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Indecomposable modules over multicoil algebras. (English) Zbl 0796.16014
The authors generalize the notion of coil-algebras [introduced in Manuscr. Math. 67, No. 3, 305-331 (1990; Zbl 0696.16023)] and initiate the study of the so-called multicoil algebras. Let $$k$$ be an algebraically closed field and $$A$$ a finite-dimensional, basic and connected $$k$$-algebra. Moreover let $$\bmod A$$ be the category of finitely generated right $$A$$-modules and $$\Gamma_ A$$ the Auslander-Reiten quiver of $$A$$. A coil is, roughly speaking, a translation quiver which is obtained from a stable tube by some admissible operations. A component $$\Gamma$$ of $$\Gamma_ A$$ is called a multicoil if it contains a full subquiver $$\Gamma'$$ which is a finite union of standard coils and has the property that the indecomposables not belonging to $$\Gamma'$$ are directing in $$\bmod A$$. Finally, $$A$$ is said to be a multicoil algebra if any cycle of non-zero non-isomorphisms $$M = M_ 0 \to M_ 1 \to \dots \to M_ t = M$$ in $$\bmod A$$ lies in one multicoil of $$\Gamma_ A$$. In particular multicoil algebras are cycle-finite and hence tame.
The main result proved in the paper is that for a multicoil algebra $$A$$ the following conditions are equivalent: (i) $$A$$ is either tame concealed or tubular, (ii) there exists a sincere indecomposable $$A$$-module lying in a stable tube of $$\Gamma_ A$$, (iii) there exist infinitely many non- isomorphic sincere modules of the same dimension type lying in homogeneous tubes of $$\Gamma_ A$$.
It follows that the indecomposable modules lying in a stable tube of a multicoil algebra have as their support a tame concealed or a tubular full convex subcategory of the algebra. Moreover, it is proved that a multicoil algebra is of polynomial growth. As a consequence, a multicoil algebra is domestic if and only if it does not contain a tubular algebra as a full convex subcategory.

##### MSC:
 16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers 16G60 Representation type (finite, tame, wild, etc.) of associative algebras
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