Quiver Varieties and Crystals in Symmetrizable Type via Modulated Graphs

Vinoth Nandakumar; Peter Tingley

doi:10.4310/MRL.2018.v25.n1.a7

Quiver Varieties and Crystals in Symmetrizable Type via Modulated Graphs

Vinoth Nandakumar, Peter Tingley

Department of Mathematics and Statistics

University of Sydney, Australia

Research output: Contribution to journal › Article › peer-review

Abstract

Kashiwara and Saito have a geometric construction of the infinity crystal for any symmetric Kac–Moody algebra. The underlying set consists of the irreducible components of Lusztig’s quiver varieties, which are varieties of nilpotent representations of a pre-projective algebra. We generalize this to symmetrizable Kac–Moody algebras by replacing Lusztig’s preprojective algebra with a more general one due to Dlab and Ringel. In non-symmetric types we are forced to work over non-algebraically-closed fields.

Original language	American English
Journal	Mathematics and Statistics: Faculty Publications and Other Works
Volume	25
DOIs	https://doi.org/10.4310/MRL.2018.v25.n1.a7
State	Published - Jan 1 2018

Keywords

crystal
pre-projective algebra
quiver variety
Kac-Moody algebra

Disciplines

Algebra
Mathematics

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N2 - Kashiwara and Saito have a geometric construction of the infinity crystal for any symmetric Kac–Moody algebra. The underlying set consists of the irreducible components of Lusztig’s quiver varieties, which are varieties of nilpotent representations of a pre-projective algebra. We generalize this to symmetrizable Kac–Moody algebras by replacing Lusztig’s preprojective algebra with a more general one due to Dlab and Ringel. In non-symmetric types we are forced to work over non-algebraically-closed fields.

AB - Kashiwara and Saito have a geometric construction of the infinity crystal for any symmetric Kac–Moody algebra. The underlying set consists of the irreducible components of Lusztig’s quiver varieties, which are varieties of nilpotent representations of a pre-projective algebra. We generalize this to symmetrizable Kac–Moody algebras by replacing Lusztig’s preprojective algebra with a more general one due to Dlab and Ringel. In non-symmetric types we are forced to work over non-algebraically-closed fields.

KW - crystal

KW - pre-projective algebra

KW - quiver variety

KW - Kac-Moody algebra

UR - https://ecommons.luc.edu/math_facpubs/21

U2 - 10.4310/MRL.2018.v25.n1.a7

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JO - Mathematics and Statistics: Faculty Publications and Other Works

JF - Mathematics and Statistics: Faculty Publications and Other Works

ER -

Quiver Varieties and Crystals in Symmetrizable Type via Modulated Graphs

Abstract

Keywords

Disciplines

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