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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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