Advertisement

Chinese Annals of Mathematics, Series B

, Volume 40, Issue 1, pp 111–116 | Cite as

Deformations on the Twisted Heisenberg-Virasoro Algebra

  • Dong Liu
  • Yufeng PeiEmail author
Article
  • 18 Downloads

Abstract

With the cohomology results on the Virasoro algebra, the authors determine the second cohomology group on the twisted Heisenberg-Virasoro algebra, which gives all deformations on the twisted Heisenberg-Virasoro algebra.

Keywords

Cohomology Deformation Virasoro algebra Heisenberg algebra 

2000 MR Subject Classification

17B56 17B68 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Arbarello, E., De Concini, C., Kac, V. G. and Procesi, C., Moduli spaces of curves and representation theory, Comm. Math. Phys., 117, 1988, 1–36.MathSciNetCrossRefGoogle Scholar
  2. [2]
    Billig, Y., Representations of the twisted Heisenberg–Virasoro algebra at level zero, Canad. Math. Bull., 46(4), 2003, 529–533.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Fialowski, A., Formal rigidity of the Witt and Virasoro algebra, J. Math. Phys., 53, 2012, 073501.Google Scholar
  4. [4]
    Fuks, D. B., Cohomology of Infinite–Dimensional Lie Algebras, Nauka, Moscow, 1984. (in Russian)zbMATHGoogle Scholar
  5. [5]
    Gao, S., Jiang, C. and Pei, Y., Low dimensional cohomology groups of the Lie algebras W(a, b), Commun. Alg., 39(2), 2011, 397–423.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Grozman, P. Ja., Classification of bilinear invariant operators on tensor fields, Functional Anal. Appl, 14(2), 1980, 127–128.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Guieu, L. and Roger, C., Algebra and the Virasoro Group, Geometric and Algebraic Aspects, Generalizations, Les Publications CRM, Montreal, QC, 2007.zbMATHGoogle Scholar
  8. [8]
    Guo, X., Lv, R. and Zhao, K., Simple Harish–Chandra modules, intermediate series modules, and Verma modules over the loop–Virasoro algebra, Forum Math., 23, 2011, 1029–1052.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Kaplansky, I. and Santharoubane, L. J., Harish–Chandra modules over the Virasoro algebra, Math. Sci. Res. Inst. Publ., Vol. 4, Springer–Verlag, New York, 1985.Google Scholar
  10. [10]
    Li, J. and Su, Y., Representations of the Schrödinger–Virasoro algebras, J. Math. Phys., 49, 2008, 053512.Google Scholar
  11. [11]
    Liu, D., Classification of Harish–Chandra modules over some Lie algebras related to the Virasoro algebra, J. Algebra, 447, 2016, 548–559.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Liu, D., Gao, S. and Zhu, L., Classification of irreducible weight modules over W–algebra W(2, 2), J. Math. Phys., 49(11), 2008, 113503.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Liu, D. and Jiang, C., Harish–Chandra modules over the twisted Heisenberg–Virasoro algebra, J. Math. Phys., 49(1), 2008, 012901.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Liu, D., Pei, Y. and Zhu, L., Lie bialgebra structures on the twisted Heisenberg–Virasoro algebra, J. Algebra, 359, 2012, 35–48.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Lv, R. and Zhao, K., Classification of irreducible weight modules over the twisted Heisenberg–Virasoro algebra, Commun. Contemp. Math. 12(2), 2010, 183–205MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Ng, S. H. and Taft, E. J., Classification of the Lie bialgebra structures on the Witt and Virasoro algebras, J. Pure Appl. Alg., 151, 2000, 67–88.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Ovsienko, V. Yu. and Rozhe, K., Extension of Virasoro group and Virasoro algebra by modules of tensor densities on S1, Functional Anal. Appl., 30, 1996, 290–291.MathSciNetCrossRefGoogle Scholar
  18. [18]
    Ovsienko, V. and Roger, C., Generalizations of the Virasoro group and Virasoro algebra through extensions by modules of tensor densities on S1, Indag. Math.(N.S.), 9(2), 1998, 277–288.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Qiu, S., The cohomology of the Virasoro algebra with coefficients in a basic Harish–Chandra modules, Northeastern Math. J., 5(1), 1989, 75–83.MathSciNetzbMATHGoogle Scholar
  20. [20]
    Roger, C. and Unterberger, J., The Schrödinger–Virasoro Lie group and algebra: From geometry to representation theory, Annales Henri Poincaré, 7, 2006, 1477–1529.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Shen, R. and Jiang, C., The derivation algebra and automorphism group of the twisted Heisenberg–Virasoro algebra, Commun. Alg., 34, 2006, 2547–2558.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Zhang, W. and Dong, C., W–algebra W(2, 2) and the vertex operator algebra L 1 2,0, Commun. Math. Phys., 285(3), 2009, 991–1004.CrossRefzbMATHGoogle Scholar

Copyright information

© Fudan University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsHuzhou UniversityHuzhou, ZhejiangChina
  2. 2.Department of MathematicsShanghai Normal UniversityShanghaiChina

Personalised recommendations