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Keywords:
extriangulated category; extriangulated equivalence; Hall algebra; quantum cluster algebra
extriangulated category; extriangulated equivalence; Hall algebra; quantum cluster algebra
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 $A_{2n}$.
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