Exact structures and degeneration of Hall algebras
Hall algebras were first defined by \textit{E. Steinitz} [Jahresber. Dtsch. Math.-Ver. 9, No. 1, 80--85 (1901; JFM 32.0149.02)] and \textit{P. Hall} [in: Proc. 4th Can. Math. Congr., Banff 1957, 147--159 (1959; Zbl 0122.03403)] for finite \(p\)-groups. Later \textit{C. M. Ringel} [Invent. Math. 101, No. 3, 583--591 (1990; Zbl 0735.16009)] defined Ringel-Hall algebras for Hom- and Ext-finite abelian categories. \textit{A. Hubery} [Contemp. Math. 406, 51--66 (2006; Zbl 1107.16021)] defined Ringel-Hall algebra for an exact category in the same way as defined by Ringel, and showed that this algebra is also unital and associative. An exact category \(\mathcal{E}\) is an additive category endowed with an extra structure, and any additive category can be endowed with many different exact structures. In the paper under review, the authors study relations between different Ringel-Hall algebras associated to different exact structures on the same additive category. They prove that the Ringel-Hall algebra of a smaller exact structure \(\mathcal{E'}< \mathcal{E}\) on the additive category \(\mathcal{A}\) is the degeneration of the Ringel-Hall algebra \(H(\mathcal{E})\) of an exact category \(\mathcal{E}\) which induced by degree functions on the set of isomorphism classes of indecomposable objects. By using these results they recover degenerations of the negative part of the quantum groups, as well as the polyhedral structure associated to the categories of representations of Dynkin quivers which is studied by \textit{X. Fang} et al. [J. Algebra 449, 321--345 (2016; Zbl 1348.17013); J. Algebr. Comb. 54, No. 3, 815--835 (2021; Zbl 1493.17013)]. Among the other results they also improve some results of \textit{H. Enomoto} [Adv. Math. 335, 838--877 (2018; Zbl 1419.18017)] and \textit{T. Brüstle} et al. [J. Pure Appl. Algebra 224, No. 4, Article ID 106212, 29 p. (2020; Zbl 1423.18034)] about the classification of exact structures on an additive category.
scientific article; zbMATH DE number 7483902
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