Hall algebras of two equivalent extriangulated categories

Shiquan Ruan; Li Wang; Haicheng Zhang

Czechoslovak Mathematical Journal (2024)

  • Issue: 1, page 95-113
  • ISSN: 0011-4642

Abstract

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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 ℳ n + 1 and ℱ n , where ℳ n + 1 and 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 ℳ n + 1 and ℱ n are isomorphic. As an application, we use the Hall algebra of ℳ 2 n + 1 to relate with the quantum cluster algebras of type A 2 n .

How to cite

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Ruan, Shiquan, Wang, Li, and Zhang, Haicheng. "Hall algebras of two equivalent extriangulated categories." Czechoslovak Mathematical Journal (2024): 95-113. <http://eudml.org/doc/299211>.

@article{Ruan2024,
abstract = {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 $A_\{2n\}$.},
author = {Ruan, Shiquan, Wang, Li, Zhang, Haicheng},
journal = {Czechoslovak Mathematical Journal},
keywords = {extriangulated category; extriangulated equivalence; Hall algebra; quantum cluster algebra},
language = {eng},
number = {1},
pages = {95-113},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Hall algebras of two equivalent extriangulated categories},
url = {http://eudml.org/doc/299211},
year = {2024},
}

TY - JOUR
AU - Ruan, Shiquan
AU - Wang, Li
AU - Zhang, Haicheng
TI - Hall algebras of two equivalent extriangulated categories
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
IS - 1
SP - 95
EP - 113
AB - 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 $A_{2n}$.
LA - eng
KW - extriangulated category; extriangulated equivalence; Hall algebra; quantum cluster algebra
UR - http://eudml.org/doc/299211
ER -

References

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