Quantum groups via Hall algebras of complexes. (English) Zbl 1268.16017

Starting from the category \(\mathcal A\) of finite dimensional representations (over a finite field) of a finite acyclic quiver \(Q\), the author describes an embedding of the quantum group associated to the derived Kac-Moody Lie algebra of \(Q\) into the reduced version of the localization of the twisted Hall algebra of the category of projective \(\mathbb Z_2\)-graded complexes in \(\mathcal A\). It is shown that this embedding is an isomorphism if and only if \(Q\) is a simply laced Dynkin diagram.


16G20 Representations of quivers and partially ordered sets
17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B05 Structure theory for Lie algebras and superalgebras
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