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Derived-tame Tree Algebras

Published online by Cambridge University Press:  04 December 2007

Thomas Brüstle
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, PO Box 100 131, D-33501 Bielefeld, Germany. E-mail: bruestle@mathematik.uni-bielefeld.de
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Abstract

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In this note we classify the derived-tame tree algebras up to derived equivalence. A tree algebra is a basic algebra A = kQ/I whose quiver Q is a tree. The algebra A is said to be derived-tame when the repetitive category  of A is tame. We show that the tree algebra A is derived-tame precisely when its Euler form χA is non-negative. Moreover, in this case, the derived equivalence class of A is determined by the following discrete invariants: The number of vertices, the corank and the Dynkin type of χA. Representatives of these derived equivalence classes of algebras are given by the following algebras: the hereditary algebras of finite or tame type, the tubular algebras and a certain class of poset algebras, the so-called semichain-algebras which we introduce below.

Type
Research Article
Copyright
© 2001 Kluwer Academic Publishers