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GRADED SUPER DUALITY FOR GENERAL LINEAR LIE SUPERALGEBRAS

  • C. LEONARDEmail author
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Abstract

We provide a new proof of the super duality equivalence between infinite-rank parabolic BGG categories of general linear Lie (super) algebras conjectured by Cheng and Wang and first proved by Cheng and Lam. We do this by establishing a new uniqueness theorem for tensor product categorifications motivated by work of Brundan, Losev, and Webster. Moreover we show that these BGG categories have Koszul graded lifts and super duality can be lifted to a graded equivalence.

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References

  1. [BLW17]
    J. Brundan, I. Losev, B. Webster, Tensor product categorifications and the super Kazhdan–Lusztig conjecture, Int. Math. Res. Not. IMRN (2017), no. 20, 6329–6410.Google Scholar
  2. [Bru03]
    J. Brundan, Kazhdan–Lusztig polynomials and character formulae for the Lie superalgebra \( \mathfrak{gl}\left(m|n\right) \), J. Amer. Math. Soc. 16 (2003), no. 1, 185–231.CrossRefzbMATHGoogle Scholar
  3. [Bru16]
    J. Brundan, On the definition of Kac–Moody 2-category, Math. Ann. 364 (2016), no. 1–2, 353–372.CrossRefzbMATHGoogle Scholar
  4. [CL10]
    S.-J. Cheng, N. Lam, Irreducible characters of general linear superalgebra and super duality, Commun. Math. Phys. 298 (2010), no. 3, 645–672.CrossRefzbMATHGoogle Scholar
  5. [CL15]
    S. Cautis, A. Lauda, Implicit structure in 2-representations of quantum groups, Selecta Math. (N.S.) 21 (2015), no. 1, 201–244.CrossRefzbMATHGoogle Scholar
  6. [CL18]
    C.-W. Chen, N. Lam, Projective modules over classical Lie algebras of infinite rank in the parabolic category, arXiv:1802.02112 (2018).Google Scholar
  7. [CLW15]
    S.-J. Cheng, N. Lam, W. Wang, The Brundan–Kazhdan–Lusztig conjecture for general linear Lie superalgebras, Duke Math. J. 164 (2015), no. 4, 617–695.CrossRefzbMATHGoogle Scholar
  8. [CPS88]
    E. Cline, B. Parshall, L. Scott, Finite-dimensional algebras and highest weight categories, J. Reine Angew. Math. 391 (1998), 85–99.zbMATHGoogle Scholar
  9. [CR08]
    J. Chuang, R. Rouquier, Derived equivalences for symmetric groups and \( \mathfrak{s}{\mathfrak{l}}_2 \)-categorification, Ann. of Math. (2) 167 (2008), no. 1, 245–298.CrossRefzbMATHGoogle Scholar
  10. [CW08]
    S.-J. Cheng, W. Wang, Brundan–Kazhdan–Lusztig and super duality conjectures, Publ. Res. Inst. Math. Sci. 44 (2008), no. 4, 1219–1272.CrossRefzbMATHGoogle Scholar
  11. [CWZ08]
    S.-J. Cheng, W. Wang, R. Zhang, Super duality and Kazhdan–Lusztig polynomials, Trans. Amer. Math. Soc. 360 (2008), no. 11, 5883–5924.CrossRefzbMATHGoogle Scholar
  12. [KL09]
    M. Khovanov, A. Lauda, A diagrammatic approach to categorification of quantum groups. I, Represent. Theory, 13:309–347, 2009.CrossRefzbMATHGoogle Scholar
  13. [KL10]
    M. Khovanov, A. Lauda, A categorification of quantum \( \mathfrak{sl}(n) \), Quantum Topol. 1 (2010), no. 1, 1–92.CrossRefzbMATHGoogle Scholar
  14. [LW15]
    I. Losev, B. Webster, On uniqueness of tensor products of irreducible categorifications, Selecta Math. (N.S.) 21 (2015), no. 2, 345–377.CrossRefzbMATHGoogle Scholar
  15. [Rou08]
    R. Rouquier, 2-Kac–Moody algebras, arXiv:0812.5023 (2008).Google Scholar
  16. [Rou12]
    R. Rouquier, Quiver Hecke algebras and 2-Lie algebras, Algebra Colloq. 19 (2012), no. 2, 359–410.CrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

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