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Let $𝒜$ be a finitary hereditary abelian category. We give a Hall algebra presentation of Kashaev’s theorem on the relation between Drinfeld double and Heisenberg double. As applications, we obtain realizations of the Drinfeld double Hall algebra of $𝒜$ via its derived Hall algebra and Bridgeland’s Hall algebra of $m$-cyclic complexes.

Let $𝒳$ be a semibrick in an extriangulated category. If $𝒳$ is a $\tau$-semibrick, then the Auslander-Reiten quiver $\Gamma \left(ℱ\right(𝒳\left)\right)$ of the filtration subcategory $ℱ\left(𝒳\right)$ generated by $𝒳$ is $\mathbb{Z}{𝔸}_{\infty }$. If $𝒳={\left\{X_i\right\}}_{i=1}^t$ is a $\tau$-cycle semibrick, then $\Gamma \left(ℱ\right(𝒳\left)\right)$ is $\mathbb{Z}{𝔸}_{\infty }/{\tau }_{𝔸}^t$.

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,...