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Quivers with loops and generalized crystals

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

Published online by Cambridge University Press:  20 July 2016

Tristan Bozec
Tristan Bozec*
Affiliation:
Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA email tbozec@mit.edu
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Abstract

In the context of varieties of representations of arbitrary quivers, possibly carrying loops, we define a generalization of Lusztig Lagrangian subvarieties. From the combinatorial study of their irreducible components arises a structure richer than the usual Kashiwara crystals. Along with the geometric study of Nakajima quiver varieties, in the same context, this yields a notion of generalized crystals, coming with a tensor product. As an application, we define the semicanonical basis of the Hopf algebra generalizing quantum groups, which was already equipped with a canonical basis. The irreducible components of the Nakajima varieties provide the family of highest weight crystals associated to dominant weights, as in the classical case.

Keywords

quiver varietiesquantum groupsKashiwara crystals

MSC classification

Primary: 16G20: Representations of quivers and partially ordered sets 17B37: Quantum groups (quantized enveloping algebras) and related deformations
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 10 , October 2016 , pp. 1999 - 2040
Copyright
© The Author 2016 

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