Quantum Grothendieck rings and derived Hall algebras
Abstract
We obtain a presentation of the t-deformed Grothendieck ring of a quantum loop algebra of Dynkin type A, D, E. Specializing t at the the square root of the cardinality of a finite field F, we obtain an isomorphism with the derived Hall algebra of the derived category of a quiver Q of the same Dynkin type. Along the way, we study for each choice of orientation Q a tensor subcategory whose t-deformed Grothendieck ring is isomorphic to the positive part of a quantum enveloping algebra of the same Dynkin type, where the classes of simple objects correspond to Lusztig's dual canonical basis.
Keywords
Cite
@article{arxiv.1109.0862,
title = {Quantum Grothendieck rings and derived Hall algebras},
author = {David Hernandez and Bernard Leclerc},
journal= {arXiv preprint arXiv:1109.0862},
year = {2020}
}
Comments
version 2: minor corrections; some examples added; section 9 on quiver varieties has been improved. version 3: minor corrections; final version to appear in Crelle