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Transformation Groups

, Volume 17, Issue 2, pp 547–570 | Cite as

Classification of irreducible quasifinite modules over map Virasoro algebras

  • Alistair Savage
Article

Abstract

We give a complete classification of the irreducible quasifinite modules for algebras of the form Vir ⊗ A, where Vir is the Virasoro algebra and A is a finitely generated commutative associative unital algebra over the complex numbers. It is shown that all such modules are tensor products of generalized evaluation modules. We also give an explicit sufficient condition for a Verma module of Vir ⊗ A to be reducible. In the case that A is an infinite-dimensional integral domain, this condition is also necessary.

Keywords

Weight Space Weight Module Verma Module High Weight Vector High Weight Module 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [AM69]
    M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, Mass., 1969. Russian transl.: М. Атья, И. Макдональд, Введенuе в коммуmаmuвную, Мир, 1972.Google Scholar
  2. [Bou58]
    N. Bourbaki. Éléments de mathématique. 23. Premiére partie: Les structures fondamentales de l’analyse. Livre II: Algébre. Chapitre 8: Modules et anneaux semi-simples, Actualités Sci. Ind., no. 1261, Hermann, Paris, 1958. Russian transl.: Н. Бурбаки, Алгеьра. Модули, кольца, формы, Наука, М., 1966.Google Scholar
  3. [BZ04]
    Y. Billig, K. Zhao. Weight modules over exp-polynomial Lie algebras, J. Pure Appl. Algebra 191 (2004), no. 1–2, 23–42.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [CFK10]
    V, Chari, G, Fourier, T, Khandai, A categorical approach to Weyl modules, Transform. Groups 15 (2010), no. 3, 517–549.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [CP88]
    V. Chari, A. Pressley, Unitary representations of the Virasoro algebra and a conjecture of Kac, Compositio Math. 67 (1988), no. 3, 315–342.MathSciNetzbMATHGoogle Scholar
  6. [GLZa]
    X. Guo, R. Lu, K. Zhao, Classification of irreducible Harish-Chandra modules over generalized Virasoro algebras, to appear in Proc. Edinburgh Math. Soc., arXiv:math/0607614.Google Scholar
  7. [GLZb]
    X. Guo, R. Lu, K. Zhao, Simple Harish-Chandra modules, intermediate series modules, and Verma modules over the loop-Virasoro algebra, Forum Math. 23 (2011), no. 5, 1029–1052.MathSciNetCrossRefGoogle Scholar
  8. [HWZ03]
    J. Hu, X. Wang, K. Zhao, Verma modules over generalized Virasoro algebras Vir[G], J. Pure Appl. Algebra 177 (2003), no. 1, 61–69.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [KR87]
    V. G. Kac, A. K. Raina, Bombay Lectures on Highest Weight Representations of Infinite-Dimensional Lie Algebras, Advanced Series in Mathematical Physics, Vol. 2, World Scientific, Teaneck, NJ, 1987.Google Scholar
  10. [Li04]
    H. Li. On certain categories of modules for affine Lie algebras, Math. Z. 248 (2004), no. 3, 635–664.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [LZ06]
    R. Lu, K. Zhao, Classification of irreducible weight modules over higher rank Virasoro algebras, Adv. Math. 206 (2006), no. 2, 630–656.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [LZ10]
    R. Lü, K. Zhao, Classification of irreducible weight modules over the twisted Heisenberg–Virasoro algebra, Commun. Contemp. Math. 12 (2010), no. 2, 183–205.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [Mat92]
    O. Mathieu, Classification of Harish-Chandra modules over the Virasoro Lie algebra, Invent. Math. 107 (1992), no. 2, 225–234.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [Maz99]
    V. Mazorchuk, Verma modules over generalized Witt algebras, Compositio Math. 115 (1999). no. 1, 21–35.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [Maz00]
    V. Mazorchuk, Classification of simple Harish-Chandra modules over Q-Virasoro algebra, Math. Nachr. 209 (2000), 171–177.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [MP91a]
    C. Martin. A. Piard, Indecomposable modules over the Virasoro Lie algebra and a conjecture of V. Kac, Comm. Math. Phys. 137 (1991), no. 1, 109–132.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [MP91b]
    C. Martin, A. Piard, Nonbounded indecomposable admissible modules over the Virasoro algebra, Lett. Math. Phys. 23 (1991), no. 4, 319–324.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [MP92]
    18 C. Martin, A, Piard, Classification of the indecomposable bounded admissible modules over the Virasoro Lie algebra with weightspaces of dimension not exceeding two, Comm. Math. Phys. 150 (1992), no. 3, 465–493.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [NS]
    E. Neher, A. Savage, Extensions and block decompositions for finite-dimensional representations of equivariant map algebras, arXiv:1103.4367.Google Scholar
  20. [NSS]
    E. Neher, A. Savage, P. Senesi, Irreducible finite-dimensional representations of equivariant map algebras, Trans. Amer. Math. Soc., to appear, arXiv:0906.5189.Google Scholar
  21. [Rao04]
    S. Eswara Rao, On representations of toroidal Lie algebras, in: Functional analysis VIII, Various Publ. Ser. (Aarhus), Vol. 47, Aarhus Univ., Aarhus, 2004, pp. 146–167.Google Scholar
  22. [Su01]
    Y. Su, Simple modules over the high rank Virasoro algebras, Comm. Algebra 29 (2001), no. 5, 2067–2080.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [Su03]
    Y. Su, Classification of Harish-Chandra modules over the higher rank Virasoro algebras, Comm. Math. Phys. 240 (2003), no. 3. 539–551.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada

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