Communications in Mathematical Physics

, Volume 296, Issue 3, pp 827–860

Highest Weight Modules Over Quantum Queer Superalgebra \({U_q(\mathfrak {q}(n))}\)

  • Dimitar Grantcharov
  • Ji Hye Jung
  • Seok-Jin Kang
  • Myungho Kim
Article

Abstract

In this paper, we investigate the structure of highest weight modules over the quantum queer superalgebra \({U_q(\mathfrak {q}(n))}\). The key ingredients are the triangular decomposition of \({U_q(\mathfrak {q}(n))}\) and the classification of finite dimensional irreducible modules over quantum Clifford superalgebras. The main results we prove are the classical limit theorem and the complete reducibility theorem for \({U_q(\mathfrak {q}(n))}\)-modules in the category \({\mathcal {O}_{q}^{\geq 0}}\).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. ABS.
    Atiyah M.F., Bott R., Shapiro A.: Clifford modules. Topology 3, 3–38 (1964)CrossRefMathSciNetGoogle Scholar
  2. B.
    Brundan J.: Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra \({\mathfrak {q}(n)}\). Adv. Math. 182, 28–77 (2004)MATHCrossRefMathSciNetGoogle Scholar
  3. BKM.
    Benkart G., Kang S.-J., Melville D.: Quantized enveloping algebras for Borcherds superalgebras. Trans. Amer Math. Soc. 350, 3297–3319 (1998)MATHCrossRefMathSciNetGoogle Scholar
  4. Dr.
    Drinfel’d, V.: Quantum groups. In: Proceedings of the International Congress of Mathematicians, Vol. 1 (Berkeley, Calif., 1986), Providence, RI: Amer. Math. Soc., 1987, pp. 798–820Google Scholar
  5. G.
    Gorelik M.: Shapovalov determinants of Q-type Lie superalgebras. Int. Math. Res. Pap., Article ID 96895, 1–71 (2006)CrossRefGoogle Scholar
  6. Har.
    Harris, J.: Algebraic Geometry, A first course. Corrected reprint of the 1992 original. Graduate Texts in Mathematics 133, New York: Springer-Verlag, 1995Google Scholar
  7. HK.
    Hong, J., Kang, S.-J.: Introduction to Quantum Groups and Crystal Bases. Graduate Studies in Mathematics 42, Providence, RI: Amer. Math. Soc., 2002Google Scholar
  8. IR.
    Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory. 2nd ed., Graduate Texts in Mathematics 84, New York: Springer-Verlag, 1990Google Scholar
  9. K.
    Kac V.: Lie superalgebras. Adv. Math. 26, 8–96 (1977)MATHCrossRefGoogle Scholar
  10. Lam.
    Lam, T.Y.: Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics 67, Providence, RI: Amer. Math. Soc., 2005Google Scholar
  11. Lang.
    Lang, S.: Algebra, Revised third edition, Graduate Texts in Mathematics 211, New York: Springer-Verlag, 2002Google Scholar
  12. LS.
    Leites, D., Serganova, V.: Defining relations for classical Lie superalgebras I. Superalgebras with Cartan matrix or Dynkin-type diagram. In: Proc. Topological and Geometrical Methods in Field Theory eds. J. Mickelson, et al., Singapore: World Sci., 1992, pp. 194–201Google Scholar
  13. N.
    Nazarov, M.: Capelli identities for Lie superalgebras, Ann. Sci. Ecole Norm. Sup. (4) 30, 6, 847–872 (1997)Google Scholar
  14. O.
    Olshanski G.: Quantized universal enveloping superalgebra of type Q and a super-extension of the Hecke alegbra. Lett. Math. Phys. 24, 93–102 (1992)MATHCrossRefMathSciNetADSGoogle Scholar
  15. P.
    Penkov I.: Characters of typical irreducible finite-dimensional \({\mathfrak {q}(n)}\)-modules. Funct. Anal. Appl. 20, 30–37 (1986)MATHCrossRefMathSciNetGoogle Scholar
  16. PS1.
    Penkov I., Serganova V.: Generic irreducible representations of finite-dimensional Lie superalgebras. Int. J. Math. 5, 389–419 (1994)MATHCrossRefMathSciNetGoogle Scholar
  17. PS2.
    Penkov I., Serganova V.: Characters of irreducible G-modules and cohomology of G/P for the Lie supergroup G = Q(N). J. Math. Sci. (New York) 84, 1382–1412 (1997)MATHCrossRefMathSciNetGoogle Scholar
  18. PS3.
    Penkov I., Serganova V.: Characters of finite-dimensional irreducible \({\mathfrak {q} (n)}\)-modules. Lett. Math. Phys. 40, 147–158 (1997)MATHCrossRefMathSciNetGoogle Scholar
  19. RTF.
    Reshetikhin, N., Takhtadzhyan, L., Faddeev, L.: Quantization of Lie groups and Lie algebras. (Russian) Algebra i Analiz 1, 178–206 (1989); translation in Leningrad Math. J. 1, 193–225 (1990)Google Scholar
  20. Se1.
    Sergeev A.: The centre of enveloping algebra for Lie superalgebra Q(n, C). Lett. Math. Phys. 7, 177–179 (1983)MATHCrossRefMathSciNetADSGoogle Scholar
  21. Se2.
    Sergeev A.: Tensor algebra of the identity representation as a module over the Lie superalgebras Gl(n, m) and Q(n). (Russian), Mat. Sb. (N.S.) 123(165), 422–430 (1984)MathSciNetGoogle Scholar
  22. Sh.
    Shimura, G.: Arithmetic and Analytic Theories of Quadratic Forms and Clifford Groups. Mathematical Surveys and Monographs 109, Providence, RI: Amer. Math. Soc., 2004Google Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Dimitar Grantcharov
    • 1
  • Ji Hye Jung
    • 2
  • Seok-Jin Kang
    • 2
  • Myungho Kim
    • 2
  1. 1.Department of MathematicsUniversity of Texas at ArlingtonArlingtonUSA
  2. 2.Department of Mathematical Sciences and Research Institute of MathematicsSeoul National UniversitySeoulKorea

Personalised recommendations