Quantum Grothendieck rings and derived Hall algebras

Details Quantum Grothendieck rings and derived Hall algebras Quantum Grothendieck rings and derived Hall algebras

2011-09-05

arxiv.org/abs/1109.0862v2

Mathematics

Quantum Algebra

version 2: minor corrections; some examples added; section 9 on quiver varieties has been improved

Scientific paper

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.

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