Abstract
In this paper, we first obtain a general result on sufficient conditions for tensor product modules to be simple over an arbitrary Lie algebra. We classify simple smooth modules over the infinite-dimensional Heisenberg algebra \(\mathfrak {H}\), and then obtain a lot of simple modules over the twisted Heisenberg-Virasoro algebra \(\widetilde {\mathcal {V}}\) from generalized oscillator representations of \(\widetilde {\mathcal {V}}\) by extending these \(\mathfrak {H}\)-modules. Using generalized oscillator representations we give the necessary and sufficient conditions for Whittaker modules over \(\widetilde {\mathcal {V}}\) (in the more general setting) to be simple. We use the “shifting technique” to determine the necessary and sufficient conditions for the tensor products of highest weight modules and modules of intermediate series over \(\widetilde {\mathcal {V}}\) to be simple. At last we establish the “embedding trick” to obtain a lot more simple \(\widetilde {\mathcal {V}}\)-modules.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Arbarello, E., De Concini, C., Kac, V.G., Procesi, C.: Moduli spaces of curves and representation theory. Commun. Math. Phys. 117, 1–36 (1988)
Batra, P., Mazorchuk, V.: Blocks and modules for Whittaker pairs. J. Pure Appl. Algebra 215(7), 1552–1568 (2011)
Bekkert, V., Benkart, G., Futorny, V., Kashuba, I.: New irreducible modules for Heisenberg and affine Lie algebras. J. Algebra 373, 284–298 (2013)
Billig, Y.: Representations of the twisted Heisenberg-Virasoro algebra at level zero. Can. Math. Bullet. 46, 529–537 (2003). arXiv:math/0201314v2
Block, R.: The irreducible representations of the Lie algebra \(\mathfrak {sl}(2)\) and of the Weyl algebra. Adv. Math. 139(1), 69–110 (1981)
Dixmier, J., algebras, Enveloping: Revised Reprint of the 1977 Translation Graduate Studies in Mathematics, vol. 11. American Mathematical Society, Providence (1996)
Chen, H., Guo, X.: New simple modules for the Heisenberg-Virasoro algebra. J. Algebra 390, 77–86 (2013)
Chen, H., Guo, X., Zhao, K.: Tensor product weight modules over the Virasoro algebra. J. Lond. Math. Soc. 88(3), 829–844 (2013)
Christodoulopoulou, K.: Whittaker modules for Heisenberg algebras and imaginary Whittaker modules for affine Lie algebras. J Algebra 320, 2871–2890 (2008)
Feigin, B., Fuks, D.: Verma modules over a Virasoro algebra. Funktsional. Anal. Prilozhen. 17(3), 91–92 (1983)
Guo, X., Zhao, K.: Irreducible representations of non-twisted affine Kac-Moody algebras. arXiv:1305.4059
Jacbson, N.: Basic algebra, II. W.H. Freeman and Company, San Francisco (1980)
Kac, V., Raina, A.: Bombay Lectures on Highest Weight Representations of Infinite-Dimensional Lie Algebras Advanced Series in Mathematical Physics, vol. 2. World Scientific Publishing Co., Inc., Teaneck (1987)
Kazhdan, D., Lusztig, G.: Tensor structures arising from affine Lie algebras. I. J. Amer. Math. Soc. 6, 90–947 (1993)
Kazhdan, D., Lusztig, G.: Tensor structures arising from affine Lie algebras, II. J. Amer. Math. Soc. 6, 949–1011 (1994)
Liu, D., Wu, Y., Zhu, L.: Whittaker modules for the twisted Heisenberg-Virasoro algebra. J. Math. Phys. 51(2), 023524, 12 (2010)
Lü, R., Guo, X., Zhao, K.: Irreducible modules over the Virasoro algebra. Doc. Math. 16, 709–721 (2011)
Lü, R., Zhao, K.: Classification of irreducible weight modules over the twisted Heisenberg-Virasoro algebra. Commun. Contem. Math. 12(2), 183–205 (2010)
Lü, R., Zhao, K.: Irreducible Virasoro modules from irreducible Weyl modules. J. Algebra 414, 271–287 (2014)
Lü, R., Zhao, K.: A family of simple weight Virasoro modules. J. Algebra 479, 437–460 (2017)
Mazorchuk, V., Zhao, K.: Simple Virasoro modules which are locally finite over a positive part. Select. Math. (N.S.) 20(3), 839–854 (2014)
Ondrus, M., Wiesner, E.: Whittaker modules for the Virasoro algebra. J. Algebra Appl. 8(3), 363–377 (2009)
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, 071701 (2013)
Tan, H., Zhao, K.: Irreducible modules from tensor products. Ark. Mat. 54 (1), 181–200 (2016)
Tan, H., Zhao, K.: Irreducible modules from tensor products (II). J. Algebra 394, 357–373 (2013)
Acknowledgments
K.Z. is partially supported by NSF of China (Grant No. 11871190) and NSERC (Grant 311907-2015). R.L. is partially supported by NSF of China (Grant 11471233, 11771122) and Jiangsu Government Scholarship for Overseas Studies (JS-2013-313). The authors are grateful to the referees for nice suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by: Vyjayanthi Chari
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Lü, R., Zhao, K. Generalized Oscillator Representations of the Twisted Heisenberg-Virasoro Algebra. Algebr Represent Theor 23, 1417–1442 (2020). https://doi.org/10.1007/s10468-019-09897-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-019-09897-1
Keywords
- Heisenberg algebra
- Virasoro algebra
- Twisted Heisenberg-Virasoro algebra
- Whittaker module
- Weight module
- Simple module