Quantum Grothendieck rings and derived Hall algebras

TL;DR

This paper establishes a connection between t-deformed Grothendieck rings of quantum loop algebras and derived Hall algebras of quivers, revealing new algebraic structures and bases in representation theory.

Contribution

It provides a presentation of the t-deformed Grothendieck ring and shows its isomorphism with derived Hall algebras for Dynkin types A, D, E, linking quantum groups and Hall algebras.

Findings

Isomorphism between t-deformed Grothendieck rings and derived Hall algebras.

Identification of tensor subcategories with positive parts of quantum enveloping algebras.

Simple objects correspond to Lusztig's dual canonical basis.

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.