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Canonical bases and higher representation theory

Part of: Lie algebras and Lie superalgebras Categories with structure

Published online by Cambridge University Press:  07 October 2014

Ben Webster
Ben Webster*
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA 22908, USA email bwebster@virginia.edu
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Abstract

This paper develops a general theory of canonical bases and how they arise naturally in the context of categorification. As an application, we show that Lusztig’s canonical basis in the whole quantized universal enveloping algebra is given by the classes of the indecomposable 1-morphisms in a categorification when the associated Lie algebra is of finite type and simply laced. We also introduce natural categories whose Grothendieck groups correspond to the tensor products of lowest- and highest-weight integrable representations. This generalizes past work of the author’s in the highest-weight case.

Keywords

canonical basescategorification

MSC classification

Primary: 17B35: Universal enveloping (super)algebras 17B37: Quantum groups (quantized enveloping algebras) and related deformations 18D05: Double categories, $2$-categories, bicategories and generalizations
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 1 , January 2015 , pp. 121 - 166
Copyright
© The Author 2014 

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