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Braid group actions via categorifiedHeisenberg complexes

Part of: Homological algebra Special aspects of infinite or finite groups Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  09 October 2013

Sabin Cautis ,
Anthony Licata  and
Joshua Sussan
Sabin Cautis*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V6T 1Z2
Anthony Licata
Affiliation:
Mathematical Sciences Institute, Australian National University, Canberra, Australia email anthony.licata@anu.edu.au
Joshua Sussan
Affiliation:
Department of Mathematics, CUNY Medgar Evers, Brooklyn, NY, USA email jsussan@mec.cuny.edu
*
cautis@math.ubc.ca
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Abstract

We construct categorical braid group actions from 2-representations of a Heisenberg algebra. These actions are induced by certain complexes which generalize spherical (Seidel–Thomas) twists and are reminiscent of the Rickard complexes defined by Chuang–Rouquier. Conjecturally, one can relate our complexes to Rickard complexes using categorical vertex operators.

MSC classification

Secondary: 17B65: Infinite-dimensional Lie (super)algebras 18G10: Resolutions; derived functors 20F36: Braid groups; Artin groups

Information

Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 1 , January 2014 , pp. 105 - 142
Copyright
© The Author(s) 2013 

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References

Anno, R., Spherical functors, Preprint (2007), arXiv:0711.4409 [math.CT].Google Scholar
Bar-Natan, D., Fast Khovanov homology computations, J. Knot Theory Ramifications 16 (2007), 243–255.Google Scholar
Cautis, S. and Kamnitzer, J., Knot homology via derived categories of coherent sheaves I, sl(2) case, Duke Math. J. 142 (2008), 511–588.CrossRefGoogle Scholar
Cautis, S. and Kamnitzer, J., Braiding via geometric Lie algebra actions, Compositio Math. 148 (2012), 464–506.CrossRefGoogle Scholar
Cautis, S., Kamnitzer, J. and Licata, A., Derived equivalences for cotangent bundles of Grassmannians via categorical ${\mathrm{sl} }_{2} $ actions, J. Reine Angew. Math. 675 (2013), 53–99.Google Scholar
Cautis, S. and Lauda, A., Implicit structure in 2-representations of quantum groups, Preprint (2011), arXiv:1111.1431.Google Scholar
Cautis, S. and Licata, A., Vertex operators and 2-representations of quantum affine algebras, Preprint (2011), arXiv:1112.6189.Google Scholar
Cautis, S. and Licata, A., Heisenberg categorification and Hilbert schemes, Duke Math. J. 161 (2012), 2469–2547.CrossRefGoogle Scholar
Chuang, J. and Rouquier, R., Derived equivalences for symmetric groups and ${\mathfrak{s}\mathfrak{l}}_{2} $-categorification, Ann. of Math. (2) 167 (2008), 245–298.Google Scholar
Horja, R. P., Derived category automorphisms from mirror symmetry, Duke Math. J. 127 (2005), 1–34.Google Scholar
Huerfano, R. S. and Khovanov, M., A category for the adjoint representation, J. Algebra 246 (2001), 514–542.Google Scholar
Khovanov, M., A functor-valued invariant of tangles, Trans. Amer. Math. Soc. 358 (2006), 315–327.CrossRefGoogle Scholar
Khovanov, M. and Lauda, A., A diagrammatic approach to categorification of quantum groups I, Represent. Theory 13 (2009), 309–347.Google Scholar
Khovanov, M. and Lauda, A., A diagrammatic approach to categorification of quantum groups III, Quantum Topol. 1 (2010), 1–92.Google Scholar
Khovanov, M. and Lauda, A., A diagrammatic approach to categorification of quantum groups II, Trans. Amer. Math. Soc. 363 (2011), 2685–2700.Google Scholar
Khovanov, M. and Seidel, P., Quivers, Floer cohomology, and braid group actions, J. Amer. Math. Soc. 15 (2002), 203–271.Google Scholar
Lauda, A., A categorification of quantum ${\mathfrak{s}\mathfrak{l}}_{2} $, Adv. Math. 225 (2010), 3327–3424.CrossRefGoogle Scholar
Ploog, D., Equivariant autoequivalences for finite group actions, Adv. Math. 216 (2007), 62–74.Google Scholar
Ringel, C. M., Tame Algebras and Integral Quadratic Forms, Lecture Notes in Mathematics, vol. 1099 (Springer, Berlin, 1984).Google Scholar
Rouquier, R., Categorification of braid groups, Preprint (2004), arXiv:math/0409593 [math.RT].Google Scholar
Rouquier, R., 2-Kac–Moody algebras, Preprint (2008), arXiv:0812.5023 [math.RT].Google Scholar
Seidel, P. and Thomas, R., Braid group actions on derived categories of coherent sheaves, Duke Math. J. 108 (2001), 37–108.Google Scholar