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Non‐Strictly Wild Algebras
Author(s) -
Nagase Hiroshi
Publication year - 2003
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610702003885
Subject(s) - indecomposable module , converse , mathematics , invariant (physics) , disjoint sets , pure mathematics , algebraically closed field , finitely generated abelian group , algebra over a field , combinatorics , discrete mathematics , geometry , mathematical physics
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 τ‐invariant, for all natural numbers n , and conjectured that the converse would be true, where τ := DTr is the Auslander–Reiten translation (see [ 1 ]) and we call an indecomposable module X τ‐ invariant if X ≅ τ X .