PUB - Publikationen an der Universität Bielefeld

The indecomposable modules over special biserial algebras are known to be of a specially simple form, so-called “tree” and “band” modules (cf. [6,2]). Crawley-Boevey has shown how to describe the homo- morphisms between tree modules (cf. [3]) and our objective in this paper is to extend this result to band modules. Tree and band modules occur as indecomposable representations of arbitrary algebras. Nevertheless we follow Crawley-Boevey and consider zero-relation algebras as the appropriate context. Given two tree or band modules we first study certain quiver homomorphisms between their underlying trees or bands and it turns out that a map between the representations is completely described by k-linear maps which are associated to these quiver homomorphisms. It is interesting to note that in general the classification of maps between representations of a special biserial algebra A is a wild problem. In fact the maps are equivalent to the representations of T,(A). For instance take for A the path algebra of an A,, n 2 6 then T,(A) is wild although the maps between indecomposable representations of A are easily to describe. Throughout, k will be a fixed algebraically closed field. Maps are written on the left. This paper results from a stay at the University of Liverpool supported by the Deutscher Akademischer Austauschdienst. I am very grateful to Sheila Brenner for her suggestion