9 Higher Auslander Algebras Admitting Trivial Maximal Orthogonal Subcategories

For an Artinian (n− 1)-Auslander algebra Λ with global dimension n(≥ 2), we show that if Λ admits a trivial maximal (n − 1)-orthogonal subcategory of modΛ, then Λ is a Nakayama algebra and the projective or injective dimension of any indecomposable module in modΛ is at most n− 1. As a result, for an Artinian Auslander algebra with global dimension 2, if Λ admits a trivial maximal 1-orthogonal subcategory of modΛ, then Λ is a tilted algebra of finite representation type. Further, for a finite-dimensional algebra Λ over an algebraically closed field K, we show that Λ is a basic and connected (n − 1)-Auslander algebra Λ with global dimension n(≥ 2) admitting a trivial maximal (n− 1)-orthogonal subcategory of modΛ if and only if Λ is given by the quiver: 1 2 β1 oo 3 β2 oo · · · β3 oo n+ 1 βn oo modulo the ideal generated by {βiβi+1|1 ≤ i ≤ 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(≥ 2) admitting a trivial maximal (n − 1)-orthogonal subcategory if and only if it is a finite direct product of K and Λ as above. Moreover, we give some necessary condition for an Artinian Auslander algebra admitting a non-trivial maximal 1-orthogonal subcategory.