Commun. Algebra 43(2015), 3754–3775.CrossRefGoogle Scholar