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Quantum groups via cyclic quiver varieties I

Part of: Representation theory of rings and algebras Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  07 September 2015

Fan Qin
Fan Qin*
Affiliation:
Institut de Recherche Mathématique Avancée (IRMA), 7 rue René Descartes, 67084 Strasbourg Cedex, France email qin@math.unistra.fr
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Abstract

We construct the quantized enveloping algebra of any simple Lie algebra of type $\mathbb{A}\mathbb{D}\mathbb{E}$ as the quotient of a Grothendieck ring arising from certain cyclic quiver varieties. In particular, the dual canonical basis of a one-half quantum group with respect to Lusztig’s bilinear form is contained in the natural basis of the Grothendieck ring up to rescaling. This paper expands the categorification established by Hernandez and Leclerc to the whole quantum groups. It can be viewed as a geometric counterpart of Bridgeland’s recent work for type $\mathbb{A}\mathbb{D}\mathbb{E}$.

Keywords

quantum groupquiver varietycategorificationdual canonical basis

MSC classification

Primary: 16G20: Representations of quivers and partially ordered sets 17B37: Quantum groups (quantized enveloping algebras) and related deformations

Information

Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 2 , February 2016 , pp. 299 - 326
Copyright
© The Author 2015 

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