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Algebras of bounded finite dimensional representation type

Published online by Cambridge University Press:  18 May 2009

Allen D. Bell  and
K. R. Goodearl
Allen D. Bell
Affiliation:
Department of Mathematics, University of Wisconsin—Milwaukee, Milwaukee, Wisconsin 53201, U.S.A. E-mail, adbell@archimedes.math.uwm.edu
K. R. Goodearl
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, U.S.A.
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It is well known that for finite dimensional algebras, “bounded representation type” implies “finite representation type”; this is the assertion of the First Brauer-Thrall Conjecture (hereafter referred to as Brauer-Thrall I), proved by Roiter [26] (see also [23]). More precisely, it states that if R is a finite dimensional algebra over a field k, such that there is a finite upper bound on the k-dimensions of the finite dimensional indecomposable right R-modules, then up to isomorphism R has only finitely many (finite dimensional) indecomposable right modules. The hypothesis and conclusion are of course left-right symmetric in this situation, because of the duality between finite dimensional left and right R-modules, given by Homk(−, k). Furthermore, it follows from finite representation type that all indecomposable R modules are finite dimensional [25].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 37 , Issue 3 , September 1995 , pp. 289 - 302
Copyright
Copyright © Glasgow Mathematical Journal Trust 1995

References

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