Article
| Title: | Hall algebras of two equivalent extriangulated categories (English) |
| Author: | Ruan, Shiquan |
| Author: | Wang, Li |
| Author: | Zhang, Haicheng |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 74 |
| Issue: | 1 |
| Year: | 2024 |
| Pages: | 95-113 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | For any positive integer $n$, let $A_n$ be a linearly oriented quiver of type $A$ with $n$ vertices. It is well-known that the quotient of an exact category by projective-injectives is an extriangulated category. We show that there exists an extriangulated equivalence between the extriangulated categories $\mathcal {M}_{n+1}$ and $\mathcal {F}_n$, where $\mathcal {M}_{n+1}$ and $\mathcal {F}_n$ are the two extriangulated categories corresponding to the representation category of $A_{n+1}$ and the morphism category of projective representations of $A_n$, respectively. As a by-product, the Hall algebras of $\mathcal {M}_{n+1}$ and $\mathcal {F}_n$ are isomorphic. As an application, we use the Hall algebra of $\mathcal {M}_{2n+1}$ to relate with the quantum cluster algebras of type $A_{2n}$. (English) |
| Keyword: | extriangulated category |
| Keyword: | extriangulated equivalence |
| Keyword: | Hall algebra |
| Keyword: | quantum cluster algebra |
| MSC: | 17B37 |
| MSC: | 18E05 |
| MSC: | 18E10 |
| DOI: | 10.21136/CMJ.2023.0344-22 |
| . | |
| Date available: | 2024-03-13T10:04:43Z |
| Last updated: | 2024-03-18 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152270 |
| . | |
| Reference: | [1] Bautista, R.: The category of morphisms between projective modules.Commun. Algebra 32 (2004), 4303-4331. Zbl 1081.16025, MR 2102451, 10.1081/AGB-200034145 |
| Reference: | [2] Bennett-Tennenhaus, R., Shah, A.: Transport of structure in higher homological algebra.J. Algebra 574 (2021), 514-549. Zbl 1461.18004, MR 4213630, 10.1016/j.jalgebra.2021.01.019 |
| Reference: | [3] Berenstein, A., Zelevinsky, A.: Quantum cluster algebras.Adv. Math. 195 (2005), 405-455. Zbl 1124.20028, MR 2146350, 10.1016/j.aim.2004.08.003 |
| Reference: | [4] Caldero, P., Keller, B.: From triangulated categories to cluster algebras.Invent. Math. 172 (2008), 169-211. Zbl 1141.18012, MR 2385670, 10.1007/s00222-008-0111-4 |
| Reference: | [5] Chaio, C., Pratti, I., Souto-Salorio, M. J.: On sectional paths in a category of complexes of fixed size.Algebr. Represent. Theory 20 (2017), 289-311. Zbl 1393.16009, MR 3638350, 10.1007/s10468-016-9643-2 |
| Reference: | [6] Chen, X., Ding, M., Zhang, H.: The cluster multiplication theorem for acyclic quantum cluster algebras.(to appear) in Int. Math. Res. Not. MR 4675077, 10.1093/imrn/rnad172 |
| Reference: | [7] Ding, M., Xu, F., Zhang, H.: Acyclic quantum cluster algebras via Hall algebras of morphisms.Math. Z. 296 (2020), 945-968. Zbl 1509.17010, MR 4159816, 10.1007/s00209-020-02465-0 |
| Reference: | [8] Fomin, S., Zelevinsky, A.: Cluster algebras. I: Foundations.J. Am. Math. Soc. 15 (2002), 497-529. Zbl 1021.16017, MR 1887642, 10.1090/S0894-0347-01-00385-X |
| Reference: | [9] Fu, C., Peng, L., Zhang, H.: Quantum cluster characters of Hall algebras revisited.Sel. Math., New Ser. 29 (2023), Article ID 4, 29 pages. Zbl 07612807, MR 4502761, 10.1007/s00029-022-00811-0 |
| Reference: | [10] Gorsky, M., Nakaoka, H., Palu, Y.: Positive and negative extensions in extriangulated categories.Available at https://arxiv.org/abs/2103.12482 (2021), 51 pages. 10.48550/arXiv.2103.12482 |
| Reference: | [11] Hubery, A.: From triangulated categories to Lie algebras: A theorem of Peng and Xiao.Trends in Representations Theory of Algebras and Related Topics Contemporary Mathematics 406. AMS, Providence (2006), 51-66. Zbl 1107.16021, MR 2258041, 10.1090/conm/406 |
| Reference: | [12] Nakaoka, H., Palu, Y.: Extriangulated categories, Hovey twin cotorsion pairs and model structures.Cah. Topol. Géom. Différ. Catég. 60 (2019), 117-193. Zbl 1451.18021, MR 3931945 |
| Reference: | [13] Ringel, C. M.: Hall algebras and quantum groups.Invent. Math. 101 (1990), 583-591. Zbl 0735.16009, MR 1062796, 10.1007/BF01231516 |
| Reference: | [14] Sheng, J., Xu, F.: Derived Hall algebras and lattice algebras.Algebra Colloq. 19 (2012), 533-538. Zbl 1250.18013, MR 2999262, 10.1142/S1005386712000399 |
| Reference: | [15] Toën, B.: Derived Hall algebras.Duke Math. J. 135 (2006), 587-615. Zbl 1117.18011, MR 2272977, 10.1215/S0012-7094-06-13536-6 |
| Reference: | [16] Wang, L., Wei, J., Zhang, H.: Hall algebras of extriangulated categories.J. Algebra 610 (2022), 366-390. Zbl 1502.18019, MR 4466102, 10.1016/j.jalgebra.2022.07.023 |
| Reference: | [17] Xiao, J., Xu, F.: Hall algebras associated to triangulated categories.Duke Math. J. 143 (2008), 357-373. Zbl 1168.18006, MR 2420510, 10.1215/00127094-2008-021 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
