Advertisement

Results in Mathematics

, 74:183 | Cite as

Irreducible Twisted Heisenberg–Virasoro Modules from Tensor Products

  • Haibo ChenEmail author
  • Yucai Su
Article
  • 71 Downloads

Abstract

In this paper, we realize polynomial \(\mathcal {H}\)-modules \(\Omega (\lambda ,\alpha ,\beta )\) from irreducible twisted Heisenberg–Virasoro modules \(\mathcal {A}_{\alpha ,\beta }\). It follows from \(\mathcal {H}\)-modules \(\Omega (\lambda ,\alpha ,\beta )\) and \(\mathrm {Ind}(M)\) that we obtain a class of tensor product modules \(\big (\bigotimes _{i=1}^m\Omega (\lambda _i,\alpha _i,\beta _i)\big )\otimes \mathrm {Ind}(M)\). We give the necessary and sufficient conditions under which these modules are irreducible and isomorphic, and also give that the irreducible modules in this class are new.

Keywords

Twisted Heisenberg–Virasoro algebra tensor product module irreducible module 

Mathematics Subject Classification

17B10 17B65 17B68 

Notes

Acknowledgements

This work was partially supported by the NSFC (11801369, 11431010, 11971350). The authors thank the referee for helpful suggestions. The authors also thank Prof. Jianzhi Han for his useful discussions.

References

  1. 1.
    Arbarello, E., DeConcini, C., Kac, V.G., Procesi, C.: Moduli spaces of curves and representation theory. Commun. Math. Phys. 117, 1–36 (1988)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chen, H., Guo, X.: New simple modules for the Heisenberg–Virasoro algebra. J. Algebra 390, 77–86 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    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)MathSciNetCrossRefGoogle Scholar
  4. 4.
    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 ppMathSciNetCrossRefGoogle Scholar
  5. 5.
    Chen, H., Han, J., Su, Y., Yue, X.: Two classes of non-weight modules over the twisted Heisenberg-Virasoro algebra. Manuscr. Math. 160, 265–284 (2019)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Han, J., Chen, Q., Su, Y.: Modules over the algebra \({\cal{V}} ir(a, b)\). Linear Algebra Appl. 515, 11–23 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kaplansky, I., Santharoubane, L.J.: Harish–Chandra modules over the Virasoro algebra. Math. Sci. Res. Inst. Publ. 4, pp. 217–231. Springer, New York (1985)Google Scholar
  8. 8.
    Lü, R., Guo, X., Zhao, K.: Irreducible modules over the Virasoro algebra. Doc. Math. 16, 709–721 (2011)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Liu, D., Jiang, C.: Harish–Chandra modules over the twisted Heisenberg–Virasoro algebra. J. Math. Phys. 49, 012901 (2008). 13 ppMathSciNetCrossRefGoogle Scholar
  10. 10.
    Liu, D., Wu, Y., Zhu, L.: Whittaker modules for the twisted Heisenberg–Virasoro algebra. J. Math. Phys. 51, 023524 (2010). 12 ppMathSciNetCrossRefGoogle Scholar
  11. 11.
    Lü, R., Zhao, K.: Irreducible Virasoro modules from irreducible Weyl modules. J. Algebra 414, 271–287 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Lü, R., Zhao, K.: Classification of irreducible weight modules over the twisted Heisenberg–Virasoro algebra. Commun. Contemp. Math. 12, 183–205 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Lü, R., Zhao, K.: A family of simple weight Virasoro modules. J. Algebra 479, 437–460 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lü, R., Zhao, K.: Generalized oscillator representations of the twisted Heisenberg–Virasoro algebra. Algebr. Represent. Theory (2019).  https://doi.org/10.1007/s10468-019-09897-1
  15. 15.
    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 ppMathSciNetCrossRefGoogle Scholar
  16. 16.
    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)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Tan, H., Zhao, K.: Irreducible Virasoro modules from tensor products (II). J. Algebra 394, 357–373 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Statistics and MathematicsShanghai Lixin University of Accounting and FinanceShanghaiChina
  2. 2.School of Mathematical SciencesTongji UniversityShanghaiChina

Personalised recommendations