Hochschild cohomology related to graded down-up algebras with weights $(1,n)$

Abstract: Let $A=A(\alpha, \beta)$ be a graded down-up algebra with $({\rm deg}\,x, {\rm deg}\,y)=(1,n)$ and $\beta \neq 0$, and let $\nabla A$ be the Beilinson algebra of $A$. If $n=1$, then a description of the Hochschild cohomology group of $\nabla A$ is known. In this paper, we calculate the Hochschild cohomology group of $\nabla A$ for the case $n \geq 2$. As an application, we see that the structure of the bounded derived category of the noncommutative projective scheme of $A$ is different depending on whether $\left(\begin{smallmatrix} 1&0 \end{smallmatrix}\right)\left(\begin{smallmatrix} \alpha &1 \ \beta &0 \end{smallmatrix}\right)^n\left(\begin{smallmatrix} 1 \ 0 \end{smallmatrix}\right)$ is zero or not. Moreover, it turns out that there is a difference between the cases $n=2$ and $n\geq 3$ in the context of Grothendieck groups.