Hostname: page-component-74656f9884-54qdv Total loading time: 0 Render date: 2026-11-09T18:39:39.343Z Has data issue: false hasContentIssue false
  • English
  • Français

Braiding via geometric Lie algebra actions

Part of: Lie algebras and Lie superalgebras Special varieties (Co)homology theory

Published online by Cambridge University Press:  24 January 2012

Sabin Cautis  and
Joel Kamnitzer
Sabin Cautis
Affiliation:
Department of Mathematics, Columbia University, New York, NY, USA (email: scautis@math.columbia.edu)
Joel Kamnitzer
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON, Canada (email: jkamnitz@math.toronto.edu)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We introduce the idea of a geometric categorical Lie algebra action on derived categories of coherent sheaves. The main result is that such an action induces an action of the braid group associated to the Lie algebra. The same proof shows that strong categorical actions in the sense of Khovanov–Lauda and Rouquier also lead to braid group actions. As an example, we construct an action of Artin’s braid group on derived categories of coherent sheaves on cotangent bundles to partial flag varieties.

Keywords

categorificationbraid groupsderived categories of coherent sheaves

MSC classification

Secondary: 14F05: Sheaves, derived categories of sheaves and related constructions 14M15: Grassmannians, Schubert varieties, flag manifolds 17B37: Quantum groups (quantized enveloping algebras) and related deformations

Information

Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 2 , March 2012 , pp. 464 - 506
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[Ann07]Anno, R., Spherical functors, math.CT/0711.4409.Google Scholar
[BS01]Balmer, P. and Schlichting, M., Idempotent completion of triangulated categories, J. Algebra 236 (2001), 819–834.CrossRefGoogle Scholar
[Bar07]Bar-Natan, D., Fast Khovanov homology computations, J. Knot Theory Ramifications 16 (2007), 243–255, math.GT/0606318.CrossRefGoogle Scholar
[BBD82]Beilinson, A., Bernstein, J. and Deligne, P., Faisceaux pervers. Analyse et topologie sur les espaces singuliers (I), Astérisque 100 (1982).Google Scholar
[BMR06]Bezrukavnikov, R., Mirkovic, I. and Rumynin, D., Singular localization and intertwining functors for semisimple Lie algebras in prime characteristic, Nagoya Math. J. 184 (2006), 1–55.Google Scholar
[Cau09]Cautis, S., Equivalences and stratified flops, Compos. Math., to appear, math.AG/0909.0817.Google Scholar
[CK08]Cautis, S. and Kamnitzer, J., Knot homology via derived categories of coherent sheaves II, sl(m) case, Invent. Math. 174 (2008), 165–232, math.AG/0710.3216.CrossRefGoogle Scholar
[CKL09]Cautis, S., Kamnitzer, J. and Licata, A., Derived equivalences for cotangent bundles of Grassmannians via categorical sl2 actions, J. Reine Angew. Math., to appear, math.AG/0902.1797.Google Scholar
[CKL11]Cautis, S., Kamnitzer, J. and Licata, A., Coherent sheaves on quiver varieties and categorification, arXiv:1104.0352.Google Scholar
[CKL10a]Cautis, S., Kamnitzer, J. and Licata, A., Categorical geometric skew Howe duality, Invent. Math. 180 (2010), 111–159, math.AG/0902.1795.CrossRefGoogle Scholar
[CKL10b]Cautis, S., Kamnitzer, J. and Licata, A., Coherent sheaves and categorical $\mathfrak {sl}_2$ actions, Duke Math. J. 154 (2010), 135–179, math.AG/0902.1796.Google Scholar
[CR08]Chuang, J. and Rouquier, R., Derived equivalences for symmetric groups and $\mathfrak {sl}_2$-categorification, Ann. of Math. (2) 167 (2008), 245–298, math.RT/0407205.Google Scholar
[GM03]Gelfand, S. and Manin, Y., Methods of homological algebra, second edition, Springer Monographs in Mathematics (Springer, Berlin, 2003).CrossRefGoogle Scholar
[Hor05]Horja, R. P., Derived category automorphisms from mirror symmetry, Duke Math. J. 127 (2005), 1–34, math.AG/0103231.CrossRefGoogle Scholar
[Huy06]Huybrechts, D., Fourier–Mukai transforms in algebraic geometry (Oxford University Press, Oxford, 2006).CrossRefGoogle Scholar
[HT06]Huybrechts, D. and Thomas, R., ℙ-objects and autoequivalences of derived categories, Math. Res. Lett. 13 (2006), 87–98, math.AG/0507040.Google Scholar
[KL09]Khovanov, M. and Lauda, A., A diagrammatic approach to categorification of quantum groups I, Represent. Theory 13 (2009), 309–347, math.QA/0803.4121.CrossRefGoogle Scholar
[KL10]Khovanov, M. and Lauda, A., A diagrammatic approach to categorification of quantum groups III, Quantum Topol. 1 (2010), 1–92, math.QA/0807.3250.CrossRefGoogle Scholar
[KL11]Khovanov, M. and Lauda, A., A diagrammatic approach to categorification of quantum groups II, Trans. Amer. Math. Soc. 363 (2011), 2685–2700, math.QA/0804.2080.CrossRefGoogle Scholar
[KT07]Khovanov, M. and Thomas, R., Braid cobordisms, triangulated categories, and flag varieties, Homology, Homotopy Appl. 9 (2007), 19–94, math.QA/0609335.Google Scholar
[Ric08]Riche, S., Geometric braid group action on derived category of coherent sheaves, Represent. Theory 12 (2008), 131–169.Google Scholar
[Rin84]Ringel, C. M., Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099 (Springer, Berlin, 1984).CrossRefGoogle Scholar
[Rou06]Rouquier, R., Categorification of $\mathfrak {sl}_2$ and braid groups, in Trends in representation theory of algebras and related topics, Contemporary Mathematics, vol. 406 (American Mathematical Society, Providence, RI, 2006), 137–167.CrossRefGoogle Scholar
[Rou08]Rouquier, R., 2-Kac-Moody algebras, math.RT/0812.5023.Google Scholar
[ST01]Seidel, P. and Thomas, R., Braid group actions on derived categories of coherent sheaves, Duke Math J. 108 (2001), 37–108, math.AG/0001043.CrossRefGoogle Scholar