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Science China Mathematics

, Volume 61, Issue 4, pp 685–694 | Cite as

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

  • Wei Jiang
  • Yufeng Pei
  • Wei Zhang
Articles
  • 49 Downloads

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).

Keywords

Lie algebra Verma module highest weight representation W(2, 2) algebra vertex algebra 

MSC(2010)

17B68 17B69 

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Notes

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.

References

  1. 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: 4322–4342MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Adamovic D, Radobolja G. On free field realization of W(2, 2)-modules. SIGMA Symmetry Integrability Geom Methods Appl, 2016, 113: 1–13MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bagchi A, Gopakumar R, Mandal I, et al. GCA in 2d. J High Energy Phys, 2010, 2010: 004MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bagchi A. The BMS/GCA correspondence. Phys Rev Lett, 2010, 105: 171601MathSciNetCrossRefGoogle Scholar
  5. 5.
    Barnich G, Oblak B. Notes on the BMS group in three dimensions, I: Induced representations. J High Energ, 2014, 1406: 129CrossRefGoogle Scholar
  6. 6.
    Barnich G, Oblak B. Notes on the BMS group in three dimensions: II. Coadjoint representation. J High Energy Phys, 2015, 2015: 033MathSciNetCrossRefGoogle Scholar
  7. 7.
    Feigin B, Frenkel E. Bosonic ghost system and the Virasoro algebra. Phys Lett B, 1990, 246: 71–74MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Feigin B, Frenkel E. Semi-Infinite Weil complex and the Virasoro algebra. Comm Math Phys, 1991, 137: 617–639MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Feigin B, Frenkel E. Determinant formula for the free field representations of the Virasoro and Kac-Moody algebras. Phys Lett B, 1992, 286: 71–77MathSciNetCrossRefGoogle Scholar
  10. 10.
    Feigin B, Fuchs D. Verma modules over the Virasoro algebra. Lecture Notes in Math, 1984, 1060: 230–245MathSciNetCrossRefGoogle Scholar
  11. 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: 1250006CrossRefzbMATHGoogle Scholar
  12. 12.
    Iohara K, Koga Y. Representation theory of the Virasoro algebra. In: Springer Monographs in Mathematics, vol. 17. London: Springer, 2011, 1–236MathSciNetzbMATHGoogle Scholar
  13. 13.
    Jiang W, Pei Y F.On the structure of Verma modules over the W-algebra W(2, 2). J Math Phys, 2010, 51: 022303MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Jiang W, Zhang W. Verma modules over the W(2, 2) algebras. J Geom Phys, 2015, 98: 118–127MathSciNetCrossRefzbMATHGoogle Scholar
  15. 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, 441–445CrossRefzbMATHGoogle Scholar
  16. 16.
    Oblak B. Characters of the BMS group in three dimensions. Comm Math Phys, 2015, 340: 413–432MathSciNetCrossRefzbMATHGoogle Scholar
  17. 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: 071701MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Tsuchiya A, Kanie Y. Fock space representations of the Virasoro algebra-Intertwining operators. Publ Res Inst Math Sci, 1986, 22: 259–327MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Wilson B. Highest-weight theory for truncated current Lie algebras. J Algebra, 2011, 336: 1–27MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Wilson B. Contractions and polynomial Lie algebras. In: Highlights in Lie Algebraic Methods. Progress in Mathematics, vol. 295. New York: Birkhäuser/Springer, 2012, 213–227CrossRefGoogle Scholar
  21. 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: 991–1004MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsChangshu Institute of TechnologyChangshuChina
  2. 2.Department of MathematicsShanghai Normal UniversityShanghaiChina
  3. 3.School of Mathematics and StatisticsBeijing Institute of TechnologyBeijingChina

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