Abstract
We know that in Ringel-Hall algebra of Dynkin type, the set of all skew commutator relations between the iso-classes of indecomposable modules forms a minimal Gröbner-Shirshov basis, and the corresponding irreducible elements forms a PBW type basis of the Ringel-Hall algebra. We aim to generalize this result to the derived Hall algebra DH(An) of type An. First, we compute all skew commutator relations between the iso-classes of indecomposable objects in the bounded derived category Db(An) using the Auslander-Reiten quiver of Db(An), and then we prove that all possible compositions between these skew commutator relations are trivial. As an application, we give a PBW type basis of DH(An).
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We are deeply impressed by the seriousness of the referee about the computations in our paper and very grateful for the useful suggestions which improved this paper.
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Supported by the Natural Science Foundation of China (Grant No. 11861061)
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He, Z., Obul, A. Gröbner–Shirshov Basis of Derived Hall Algebra of Type An. Acta. Math. Sin.-English Ser. 36, 929–942 (2020). https://doi.org/10.1007/s10114-020-9547-2
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DOI: https://doi.org/10.1007/s10114-020-9547-2