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arXiv · 1109.0862

Quantum Grothendieck rings and derived Hall algebras

David Hernandez·Bernard Leclerc
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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.

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David Hernandez, Bernard Leclerc. 2013-03-02. Quantum Grothendieck rings and derived Hall algebras. https://doi.org/10.1515/crelle-2013-0020

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