Higher Auslander Algebras Admitting Trivial Maximal Orthogonal Subcategories
| dc.creator | Huang, Zhaoyong | |
| dc.creator | Zhang, Xiaojin | |
| dc.date | 2009-03-04 | |
| dc.date | 2009-03-24 | |
| dc.date.accessioned | 2026-07-07T12:55:28Z | |
| dc.date.available | 2026-07-07T12:55:28Z | |
| dc.description | For an Artinian $(n-1)$-Auslander algebra $Λ$ with global dimension $n(\geq 2)$, 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 $n(\geq 2)$ admitting a trivial maximal $(n-1)$-orthogonal subcategory of $\modΛ$ if and only if $Λ$ is given by the quiver: $$\xymatrix{1 & \ar[l]_{β_{1}} 2 & \ar[l]_{β_{2}} 3 & \ar[l]_{β_{3}} ... & \ar[l]_{β_{n}} n+1} $$ modulo the ideal generated by $\{β_{i}β_{i+1}| 1\leq i\leq n-1 \}$. 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 $n(\geq 2)$ 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. | |
| dc.description | 25 pages. This version is a combination of the orginal version of this paper with "From Auslander Algebras to Tilted Algebras" (arXiv:0903.0760). The latter paper has been withdrawn | |
| dc.identifier | https://arxiv.org/abs/0903.0761 | |
| dc.identifier | http://arxiv.org/abs/0903.0761 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/224260 | |
| dc.subject | Representation Theory | |
| dc.subject | Rings and Algebras | |
| dc.subject | 16G10; 16G70; 16E10 | |
| dc.title | Higher Auslander Algebras Admitting Trivial Maximal Orthogonal Subcategories | |
| dc.type | text |