Skip to main content
SpringerLink
Log in

Type C blocks of super category \(\mathcal {O}\)

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We show that the blocks of category \(\mathcal {O}\) for the Lie superalgebra \({\mathfrak {q}}_n({\mathbb {C}})\) associated to half-integral weights carry the structure of a tensor product categorification for the infinite rank Kac-Moody algebra of type \(\hbox {C}_\infty \). This allows us to prove two conjectures formulated by Cheng, Kwon and Wang. We then focus on the full subcategory consisting of finite-dimensional representations, which we show is a highest weight category with blocks that are Morita equivalent to certain generalized Khovanov arc algebras.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Brundan, J., Losev, I., Webster, B.: Tensor product categorifications and the super Kazhdan–Lusztig Conjecture. Int. Math. Res. Notices 20, 6329–6410 (2017)

  2. Brundan, J.: Kazhdan–Lusztig polynomials and character formulae for the Lie superalgebra \({{\mathfrak{q}}}(n)\). Adv. Math. 182, 28–77 (2004)

    Article  MathSciNet  Google Scholar 

  3. Brundan, J.: Tilting modules for Lie superalgebras. Commun. Algebra 32, 2251–2268 (2004)

    Article  MathSciNet  Google Scholar 

  4. Brundan, J.: Representations of the general linear Lie superalgebra in the BGG category \({\cal{O}}\). In: Mason, G., et al. (eds.) Developments and Retrospectives in Lie Theory: Algebraic Methods Developments in Mathematics, vol. 38, pp. 71–98. Springer, Berlin (2014)

    Google Scholar 

  5. Brundan, J., Davidson, N.: Categorical actions and crystals. Contemp. Math. 684, 116–159 (2017)

    MATH  Google Scholar 

  6. Brundan, J., Davidson, N.: Type A blocks of super category \({\cal{O}}\). J. Algebra 473, 447–480 (2017)

    Article  MathSciNet  Google Scholar 

  7. Brundan, J., Ellis, A.: Monoidal supercategories. Commun. Math. Phys. 351, 1045–1089 (2017)

    Article  MathSciNet  Google Scholar 

  8. Brundan, J., Kleshchev, A.: Hecke–Clifford superalgebras, crystals of type \(A_{2\ell }^{(2)}\) and modular branching rules for \({\widehat{S}}_n\). Represent. Theory 5, 317–403 (2001)

    Article  MathSciNet  Google Scholar 

  9. Brundan, J., Stroppel, C.: Highest weight categories arising from Khovanov’s diagram algebras I: cellularity. Mosc. Math. J. 11, 685–722 (2011)

    Article  MathSciNet  Google Scholar 

  10. Brundan, J., Stroppel, C.: Highest weight categories arising from Khovanov’s diagram algebras IV: the general linear supergroup. JEMS 14, 373–419 (2012)

    Article  MathSciNet  Google Scholar 

  11. Chen, C.-W.: Reduction method for representations of queer Lie superalgebras. J. Math. Phys. 57(5), 051703–12 (2016)

    Article  MathSciNet  Google Scholar 

  12. Cheng, S.-J., Kwon, J.-H., Wang, W.: Character formulae for queer Lie superalgebras and canonical bases of types A/C. Commun. Math. Phys. 352, 1091–1119 (2017)

  13. Cheng, S.-J., Kwon, J.-H.: Finite-dimensional half-integer weight modules over queer Lie superalgebras. Commun. Math. Phys. 346, 945–965 (2016)

    Article  MathSciNet  Google Scholar 

  14. Cline, E., Parshall, B., Scott, L.: Finite dimensional algebras and highest weight categories. J. Reine Angew. Math. 391, 85–99 (1988)

    MathSciNet  MATH  Google Scholar 

  15. Davidson, N.: Type B blocks of super category \({\cal{O}}\) (in preparation)

  16. Hill, D., Kujawa, J., Sussan, J.: Degenerate affine Hecke–Clifford algebras and type Q Lie superalgebras. Math. Z. 268, 1091–1158 (2011)

    Article  MathSciNet  Google Scholar 

  17. Jantzen, J.C.: Lectures on Quantum Groups, AMS (1995)

  18. Kac, V.: Characters of typical representations of classical Lie superalgebras. Commun. Algebra 5, 889–897 (1977)

    Article  MathSciNet  Google Scholar 

  19. Kang, S.-J., Kashiwara, M., Tsuchioka, S.: Quiver Hecke superalgebras. J. Reine Angew. Math. 711, 1–54 (2016)

    Article  MathSciNet  Google Scholar 

  20. Khovanov, M., Lauda, A.: A diagrammatic approach to categorification of quantum groups I. Represent. Theory 13, 309–347 (2009)

    Article  MathSciNet  Google Scholar 

  21. Khovanov, M., Lauda, A.: A diagrammatic approach to categorification of quantum groups II. Trans. Am. Math. Soc. 363, 2685–2700 (2011)

    Article  MathSciNet  Google Scholar 

  22. Losev, I., Webster, B.: On uniqueness of tensor products of irreducible categorifications. Sel. Math. 21, 345–377 (2015)

    Article  MathSciNet  Google Scholar 

  23. Lusztig, G.: Introduction to Quantum Groups. Birkhäuser, Basel (1993)

    MATH  Google Scholar 

  24. Nazarov, M.: Young’s symmetrizers for projective representations of the symmetric group. Adv. Math. 127, 190–257 (1997)

    Article  MathSciNet  Google Scholar 

  25. Penkov, I.: Characters of typical irreducible finite-dimensional \(\mathfrak{q}(n)\)-modules. Funct. Anal. Appl. 20, 30–37 (1986)

    Article  MathSciNet  Google Scholar 

  26. Rouquier, R.: 2-Kac–Moody algebras. arXiv:0812.5023

  27. Webster, B.: Canonical bases and higher representation theory. Compos. Math. 151, 121–166 (2015)

    Article  MathSciNet  Google Scholar 

  28. Webster, B.: Knot invariants and higher representation theory. Mem. Am. Math. Soc. 1191, 133 (2017)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank Shunsuke Tsuchioka for allowing us to include his counterexamples to positivity in Example 2.12.

Author information

Author notes

  1. Nicholas Davidson

    Present address: Department of Mathematics, Reed College, Portland, OR, 97202, USA

Authors and Affiliations

  1. Department of Mathematics, University of Oregon, Eugene, OR, 97403, USA

    Jonathan Brundan

  2. Department of Mathematics, University of Oklahoma, Norman, OK, 73019, USA

    Nicholas Davidson

Authors
  1. Jonathan Brundan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicholas Davidson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicholas Davidson.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Research supported in part by NSF Grants DMS-1161094 and DMS-1700905.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Brundan, J., Davidson, N. Type C blocks of super category \(\mathcal {O}\). Math. Z. 293, 867–901 (2019). https://doi.org/10.1007/s00209-019-02400-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-019-02400-y

Search

Navigation