## A criterion for finite representation type.(English)Zbl 0552.16012

The author calls a k-algebra R standard if R is finite dimensional, distributive, basic and admits a Galois covering $$\tilde R\to R$$ with $$\tilde R$$ simply connected. The main result of the paper is the following useful characterization of representation-finite standard algebras R in terms of their Galois coverings $$\tilde R\to R$$ and partially ordered sets $$\tilde R_ s$$, $$\tilde R^ s$$, $$s\in R$$, associated to $$\tilde R$$ in a natural way. Let R be a standard algebra of dimension d with a Galois covering $$\tilde R\to R$$, where $$\tilde R$$ is simply connected. Then R is representation-finite if and only if $$\tilde R$$ has the following properties: (a) the path algebras k\~R$${}_ s$$, k\~R$${}^ s$$, $$s\in \tilde R$$, do not contain a full subcategory $$B\overset \sim \rightarrow kQ_ B$$ with $$| Q_ B|$$ of the form $1 - 2 - \cdots - n - n+1 - 1$
$1 - 2 - \cdots - 2d+1$ $$\tilde R$$ does not contain a full convex subcategory $$B\overset \sim \rightarrow kQ_ B$$ with $$| Q_ B|$$ of the form $$1 -- 2 -- \cdots -- 2d+1$$, $$\tilde R$$ does not contain as a full convex subcategory an algebra of one of three quiver types (described in the paper) with at most $$2d+2$$ points, (d) any full convex subcategory of $$\tilde R$$ with at most 9 points is representation-finite.
Reviewer: D.Simson

### MSC:

 16Gxx Representation theory of associative rings and algebras 16P10 Finite rings and finite-dimensional associative algebras 16B50 Category-theoretic methods and results in associative algebras (except as in 16D90)
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### References:

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