Algebras and Representation Theory

, Volume 20, Issue 2, pp 469–486 | Cite as

A Categorification of \(\displaystyle {\mathfrak q} (2)\)-Crystals

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

Abstract

We provide a categorification of \(\mathfrak {q}(2)\)-crystals on the singular \(\mathfrak {gl}_{n}\)-category \({\mathcal O}_{n}\). Our result extends the \(\mathfrak {gl}_{2}\)-crystal structure on \(\text {Irr} ({\mathcal O}_{n})\) induced from the work of Bernstein-Frenkel-Khovanov. Further properties of the \({\mathfrak q}(2)\)-crystal \(\text {Irr} ({\mathcal O}_{n})\) are also discussed.

Keywords

Crystal bases Odd Kashiwara operators Quantum queer superalgebras \(\displaystyle {\mathfrak q} (2)\)-categorification 

Mathematics Subject Classification (2010)

17B37 81R50 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Benkart, G., Kang, S.-J., Kashiwara, M.: Crystal bases for the quantum superalgebra \(U_{q}(\mathfrak {gl}(m,n))\). J. Amer. Math. Soc. 13, 293–331 (2000)CrossRefGoogle Scholar
  2. 2.
    Bernstein, J., Frenkel, I., Khovanov, M.: A categorification of the Temperley-Lieb algebra and Schur quotients of U(s l 2) via projective and Zuckerman functors. Selecta. Math. (N.S.) 5(2), 199–241 (1999)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Brundan, J., Kleshchev, A.: Representations of shifted Yangians and finite W-algebras. Mem. Amer. Math. Soc. 196(918), viii+107 (2008)MathSciNetMATHGoogle Scholar
  4. 4.
    Brundan, J., Stroppel, C.: Highest weight categories arising from Khovanov’s diagram algebra III: category \(\mathcal O\). Represent. Theory 15, 170–243 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chuang, J., Rouquier, R.: Derived equivalences for symmetric groups and \(\mathfrak {sl}_{2}\)-categorification. Ann. Math. 1(2), 245–298 (2008)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Fiebig, P.: Centers and translation functors for the category \(\mathcal O\) over Kac-Moody algebras. Math. Z 243(4), 689–717 (2003)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Frenkel, I.B., Khovanov, M.G., Kirillov Jr., A. A.: Kazhdan-Lusztig polynomials and canonical basis. Transform. Groups 3(4), 321–336 (1998)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Frenkel, I., Khovanov, M., Stroppel, C.: A categorification of finite-dimensional irreducible representations of quantum s l 2 and their tensor products. Selecta. Math. (N.S.) 12, 379–431 (2006)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Grantcharov, D., Jung, J.H., Kang, S.-J., Kashiwara, M., Kim, M.: Quantum queer superalgebra and crystal bases. Proc. Japan Acad. Ser. A Math. Sci. 86, 177–182 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Grantcharov, D., Jung, J.H., Kang, S.-J., Kashiwara, M., Kim, M.: Crystal bases for the quantum queer superalgebra. J. Eur. Math. Soc. 17(7), 1593–1627 (2015)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Grantcharov, D., Jung, J.H., Kang, S.-J., Kashiwara, M., Kim, M.: Crystal bases for the quantum queer superalgebra and semistandard decomposition tableaux. Trans. Amer. Math. Soc. 366, 457–489 (2014)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Grantcharov, D., Jung, J.H., Kang, S.-J., Kim, M.: Highest weight modules over quantum queer superalgebra \(U_{q}({\mathfrak q}(n))\). Commun. Math. Phys. 296, 827–860 (2010)CrossRefMATHGoogle Scholar
  13. 13.
    Hong, J., Kang, S.-J.: Introduction to Quantum Groups and Crystal Bases, Graduate Studies in Mathematics, vol. 42. American Mathematical Society (2002)Google Scholar
  14. 14.
    Humphreys, J.: Representations of Semisimple Lie Algebras in the BGG Category \(\mathcal O\), Graduate Studies in Mathematics, vol. 94. American Mathematical Society (2008)Google Scholar
  15. 15.
    Kȧhrström, J.: Tensoring with infinite-dimensional modules in \({\mathcal O}_{0}\). Algebr. Represent. Theory 13, 561–587 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kang, S.-J.: Crystal bases for quantum affine algebras and combinatorics of Young walls. Proc. Lond. Math. Soc. 86(3), 29–69 (2003)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kashiwara, M.: Crystalizing the q-analogue of universal enveloping algebras. Commun. Math. Phys 133, 249–260 (1990)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Kashiwara, M.: On crystal bases of the q-analogue of universal enveloping algebras, Duke. Math. J 63, 465–516 (1991)MathSciNetMATHGoogle Scholar
  19. 19.
    Kashiwara, M.: Crystal base and Littelmann’s refined Demazure character formula. Duke Math. J. 71, 839–858 (1993)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Kwon, J.-H.: Super duality and Crystal bases for quantum orthosymplectic superalgebras. Int. Math. Res. Notices 2015(23), 12620–12677 (2015)MathSciNetMATHGoogle Scholar
  21. 21.
    Kashiwara, M., Nakashima, T.: Crystal graphs for representations of the q-analogue of classical Lie algebras. J. Algebra 165, 295–345 (1994)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Losev, I.: Highest weight s l 2-categorifications I: crystals. Math. Z 274, 1231–1247 (2013)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Lusztig, G.: Canonical bases arising from quantized enveloping algebras. J. Amer. Math. Soc. 3, 447–498 (1990)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Lusztig, G.: Quivers, perverse sheaves and quantized enveloping algebras. J. Amer. Math. Soc. 4, 365–421 (1991)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Mathieu, O.: Classification of weight modules. Ann. Inst. Fourier 50, 537–592 (2000)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Olshanski, G.: Quantized universal enveloping superalgebra of type Q and a super-extension of the Hecke algebra. Lett. Math. Phys. 24, 93–102 (1992)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Rouquier, R.: 2-Kac-Moody algebras. arXiv:0812.5023
  28. 28.
    Sergeev, A.: The tensor algebra of the tautological representation as a module over the Lie superalgebras \(\mathfrak {gl}(n,m)\) and Q(n). Mat. Sb. 123, 422–430 (1984). (in Russian)MathSciNetGoogle Scholar
  29. 29.
    Stroppel, C.: Categorification of the Temperley-Lieb category, tangles, and cobordisms via projective functors. Duke Math. J. 126, 547–596 (2005)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Dimitar Grantcharov
    • 1
  • Ji Hye Jung
    • 2
  • Seok-Jin Kang
    • 3
  • Myungho Kim
    • 4
  1. 1.Department of MathematicsUniversity of Texas at ArlingtonArlingtonUSA
  2. 2.Department of Mathematical SciencesSeoul National UniversitySeoulKorea
  3. 3.Department of Mathematical Sciences and Research Institute of MathematicsSeoul National UniversitySeoulKorea
  4. 4.Department of MathematicsKyung Hee UniversitySeoulKorea

Personalised recommendations