Skip to main content
SpringerLink
Log in

Determinant formula and a realization for the Lie algebra W (2, 2)

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

In this paper, an explicit determinant formula is given for the Verma modules over the Lie algebra W(2, 2). We construct a natural realization of certain vaccum module for the algebra W(2, 2) via theWeyl vertex algebra. We also describe several results including the irreducibility, characters and the descending filtrations of submodules for the Verma module over the algebra W(2, 2).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adamovic D, Radobolja G. Free field realization of the twisted Heisenberg-Virasoro algebra at level zero and its applications. J Pure Appl Algebra, 2015, 219: 4322–4342

    Article  MathSciNet  MATH  Google Scholar 

  2. Adamovic D, Radobolja G. On free field realization of W(2, 2)-modules. SIGMA Symmetry Integrability Geom Methods Appl, 2016, 113: 1–13

    MathSciNet  MATH  Google Scholar 

  3. Bagchi A, Gopakumar R, Mandal I, et al. GCA in 2d. J High Energy Phys, 2010, 2010: 004

    Article  MathSciNet  MATH  Google Scholar 

  4. Bagchi A. The BMS/GCA correspondence. Phys Rev Lett, 2010, 105: 171601

    Article  MathSciNet  Google Scholar 

  5. Barnich G, Oblak B. Notes on the BMS group in three dimensions, I: Induced representations. J High Energ, 2014, 1406: 129

    Article  Google Scholar 

  6. Barnich G, Oblak B. Notes on the BMS group in three dimensions: II. Coadjoint representation. J High Energy Phys, 2015, 2015: 033

    Article  MathSciNet  Google Scholar 

  7. Feigin B, Frenkel E. Bosonic ghost system and the Virasoro algebra. Phys Lett B, 1990, 246: 71–74

    Article  MathSciNet  MATH  Google Scholar 

  8. Feigin B, Frenkel E. Semi-Infinite Weil complex and the Virasoro algebra. Comm Math Phys, 1991, 137: 617–639

    Article  MathSciNet  MATH  Google Scholar 

  9. Feigin B, Frenkel E. Determinant formula for the free field representations of the Virasoro and Kac-Moody algebras. Phys Lett B, 1992, 286: 71–77

    Article  MathSciNet  Google Scholar 

  10. Feigin B, Fuchs D. Verma modules over the Virasoro algebra. Lecture Notes in Math, 1984, 1060: 230–245

    Article  MathSciNet  Google Scholar 

  11. Henkel M, Schott R, Stoimenov S, et al. The Poincaré algebra in the context of ageing systems: Lie structure, representations, Appell systems and coherent states. Con uentes Math, 2012, 4: 1250006

    Article  MATH  Google Scholar 

  12. Iohara K, Koga Y. Representation theory of the Virasoro algebra. In: Springer Monographs in Mathematics, vol. 17. London: Springer, 2011, 1–236

    MathSciNet  MATH  Google Scholar 

  13. Jiang W, Pei Y F.On the structure of Verma modules over the W-algebra W(2, 2). J Math Phys, 2010, 51: 022303

    Article  MathSciNet  MATH  Google Scholar 

  14. Jiang W, Zhang W. Verma modules over the W(2, 2) algebras. J Geom Phys, 2015, 98: 118–127

    Article  MathSciNet  MATH  Google Scholar 

  15. Kac V. Contravariant form for Lie algebras and superalgebras. In: Group Theoretical Methods in Physics. Lecture Notes in Physics, vol. 94. Berlin: Springer-Verlag, 1979, 441–445

    Article  MATH  Google Scholar 

  16. Oblak B. Characters of the BMS group in three dimensions. Comm Math Phys, 2015, 340: 413–432

    Article  MathSciNet  MATH  Google Scholar 

  17. 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, 2013, 54: 071701

    Article  MathSciNet  MATH  Google Scholar 

  18. Tsuchiya A, Kanie Y. Fock space representations of the Virasoro algebra-Intertwining operators. Publ Res Inst Math Sci, 1986, 22: 259–327

    Article  MathSciNet  MATH  Google Scholar 

  19. Wilson B. Highest-weight theory for truncated current Lie algebras. J Algebra, 2011, 336: 1–27

    Article  MathSciNet  MATH  Google Scholar 

  20. Wilson B. Contractions and polynomial Lie algebras. In: Highlights in Lie Algebraic Methods. Progress in Mathematics, vol. 295. New York: Birkhäuser/Springer, 2012, 213–227

    Article  Google Scholar 

  21. Zhang W, Dong C. W-algebra W(2, 2) and the vertex operator algebra L, (1/2, 0)⊗L (1/2, 0). Comm Math Phys, 2009, 285: 991–1004

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11271056, 11671056 and 11101030), National Science Foundation of Jiangsu (Grant No. BK20160403), National Science Foundation of Zhejiang (Grant Nos. LQ12A01005 and LZ14A010001), National Science Foundation of Shanghai (Grant No. 16ZR1425000), Beijing Higher Education Young Elite Teacher Project and Morning- side Center of Mathematics. The authors thank the two anonymous reviewers for their helpful comments and suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, Changshu Institute of Technology, Changshu, 215500, China

    Wei Jiang

  2. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    Yufeng Pei

  3. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China

    Wei Zhang

Authors
  1. Wei Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yufeng Pei
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yufeng Pei.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jiang, W., Pei, Y. & Zhang, W. Determinant formula and a realization for the Lie algebra W (2, 2). Sci. China Math. 61, 685–694 (2018). https://doi.org/10.1007/s11425-016-9046-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-016-9046-1

Keywords

MSC(2010)

Search

Navigation