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Bridgeland, Tom

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

Ann. Math. (2) 177, No. 2, 739-759 (2013).
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.
Reviewer: Volodymyr Mazorchuk (Uppsala)

Cited in 5 Reviews
Cited in 44 Documents

MSC:

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

Keywords:

Hall algebras; quantum groups; graded complexes; Kac-Moody Lie algebras; quantum enveloping algebras; quiver representations; Dynkin diagrams
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\textit{T. Bridgeland}, Ann. Math. (2) 177, No. 2, 739--759 (2013; Zbl 1268.16017)
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References:

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