Universität zu Köln

Fang, Xin and Gorsky, Mikhail ORCID: 0000-0003-0547-3571 (2022). Exact structures and degeneration of Hall algebras. Adv. Math., 398. SAN DIEGO: ACADEMIC PRESS INC ELSEVIER SCIENCE. ISSN 1090-2082

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Abstract

We study degenerations of the Hall algebras of exact categories induced by degree functions on the set of isomorphism classes of indecomposable objects. We prove that each such degeneration of the Hall algebra 7-l(E) of an exact category E is the Hall algebra of a smaller exact structure E, < E on the same additive category A. When E is admissible in the sense of Enomoto, for any E, < E satisfying suitable finiteness conditions, we prove that 7-l(E,) is a degeneration of 7-l(E) of this kind. In the additively finite case, all such degree functions form a simplicial cone whose face lattice reflects properties of the lattice of exact structures. For the categories of representations of Dynkin quivers, we recover degenerations of the negative part of the corresponding quantum group, as well as the associated polyhedral structure studied by Fourier, Reineke and the first author. Along the way, we give minor improvements to certain results of Enomoto and Brustle-Langford-Hassoun-Roy concerning the classification of exact structures on an additive category. We prove that for each idempotent complete additive category A, there exists an abelian category whose lattice of Serre subcategories is isomorphic to the lattice of exact structures on A. We show that every Krull-Schmidt category admits a unique maximal admissible exact structure and that the lattice of smaller exact structures of an admissible exact structure is Boolean. (c) 2022 Elsevier Inc. All rights reserved.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Fang, Xin UNSPECIFIED UNSPECIFIED UNSPECIFIED Gorsky, Mikhail UNSPECIFIED orcid.org/0000-0003-0547-3571 UNSPECIFIED
URN:	urn:nbn:de:hbz:38-668978
DOI:	10.1016/j.aim.2022.108210
Journal or Publication Title:	Adv. Math.
Volume:	398
Date:	2022
Publisher:	ACADEMIC PRESS INC ELSEVIER SCIENCE
Place of Publication:	SAN DIEGO
ISSN:	1090-2082
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language MAXIMAL EXACT STRUCTURES; GROTHENDIECK GROUPS; REPRESENTATION-THEORY; LINEAR DEGENERATIONS; EXACT SEQUENCES; QUANTUM GROUPS; FILTRATION; CATEGORIES; HOMOLOGY; MODULES Multiple languages Mathematics Multiple languages
URI:	http://kups.ub.uni-koeln.de/id/eprint/66897