Exact structures and degeneration of Hall algebras
type: Item
instance of: scholarly article
title: Exact structures and degeneration of Hall algebras
author: Yanyan Li
published in: Advances in Mathematics
publication date: 2022-03-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/2005.12130
review text: 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.
reviewed by: Alireza Nasr-Isfahani
Mathematics Subject Classification ID: 16T20
Mathematics Subject Classification ID: 18E10
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 18G80
Mathematics Subject Classification ID: 16G20
zbMATH DE Number: 7483902
zbMATH Keywords: exact structure
zbMATH Keywords: Hall algebra
zbMATH Keywords: degeneration
MaRDI profile type: Publication
OpenAlex ID: W4205897894
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Exact structures and degeneration of Hall algebras (DOI)
DOI: 10.1016/J.AIM.2022.108210

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