Frontiers of Mathematics in China

, Volume 3, Issue 1, pp 37–47 | Cite as

Generalized Verma modules over some Block algebras

Research Article

Abstract

In this paper, a class of generalized Verma modules M(V) over some Block type Lie algebra ℬ(G) are constructed, which are induced from nontrivial simple modules V over a subalgebra of ℬ(G). The irreducibility of M(V) is determined.

Keywords

Lie algebra of Block type generalized Verma module irreducible module 

MSC

17B56 17B68 

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References

  1. 1.
    Block R. On torsion-free abelian groups and Lie algebras. Proc Amer Math Soc, 1958, 9: 613–620MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Hu J, Wang X D, Zhao K M. Verma modules over generalized Virasoro algebras Vir[G]. J Pure Appl Algebra, 2003, 177: 61–69MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Kac V, Radul A. Quasi-finite highest weight modules over the Lie algebra of differential operators on the circle. Comm Math phys, 1993, 157: 429–457MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Kac V, Raina A. Highest Weight Representations of Infinite Dimensional Lie Algebras. Hong Kong: World Scientific, 1987, 1–9MATHGoogle Scholar
  5. 5.
    Khomenko A, Mazorchuk V. Generalized Verma modules over the Lie algebra of type G 2. Comm Algebra, 1999, 27: 777–783MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Khomenko A, Mazorchuk V. Generalized Verma modules induced from sl(2,ℂ) and associated Verma modules. J Algebra, 2001, 242(2): 561–576MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Mazorchuk V. Verma modules over generalized Witt algebra. Compos Math, 1999, 115(1): 21–35MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Mazorchuk V. The Structure of Generalized Verma Modules. Dissertation for the Doctoral Degree. Kyiv University, 1996, 5–12Google Scholar
  9. 9.
    Su Y. Classification of quasifinite modules over the Lie algebras of Weyl type. Adv Math, 2003, 174: 57–68MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Su Y. Quasifinite representations of a Lie algebra of Block type. J Algebra, 2004, 276: 117–128MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Su Y. Quasifinite representations of a family of Lie algebras of Block type. J Pure Appl Algebra, 2004, 192: 293–305MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Verma D. Structure of certain induced representations of complex semisimple Lie algebras. Bull Amer Math Soc, 1968, 74: 160–166MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Wang X, Zhao K. Verma modules over the Virasoro-like algebra. J Australia Math, 2006, 80: 179–191MATHMathSciNetGoogle Scholar
  14. 14.
    Wu Y, Su Y. Highest weight representations of a Lie algebra of Block type. Science in China, Ser A, 2007, 50(4): 549–560MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Xu X, Generalizations of Block algebras. Manuscripta Math, 1999, 100: 489–518MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Xu X. Quadratic conformal superalgebras. J Algebra, 2000, 231: 1–38MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Xu X. New generalized simple Lie algebras of Cartan type over a field with characteristic 0. J Algebra, 2000, 224: 23–58MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Yue X, Su Y, Xin B. Highest weight representations of a family of Lie algebras of Block type. Acta Mathematica Sinica (English Ser) (in press)Google Scholar
  19. 19.
    Zhang C, Su Y. Generalized Verma module over nongraded Witt algebra. J Univ Sci Tech China (in press)Google Scholar

Copyright information

© Higher Education Press 2008

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Jiao Tong UniversityShanghaiChina
  2. 2.College of Mathematics and Information Science Henan UniversityKaifengChina
  3. 3.Department of MathematicsUniversity of Science and Technology of ChinaHefeiChina

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