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Journal of the London Mathematical Society 2003 67(1):57-72; doi:10.1112/S0024610702003885
© 2003 by London Mathematical Society
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© The London Mathematical Society

Non-Strictly Wild Algebras

Hiroshi Nagase

Department of Mathematics, Osaka City University 3-3-138 Sugimoto, Sumiyoshiku, Osaka 558-8585, Japan, nagase{at}sci.osaka-cu.ac.jp

Received 28 November 2000.

Finite-dimensional algebras over an algebraically closed field are divided into two disjoint classes, called tame and wild respectively, by Drozd's tame and wild dichotomy (see [5] and [2]). A tame algebra, roughly speaking, has its n-dimensional indecomposable modules parametrized by finitely many one-parameter families, for all natural numbers n, but a wild algebra has more indecomposable modules and it is considered hopeless to classify them. In [2], Crawley-Boevey showed that all but finitely many n-dimensional indecomposable modules over a tame algebra are {tau}-invariant, for all natural numbers n, and conjectured that the converse would be true, where {tau} := DTr is the Auslander–Reiten translation (see [1]) and we call an indecomposable module X {tau}-invariant if X {cong} {tau}X.


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