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A characterization of finite Auslander-Reiten quivers. (English) Zbl 0538.16026

Let \(\Gamma\) be a translation quiver with translation \(\tau\) [C. Riedtmann, Comment. Math. Helv. 55, 199-224 (1980; Zbl 0444.16018)] and k a commutative field. A k-modulation on \(\Gamma\) consists of the following:
(a) A finite dimensional division algebra \(F_ x\) over k for every vertex x of \(\Gamma\).
(b) A finite dimensional \(F_ y-F_ x\)-bimodule \({}_ yM_ x\) for every arrow \(x\to y\) of \(\Gamma\).
(c) A k-algebra isomorphism \(\tau_*:F_ x\to F_{\tau x}\) for each nonprojective vertex x.
(d) A nonsingular bilinear pairing \(\sigma_*:_ yM_{\tau x}\otimes_ xM_ y\to F_ y\) for every nonprojective vertex x and \(y\in \bar x.\)
Let \(\Lambda\) be an indecomposable Artin algebra of finite representation type, R its center and \(k=R/rad R.\) Then the Auslander-Reiten quiver \(\Gamma_{\Lambda}\) of \(\Lambda\) bears in a canonical way the structure of a k-modulated translation quiver, where \(F_ x=End(x)/rad End(x)\) and \({}_ yM_ x\) is the bimodule of irreducible maps from x to y. The underlying valuation is given by the positive integers \(m_ x=\dim_ k(F_ x),\quad d_{xy}=\dim_{F_ y}(_ yM_ x)\) and \(d'\!_{xy}=\dim_{F_ x}(_ yM_ x).\)
The authors give necessary and sufficient conditions for a finite k- modulated translation quiver to be an Auslander-Reiten quiver in terms of certain homology groups associated to the quiver. They show that, as a consequence, the property to be an Auslander-Reiten quiver depends only on the underlying valuation and the characteristic of k. Moreover a finite k-modulated translation quiver is an Auslander-Reiten quiver if and only if the underlying translation quiver \(\Gamma\) with the trivial K-modulation \((K=prime\) field of k, \(F_ x=K,\quad_ yM_ x=K,\quad \tau_*=id_ K,\quad \sigma_*(a\otimes b)=a\cdot b)\) is an Auslander- Reiten quiver. Furthermore, some applications of the result are given.
Reviewer: J.Waschbüsch

MSC:

16Gxx Representation theory of associative rings and algebras
16Exx Homological methods in associative algebras
16P10 Finite rings and finite-dimensional associative algebras
16P20 Artinian rings and modules (associative rings and algebras)

Citations:

Zbl 0444.16018
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References:

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