Duke Mathematical Journal

Coherent sheaves and categorical $\mathfrak{sl}_2$ actions

Sabin Cautis,Joel Kamnitzer, and Anthony Licata

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Abstract

We introduce the concept of a geometric categorical $\mathfrak{sl}_2$ action and relate it to that of a strong categorical $\mathfrak{sl}_2$ action. The latter is a special kind of $2$-representation in the sense of Lauda and Rouquier. The main result is that a geometric categorical $\mathfrak{sl}_2$ action induces a strong categorical $\mathfrak{sl}_2$ action. This allows one to apply the theory of strong $\mathfrak{sl}_2$ actions to various geometric situations. Our main example is the construction of a geometric categorical $\mathfrak{sl}_2$ action on the derived category of coherent sheaves on cotangent bundles of Grassmannians

Article information

Source
Duke Math. J. Volume 154, Number 1 (2010), 135-179.

Dates
First available: 14 July 2010

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1279140507

Digital Object Identifier
doi:10.1215/00127094-2010-035

Mathematical Reviews number (MathSciNet)
MR2668555

Subjects
Primary: 14E05: Rational and birational maps
Secondary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23]

Citation

Cautis, Sabin; Kamnitzer, Joel; Licata, Anthony. Coherent sheaves and categorical sl 2 actions. Duke Mathematical Journal 154 (2010), no. 1, 135--179. doi:10.1215/00127094-2010-035. http://projecteuclid.org/euclid.dmj/1279140507.


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References

  • A. Beilinson, V. Ginzburg, and W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), 473--527.
  • J. Bernstein, I. Frenkel, and M. Khovanov, A categorification of the Temperley-Lieb algebra and Schur quotients of $U(\sl_2)$ via projective and Zuckerman functors, Selecta Math. (N.S.) 5 (1999), 199--241.
  • S. Cautis and J. Kamnitzer, Knot homology via derived categories of coherent sheaves, I: The $\mathfrak{sl}$(2)-case, Duke Math. J. 142 (2008), 511--588.
  • —, Knot homology via derived categories of coherent sheaves, II: $\mathfrak{sl}_m$ case, Invent. Math. 174 (2008), 165--232.
  • S. Cautis, J. KamnitzerandA. Licata, Categorical geometric skew Howe duality, Invent. Math. 180 (2010), 111--159.
  • —, Derived equivalences for cotangent bundles of Grassmannians via categorical $\sl_2$ actions, preprint.
  • —, Coherent sheaves on quiver varieties and categorification, in preparation.
  • J. Chuang and R. Rouquier, Derived equivalences for symmetric groups and $\sl_2$-categorification, Ann. of Math. (2) 167 (2008), 245--298.
  • D. Huybrechts and R. Thomas, $\p$-objects and autoequivalences of derived categories, Math. Res. Lett., 13 (2006) 87--98.
  • Y. Kawamata, ``Derived equivalence for stratified Mukai flop on $\bG(2,4)$'' in Mirror Symmetry, V, AMS/IP Stud. Adv. Math. 38, Amer. Math. Soc., Providence, 2006, 285--294.
  • M. Khovanov and A. D. Lauda, A diagrammatic approach to categorification of quantum groups I, Represent. Theory 13 (2009), 309--347.; II, preprint, math.QA/0804.2080v1 [math.QA]; III, Quantum Topology 1 (2010), 1--92.
  • A. D. Lauda, A categorification of quantum $\sl_2$, preprint.
  • I. Mirković and M. Vybornov, Quiver varieties and Beilinson-Drinfeld Grassmannians of type A, preprint.
  • Y. Namikawa, ``Mukai flops and derived categories, II'' in Algebraic Structures and Moduli Spaces, CRM Proc. Lecture Notes 38, Amer. Math. Soc., Providence, 2004, 149--175.
  • B. C. Ngo, Faisceaux pervers, homomorphisme de changement de base et lemme fondamental de Jacquet et Ye, preprint.
  • C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math. 1099, 1984, Springer, Berlin.
  • R. Rouquier, 2-Kac-Moody algebras, preprint.