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Braiding via geometric Lie algebra actions

Part of: Special varieties Lie algebras and Lie superalgebras (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)
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Abstract

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

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