Mathematical Research Letters

Volume 25 (2018)

Number 1

Quiver varieties and crystals in symmetrizable type via modulated graphs

Pages: 159 – 180



Vinoth Nandakumar (School of Mathematics and Statistics, University of Sydney, Australia)

Peter Tingley (Department of Mathematics and Statistics., Loyola University, Chicago, Illinois, U.S.A.)


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.


crystal, pre-projective algebra, quiver variety, Kac–Moody algebra

2010 Mathematics Subject Classification


The second author was partially supported by NSF grants DMS-0902649 and DMS-1265555.

Received 1 July 2016

Accepted 2 January 2017

Published 4 June 2018