Mathematics – Category Theory
Scientific paper
2006-12-06
Mathematics
Category Theory
Scientific paper
We study the problem of classifying triangulated categories with finite-dimensional morphism spaces and finitely many indecomposables over an algebraically closed field. We obtain a new proof of the following result due to Xiao and Zhu: the Auslander-Reiten quiver of such a category is of the form $\mathbb{Z}\Delta/G$ where $\Delta$ is a disjoint union of simply laced Dynkin diagrams and $G$ a weakly admissible group of automorphisms of $\mathbb{Z}\Delta$. Then we prove that for `most' groups $G$, the category $\T$ is standard, \emph{i.e.} $k$-linearly equivalent to an orbit category $\mathcal{D}^b(\modd k\Delta)/\Phi$. This happens in particular when $\T$ is maximal $d$-Calabi-Yau with $d\geq2$. Moreover, if $\T$ is standard and algebraic, we can even construct a triangle equivalence between $\T$ and the corresponding orbit category. Finally we give a sufficient condition for the category of projectives of a Frobenius category to be triangulated. This allows us to construct non standard 1-Calabi-Yau categories using deformed preprojective algebras of generalized Dynkin type.
No associations
LandOfFree
On the structure of triangulated category with finitely many indecomposables does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On the structure of triangulated category with finitely many indecomposables, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the structure of triangulated category with finitely many indecomposables will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-452784