×

On Auslander-Reiten quivers without oriented cycles. (English) Zbl 0536.16028

The covering techniques of K. Bongartz and P. Gabriel [see Invent. Math. 65, 331-378 (1982; Zbl 0482.16026)] reduce the study of the Auslander-Reiten quiver \(\Gamma_{\Lambda}\) of an algebra \(\Lambda\) of finite representation type to the universal cover \({\tilde \Gamma}{}_{\Lambda}\), i.e. the Auslander-Reiten quiver of a simply connected, locally bounded category of locally finite representation type. This paper gives a contribution in the direction of making a similar reduction to some bounded category, in case \(\Gamma_{\Lambda}\) has no oriented cycles. This is done by distinguishing a full subtranslation quiver within \({\tilde \Gamma}{}_{\Lambda}.\)
For each vertex x of a translation quiver, a subset of the projective vertices, the support of x, is determined and the concept of full injective vertex is introduced, both in a natural way. The main result is:
Theorem. Let (\(\Gamma\),\(\tau)\) be a connected, finite translation quiver without oriented cycles, whose vertices have non-empty support and whose injectives are full. Given a finite, connected subset, E, of \({\tilde \Gamma}{}_ 0\), let us denote by [-,E] the set of all \(y\in {\tilde \Gamma}_ 0\) such that there is a path from y to some vertex in E. Then, there exists a full subtranslation quiver D of \({\tilde \Gamma}\) such that \([-,E]\subset D_ 0,\quad {\mathbb{P}}_ D=[-,E]{\mathbb{P}}_{{\tilde \Gamma}}\) and D is the Auslander-Reiten quiver of some simply connected algebra. As easy consequences of this theorem, the authors are able to characterize the translation quivers (\(\Gamma\),\(\tau)\) which are Auslander-Reiten quivers of an algebra \(\Lambda\) in each of the following cases: 1) \(\Lambda\) is a finite dimensional k-algebra (k algebraically closed), 2) \(\Lambda\) is a simply connected, finite dimensional k-algebra and 3) \(\Lambda\) is a (P)-algebra in the sense of R. Bautista and F. Larrión [see J. Lond. Math. Soc., II. Ser. 26, 43-52 (1982; Zbl 0501.16030)].
Reviewer: H.A.Merklen

MSC:

16Gxx Representation theory of associative rings and algebras
16P10 Finite rings and finite-dimensional associative algebras
PDFBibTeX XMLCite
Full Text: DOI