Communications in Mathematical Physics

, Volume 352, Issue 3, pp 1091–1119 | Cite as

Character Formulae for Queer Lie Superalgebras and Canonical Bases of Types A/C

  • Shun-Jen Cheng
  • Jae-Hoon KwonEmail author
  • Weiqiang Wang


For the BGG category of \({{\mathfrak{q}}(n)}\)-modules of half-integer weights, a Kazhdan–Lusztig conjecture à la Brundan is formulated in terms of categorical canonical basis of the nth tensor power of the natural representation of the quantum group of type C. For the BGG category of \({{\mathfrak{q}}(n)}\)-modules of congruent non-integral weights, a Kazhdan–Lusztig conjecture is formulated in terms of canonical basis of a mixed tensor of the natural representation and its dual of the quantum group of type A. We also establish a character formula for the finite-dimensional irreducible \({\mathfrak{q}(n)}\)-modules of half-integer weights in terms of type C canonical basis of the corresponding q-wedge space.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. B.
    Bao, H.: Kazhdan–Lusztig theory of super type D and quantum symmetric pairs. arXiv:1603.05105
  2. BW.
    Bao, H., Wang, W.: A new approach to Kazhdan–Lusztig theory of type B via quantum symmetric pairs. arXiv:1310.0103v2
  3. Bg.
    Bergman G.: The diamond lemma for ring theory. Adv. Math. 29, 178–218 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  4. Br1.
    Brundan J.: Kazhdan–Lusztig polynomials and character formulae for the Lie superalgebra \({{\mathfrak{gl}}(m|n)}\) . J. Am. Math. Soc. 16, 185–231 (2003)CrossRefzbMATHGoogle Scholar
  5. Br2.
    Brundan J.: Kazhdan–Lusztig polynomials and character formulae for the Lie superalgebra \({{\mathfrak{q}}(n)}\) . Adv. Math. 182, 28–77 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  6. Br3.
    Brundan J.: Tilting modules for Lie superalgebras. Commun. Algebra 32, 2251–2268 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  7. BLW.
    Brundan, J., Losev, I., Webster, B.: Tensor product categorifications and the super Kazhdan–Lusztig conjecture. Preprint, arXiv:1310.0349. IMRN (to appear)
  8. BD.
    Brundan, J., Davidson, N.: Type A blocks of super category \({{\mathcal{O}}}\) . arXiv:1606.05775
  9. CC.
    Chen, C.-W., Cheng, S.-J.: Quantum group of type A and representations of queer Lie superalgebra. J. Algebra. 473, 1–28 (2017)Google Scholar
  10. Ch.
    Chen, C.-W.: Reduction method for representations of queer Lie superalgebras. J. Math. Phys. 57, 051703 (2016). arXiv:1601.03924
  11. CK.
    Cheng S.-J., Kwon J.-H.: Finite-dimensional half-integer weight modules over queer Lie superalgebras. Commun. Math. Phys. 346, 945–965 (2016)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  12. CLW.
    Cheng S.-J., Lam N., Wang W.: Brundan–Kazhdan–Lusztig conjecture for general linear Lie superalgebras. Duke Math. J. 110, 617–695 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  13. CMW.
    Cheng S.-J., Mazorchuk V., Wang W.: Equivalence of blocks for the general linear Lie superalgebra. Lett. Math. Phys. 103, 1313–1327 (2013)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  14. CW1.
    Cheng S.-J., Wang W.: Remarks on the Schur–Howe–Sergeev duality. Lett. Math. Phys. 52, 143–153 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  15. CW2.
    Cheng S.-J., Wang W.: Dualities and Representations of Lie Superalgebras. Graduate Studies in Mathematics 144. American Mathematical Society, Providence (2012)CrossRefGoogle Scholar
  16. Cl.
    Clark S.: Quantum supergroups IV. The modified form. Math. Z. 278, 493–528 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  17. CHW.
    Clark S., Hill D., Wang W.: Quantum supergroups II. Canonical basis. Represent. Theory 18, 278–309 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  18. F.
    Frisk A.: Typical blocks of the category \({{\mathcal{O}}}\) for the queer Lie superalgebra. J. Algebra Appl. 6, 731–778 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  19. FM.
    Frisk A., Mazorchuk V.: Regular strongly typical blocks of \({{\mathcal{O}}^{\mathfrak{q}}}\) . Commun. Math. Phys. 291, 533–542 (2009)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  20. Jan.
    Jantzen J.C.: Lectures on Quantum Groups, Graduate Studies in Mathematics 6. Am. Math. Soc, (1996)Google Scholar
  21. JMO.
    Jing N., Misra K., Okado M.: q-wedge modules for quantized enveloping algebra of classical type. J. Algebra 230, 518–539 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  22. Ka.
    Kashiwara M.: On crystal bases of the Q-analogue of universal enveloping algebras. Duke Math. J. 63, 456–516 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  23. KKT.
    Kang S.-J., Kashiwara M., Tsuchioka S.: Quiver Hecke superalgebras. J. Reine Angew. Math. 711, 1–54 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  24. Lu1.
    Lusztig G.: Canonical bases arising from quantized enveloping algebras. J. Am. Math. Soc. 3, 447–498 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  25. Lu2.
    Lusztig G.: Canonical bases in tensor products. Proc. Natl. Acad. Sci. 89, 8177–8179 (1992)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  26. Lu3.
    Lusztig, G.: Introduction to Quantum Groups, Progress in Math., 110. Birkhäuser, Basel (1993)Google Scholar
  27. P.
    Penkov I.: Characters of typical irreducible finite-dimensional \({{\mathfrak{q}}(n)}\) -modules. Funct. Anal. App. 20, 30–37 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  28. PS.
    Penkov I., Serganova V.: Characters of irreducible G-modules and cohomology of G/P for the supergroup G = Q(N). J. Math. Sci. 84, 1382–1412 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  29. RW.
    Riche, S., Williamson, G.: Tilting modules and the p-canonical basis. arXiv:1512.08296
  30. San.
    Santos J.: Foncteurs de Zuckerman pour les superalgébres de Lie. J. Lie Theory 9, 69–112 (1999)ADSzbMATHMathSciNetGoogle Scholar
  31. Sv.
    Sergeev A.: The centre of enveloping algebra for Lie superalgebra \({Q(n,{\mathbb{C}})}\) . Lett. Math. Phys. 7, 177–179 (1983)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  32. Tsu.
    Tsuchioka, S.: Private communications, December 2015–January 2016Google Scholar
  33. W1.
    Webster, B.: Knot invariants and higher representation theory. arXiv:1309.3796. Memoirs AMS (to appear)
  34. W2.
    Webster B.: Canonical bases and higher representation theory. Compos. Math. 151, 121–166 (2015)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institute of Mathematics, Academia SinicaTaipeiTaiwan
  2. 2.Department of Mathematical SciencesSeoul National UniversitySeoulKorea
  3. 3.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

Personalised recommendations