Classification of silted algebras for two quivers of Dynkin type $\mathbb{D}_{n}$

Abstract

Let $Q$ be the Dynkin quiver of type $\mathbb{D}_{n}$ with linear orientation and let $Q'$ be the quiver formed by reversing the arrow at the unique source in $Q$. In this paper, we present a complete classification of both silted algebras and strictly shod algebras associated with these two quivers. Based on the classification, we derive formulas for counting the number of silted algebras and strictly shod algebras. Furthermore, we establish that all strictly shod algebras corresponding to $Q$ and $Q'$ are string algebras. As an application, we provide a way to construct examples such that the realization functor which is induced from the $t$-structure does not extend to a derived equivalence.