Geodesic
Hall algebras of two equivalent extriangulated categories
Czechoslovak Mathematical Journal, Tome 74 (2024) no. 1, pp. 95-113
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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 $A_{2n}$.
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 $A_{2n}$.
DOI : 10.21136/CMJ.2023.0344-22
Classification : 17B37, 18E05, 18E10
Keywords: extriangulated category; extriangulated equivalence; Hall algebra; quantum cluster algebra 
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Ruan, Shiquan; Wang, Li; Zhang, Haicheng. Hall algebras of two equivalent extriangulated categories. Czechoslovak Mathematical Journal, Tome 74 (2024) no. 1, pp. 95-113. doi: 10.21136/CMJ.2023.0344-22

[1] Bautista, R.: The category of morphisms between projective modules. Commun. Algebra 32 (2004), 4303-4331. | DOI | MR | JFM

[2] Bennett-Tennenhaus, R., Shah, A.: Transport of structure in higher homological algebra. J. Algebra 574 (2021), 514-549. | DOI | MR | JFM

[3] Berenstein, A., Zelevinsky, A.: Quantum cluster algebras. Adv. Math. 195 (2005), 405-455. | DOI | MR | JFM

[4] Caldero, P., Keller, B.: From triangulated categories to cluster algebras. Invent. Math. 172 (2008), 169-211. | DOI | MR | JFM

[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. | DOI | MR | JFM

[6] Chen, X., Ding, M., Zhang, H.: The cluster multiplication theorem for acyclic quantum cluster algebras. (to appear) in Int. Math. Res. Not. | DOI | MR

[7] Ding, M., Xu, F., Zhang, H.: Acyclic quantum cluster algebras via Hall algebras of morphisms. Math. Z. 296 (2020), 945-968. | DOI | MR | JFM

[8] Fomin, S., Zelevinsky, A.: Cluster algebras. I: Foundations. J. Am. Math. Soc. 15 (2002), 497-529. | DOI | MR | JFM

[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. | DOI | MR | JFM

[10] Gorsky, M., Nakaoka, H., Palu, Y.: Positive and negative extensions in extriangulated categories. Available at , 51 pages. | arXiv | DOI

[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. | DOI | MR | JFM

[12] Nakaoka, H., Palu, Y.: Extriangulated categories, Hovey twin cotorsion pairs and model structures. Cah. Topol. Géom. Différ. Catég. 60 (2019), 117-193. | MR | JFM

[13] Ringel, C. M.: Hall algebras and quantum groups. Invent. Math. 101 (1990), 583-591. | DOI | MR | JFM

[14] Sheng, J., Xu, F.: Derived Hall algebras and lattice algebras. Algebra Colloq. 19 (2012), 533-538. | DOI | MR | JFM

[15] Toën, B.: Derived Hall algebras. Duke Math. J. 135 (2006), 587-615. | DOI | MR | JFM

[16] Wang, L., Wei, J., Zhang, H.: Hall algebras of extriangulated categories. J. Algebra 610 (2022), 366-390. | DOI | MR | JFM

[17] Xiao, J., Xu, F.: Hall algebras associated to triangulated categories. Duke Math. J. 143 (2008), 357-373. | DOI | MR | JFM

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