Duke Mathematical Journal

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

Sabin Cautis, Joel Kamnitzer, and Anthony Licata

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 14 July 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

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


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

Export citation


  • 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.