%0 Journal Article %T Quantum Grothendieck rings and derived Hall algebras %A David Hernandez %A Bernard Leclerc %J Mathematics %D 2011 %I arXiv %X 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. %U http://arxiv.org/abs/1109.0862v3