Higher Auslander Algebras Admitting Trivial Maximal Orthogonal Subcategories

For an Artinian $(n-1)$-Auslander algebra $\mathrm{\Lambda }$ with global dimension $n(\ge 2)$, we show that if $\mathrm{\Lambda }$ admits a trivial maximal $(n-1)$-orthogonal subcategory of $\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{\Lambda }$, then $\mathrm{\Lambda }$ is a Nakayama algebra and the projective or injective dimension of any indecomposable module in $\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{\Lambda }$ is at most $n-1$. As a result, for an Artinian Auslander algebra with global dimension 2, if $\mathrm{\Lambda }$ admits a trivial maximal 1-orthogonal subcategory of $\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{\Lambda }$, then $\mathrm{\Lambda }$ is a tilted algebra of finite representation type. Further, for a finite-dimensional algebra $\mathrm{\Lambda }$ over an algebraically closed field $K$, we show that $\mathrm{\Lambda }$ is a basic and connected $(n-1)$-Auslander algebra $\mathrm{\Lambda }$ with global dimension $n(\ge 2)$ admitting a trivial maximal $(n-1)$-orthogonal subcategory of $\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{\Lambda }$ if and only if $\mathrm{\Lambda }$ is given by the quiver: \xymatrix{1 & \ar[l]_{\beta_{1}} 2 & \ar[l]_{\beta_{2}} 3 & \ar[l]_{\beta_{3}} ... & \ar[l]_{\beta_{n}} n+1} modulo the ideal generated by $\{{\beta }_i{\beta }_{i+1}\mathrm{\mid }1\le i\le n-1\}$. As a consequence, we get that a finite-dimensional algebra over an algebraically closed field $K$ is an $(n-1)$-Auslander algebra with global dimension $n(\ge 2)$ admitting a trivial maximal $(n-1)$-orthogonal subcategory if and only if it is a finite direct product of $K$ and $\mathrm{\Lambda }$ as above. Moreover, we give some necessary condition for an Artinian Auslander algebra admitting a non-trivial maximal 1-orthogonal subcategory.