Advertisement

Parameters in Categorified Quantum Groups

  • Aaron D. LaudaEmail author
Article
  • 4 Downloads

Abstract

In this note we give explicit isomorphisms of 2-categories between various versions of the categorified quantum group associated to a simply-laced Kac-Moody algebra. These isomorphisms are convenient when working with the categorified quantum group. They make it possible to translate results from the \(\mathfrak {g}\mathfrak {l}_{n}\) variant of the 2-category to the \(\mathfrak {s}\mathfrak {l}_{n}\) variant and transfer results between various conventions in the literature. We also extend isomorphisms of finite type KLR algebras for different choices of parameters to the level of 2-categories.

Keywords

Categorification Quantum group KLR-algebra 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

The author is partially supported by the NSF grants DMS-1255334 and DMS-1664240. He would also like to thank Joshua Sussan and Hoel Queffelec for comments on an early version of this note.

References

  1. 1.
    Beliakova, A., Habiro, K., Lauda, A.D., Webster, B.: Cyclicity for categorified quantum groups. J. Algebra 452, 118–132 (2016). arXiv:1506.04671 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brundan, J.: On the definition of Kac-Moody 2-category. Math. Ann. 364(1-2), 353–372 (2016). arXiv:1501.00350 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cautis, S.: Clasp technology to knot homology via the affine Grassmannian. Math. Ann. 363(3-4), 1053–1115 (2015). arXiv:1207.2074 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cautis, S., Kamnitzer, J., Licata, A.: Categorical geometric skew Howe duality. Inventiones Math. 180(1), 111–159 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cautis, S., Lauda, A.D.: Implicit structure in 2-representations of quantum groups. Selecta Mathematica, pp. 1–44. arXiv:1111.1431 (2014)
  6. 6.
    Rose, D.E.V., Queffelec, H., Sartori, A.: Annular evaluation and link homology. arXiv:1802.04131 (2018)
  7. 7.
    Kashiwara, M.: Notes on parameters of quiver Hecke algebras. Proc. Japan Acad. Ser. A Math. Sci. 88(7), 97–102 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Khovanov, M., Lauda, A.D.: A diagrammatic approach to categorification of quantum groups III. Quantum Topology 1, 1–92 (2010). arXiv:0807.3250 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Khovanov, M., Lauda, A.D.: A diagrammatic approach to categorification of quantum groups II. Trans. Amer. Math. Soc. 363, 2685–2700 (2011). arXiv:0804.2080 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lauda, A.D.: A categorification of quantum sl(2). Adv. Math. 225, 3327–3424 (2008). arXiv:0803.3652 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lauda, A.D.: Categorified quantum sl(2) and equivariant cohomology of iterated flag varieties. Algebras and Representation Theory, pp. 1–30. arXiv:0803.3848 (2009)
  12. 12.
    Lauda, A.D.: An introduction to diagrammatic algebra and categorified quantum \({\mathfrak {sl}}_{2}\). Bullet. Inst. Math. Acad. Sin. 7, 165–270 (2012). arXiv:1106.2128 zbMATHGoogle Scholar
  13. 13.
    Lauda, A.D., Queffelec, H., Rose, D.E.V.: Khovanov homology is a skew Howe 2-representation of categorified quantum \(\mathfrak {s}\mathfrak {l}_{m}\). Algebr. Geom. Topol. 15(5), 2517–2608 (2015). arXiv:1212.6076 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mackaay, M., Stošić, M., Vaz, P.: A diagrammatic categorification of the q-Schur algebra. Quantum Topol. 4(1), 1–75 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mackaay, M., Webster, B.: Categorified skew Howe duality and comparison of knot homologies. Adv. Math. 330, 876–945 (2018). arXiv:1502.06011 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Queffelec, H., Rose, D.E.V.: The \(\mathfrak {s}\mathfrak {l}_{n}\) foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality. Adv. Math. 302, 1251–1339 (2016). arXiv:1405.5920 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Rouquier, R.: 2-Kac-Moody algebras. arXiv:0812.5023 (2008)
  18. 18.
    Webster, B.: Knot invariants and higher representation theory. Mem. Amer. Math. Soc. 250(1191), v + 141 (2017). arXiv:1001.2020 MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA

Personalised recommendations