Hall algebras of two equivalent extriangulated categories

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 .