Exact structures and degeneration of Hall algebras
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 H(E) of an exact category E is the Hall algebra of a smaller exact structure E′
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| Veröffentlicht in: | Advances in mathematics (New York. 1965) 2022-03, Vol.398, p.108210, Article 108210 |
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| container_title | Advances in mathematics (New York. 1965) |
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| creator | Fang, Xin Gorsky, Mikhail |
| description | 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 H(E) of an exact category E is the Hall algebra of a smaller exact structure E′ |
| doi_str_mv | 10.1016/j.aim.2022.108210 |
| format | Article |
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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 Brüstle-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.</description><identifier>ISSN: 0001-8708</identifier><identifier>EISSN: 1090-2082</identifier><identifier>DOI: 10.1016/j.aim.2022.108210</identifier><language>eng</language><publisher>Elsevier Inc</publisher><subject>Category Theory ; Degeneration ; Exact structure ; Hall algebra ; Mathematics ; Quantum Algebra ; Representation Theory ; Rings and Algebras</subject><ispartof>Advances in mathematics (New York. 1965), 2022-03, Vol.398, p.108210, Article 108210</ispartof><rights>2022 Elsevier Inc.</rights><rights>Distributed under a Creative Commons Attribution 4.0 International License</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c374t-65774459ce821ba172fec1040dc130738597fbcfea8f5376d6faf6e3622745b13</citedby><cites>FETCH-LOGICAL-c374t-65774459ce821ba172fec1040dc130738597fbcfea8f5376d6faf6e3622745b13</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://dx.doi.org/10.1016/j.aim.2022.108210$$EHTML$$P50$$Gelsevier$$H</linktohtml><link.rule.ids>230,314,780,784,885,3550,27924,27925,45995</link.rule.ids><backlink>$$Uhttps://hal.science/hal-03536374$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>Fang, Xin</creatorcontrib><creatorcontrib>Gorsky, Mikhail</creatorcontrib><title>Exact structures and degeneration of Hall algebras</title><title>Advances in mathematics (New York. 1965)</title><description>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 H(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 H(E′) is a degeneration of H(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 Brüstle-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.</description><subject>Category Theory</subject><subject>Degeneration</subject><subject>Exact structure</subject><subject>Hall algebra</subject><subject>Mathematics</subject><subject>Quantum Algebra</subject><subject>Representation Theory</subject><subject>Rings and Algebras</subject><issn>0001-8708</issn><issn>1090-2082</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><recordid>eNp9kD9PwzAQxS0EEqXwAdiyMqSc7SROxFRVpUWqxAKzdXHOxVWaIDut4NvjKIiR6fRO73d_HmP3HBYcePF4WKA7LgQIEXUpOFywGYcKUhHVJZsBAE9LBeU1uwnhEGWV8WrGxPoLzZCEwZ_McPIUEuyapKE9deRxcH2X9DbZYtsm2O6p9hhu2ZXFNtDdb52z9-f122qb7l43L6vlLjVSZUNa5EplWV4ZiufUyJWwZDhk0BguQckyr5StjSUsbS5V0RQWbUGyEEJlec3lnD1Mcz-w1Z_eHdF_6x6d3i53euyBzGURd51HL5-8xvcheLJ_AAc9BqQPOgakx4D0FFBkniaG4hNnR14H46gz1DhPZtBN7_6hfwCkrGur</recordid><startdate>20220326</startdate><enddate>20220326</enddate><creator>Fang, Xin</creator><creator>Gorsky, Mikhail</creator><general>Elsevier Inc</general><general>Elsevier</general><scope>AAYXX</scope><scope>CITATION</scope><scope>1XC</scope></search><sort><creationdate>20220326</creationdate><title>Exact structures and degeneration of Hall algebras</title><author>Fang, Xin ; Gorsky, Mikhail</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c374t-65774459ce821ba172fec1040dc130738597fbcfea8f5376d6faf6e3622745b13</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>Category Theory</topic><topic>Degeneration</topic><topic>Exact structure</topic><topic>Hall algebra</topic><topic>Mathematics</topic><topic>Quantum Algebra</topic><topic>Representation Theory</topic><topic>Rings and Algebras</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Fang, Xin</creatorcontrib><creatorcontrib>Gorsky, Mikhail</creatorcontrib><collection>CrossRef</collection><collection>Hyper Article en Ligne (HAL)</collection><jtitle>Advances in mathematics (New York. 1965)</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Fang, Xin</au><au>Gorsky, Mikhail</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Exact structures and degeneration of Hall algebras</atitle><jtitle>Advances in mathematics (New York. 1965)</jtitle><date>2022-03-26</date><risdate>2022</risdate><volume>398</volume><spage>108210</spage><pages>108210-</pages><artnum>108210</artnum><issn>0001-8708</issn><eissn>1090-2082</eissn><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 H(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 H(E′) is a degeneration of H(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 Brüstle-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.</abstract><pub>Elsevier Inc</pub><doi>10.1016/j.aim.2022.108210</doi><oa>free_for_read</oa></addata></record> |
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| ispartof | Advances in mathematics (New York. 1965), 2022-03, Vol.398, p.108210, Article 108210 |
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| language | eng |
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| source | Elsevier ScienceDirect Journals Complete |
| subjects | Category Theory Degeneration Exact structure Hall algebra Mathematics Quantum Algebra Representation Theory Rings and Algebras |
| title | Exact structures and degeneration of Hall algebras |
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