Classification of the indecomposable finite dimensional modules of clannish algebras

In this thesis we classify all indecomposable finite dimensional modules of clannish algebras with idempotent relations on the special loops. To this end, we start with the introduction of the notion of asymmetric and symmetric strings and bands in terms of words. The classification will be given in terms of those. We first examine directions on special letters in these words of a clannish algebra. Then we reduce the case to skewed-gentle algebras and construct a bundle of semichains for such an algebra. Thus, we are able to reduce the classification problem for skewed-gentle algebras to the matrix problem of bundles of semichains studied by Bondarenko. From this problem, we extract one classification of the indecomposable finite dimensional modules of a skewed-gentle algebra. From this classification, we can deduce a classification for clannish algebras. Finally, we adjust this classification to obtain one similar to that obtained by Crawley-Boevey, in which the symmetric band modules are indexed by a vector space equipped with a pair of idempotent endomorphisms. In contrast to Crawley-Boevey's classification, however, ours gives representations with better bases, not requiring the introduction of a fixed non-zero non-identity element of the field, and so working over the field with two elements. Applied to the algebra generated by an idempotent and a square zero element, it confirms a conjecture of Crawley-Boevey in 1988.

Level thesis:degree_level

thesis.doctoral

Grantor dc:publisher

Universität Bielefeld

Year

2022

Author dc:creator

• Hansper, Ulrike

Repository record source_url

https://pub.uni-bielefeld.de/record/2962952

OAI identifier oai:identifier

oai:pub.uni-bielefeld.de:2962952