Nakajima quiver varieties, affine crystals and combinatorics of Auslander-Reiten quivers
We obtain an explicit crystal isomorphism between two realizations of crystal bases of finite dimensional irreducible representations of simple Lie algebras of type \(A\) and \(D\). The first realization we consider is a geometric construction in terms of irreducible components of certain Nakajima q...
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| Datum: | 2023 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Lugansk National Taras Shevchenko University
2023
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| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1952 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543385395920896 |
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| author | Kus, D. Schumann, B. |
| author_facet | Kus, D. Schumann, B. |
| author_sort | Kus, D. |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2023-03-06T16:39:17Z |
| description | We obtain an explicit crystal isomorphism between two realizations of crystal bases of finite dimensional irreducible representations of simple Lie algebras of type \(A\) and \(D\). The first realization we consider is a geometric construction in terms of irreducible components of certain Nakajima quiver varieties established by Saito and the second is a realization in terms of isomorphism classes of quiver representations obtained by Reineke. We give a homological description of the irreducible components of Lusztig's quiver varieties which correspond to the crystal of a finite dimensional representation and describe the promotion operator in type A to obtain a geometric realization of Kirillov-Reshetikhin crystals. |
| first_indexed | 2025-12-02T15:42:21Z |
| format | Article |
| id | admjournalluguniveduua-article-1952 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:42:21Z |
| publishDate | 2023 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-19522023-03-06T16:39:17Z Nakajima quiver varieties, affine crystals and combinatorics of Auslander-Reiten quivers Kus, D. Schumann, B. crystal bases, Kirillov-Reshetikhin crystals, Nakajima's quiver varieties, affine Lie algebras 81R50, 81R10, 16G20 We obtain an explicit crystal isomorphism between two realizations of crystal bases of finite dimensional irreducible representations of simple Lie algebras of type \(A\) and \(D\). The first realization we consider is a geometric construction in terms of irreducible components of certain Nakajima quiver varieties established by Saito and the second is a realization in terms of isomorphism classes of quiver representations obtained by Reineke. We give a homological description of the irreducible components of Lusztig's quiver varieties which correspond to the crystal of a finite dimensional representation and describe the promotion operator in type A to obtain a geometric realization of Kirillov-Reshetikhin crystals. Lugansk National Taras Shevchenko University 2023-03-06 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1952 10.12958/adm1952 Algebra and Discrete Mathematics; Vol 34, No 2 (2022) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1952/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1952/996 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1952/1045 Copyright (c) 2023 Algebra and Discrete Mathematics |
| spellingShingle | crystal bases Kirillov-Reshetikhin crystals Nakajima's quiver varieties affine Lie algebras 81R50 81R10 16G20 Kus, D. Schumann, B. Nakajima quiver varieties, affine crystals and combinatorics of Auslander-Reiten quivers |
| title | Nakajima quiver varieties, affine crystals and combinatorics of Auslander-Reiten quivers |
| title_full | Nakajima quiver varieties, affine crystals and combinatorics of Auslander-Reiten quivers |
| title_fullStr | Nakajima quiver varieties, affine crystals and combinatorics of Auslander-Reiten quivers |
| title_full_unstemmed | Nakajima quiver varieties, affine crystals and combinatorics of Auslander-Reiten quivers |
| title_short | Nakajima quiver varieties, affine crystals and combinatorics of Auslander-Reiten quivers |
| title_sort | nakajima quiver varieties, affine crystals and combinatorics of auslander-reiten quivers |
| topic | crystal bases Kirillov-Reshetikhin crystals Nakajima's quiver varieties affine Lie algebras 81R50 81R10 16G20 |
| topic_facet | crystal bases Kirillov-Reshetikhin crystals Nakajima's quiver varieties affine Lie algebras 81R50 81R10 16G20 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1952 |
| work_keys_str_mv | AT kusd nakajimaquivervarietiesaffinecrystalsandcombinatoricsofauslanderreitenquivers AT schumannb nakajimaquivervarietiesaffinecrystalsandcombinatoricsofauslanderreitenquivers |