We determine the length of composition series of projective modules of G-transitive algebras with an Auslander-Reiten component of Euclidean tree class. We thereby correct and generalize a result of Farnsteiner [Math. Nachr. 202 (1999)]. Furthermore we show that modules with certain length of composition series are periodic. We apply these results to G-transitive blocks of the universal enveloping algebras of restricted p-Lie algebras and prove that G-transitive principal blocks only allow components with Euclidean tree class if p = 2. Finally, we deduce conditions for a smash product of a local basic algebra Γ with a commutative semisimple group algebra to have components with Euclidean tree class, depending on the components of the Auslander-Reiten quiver of Γ.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm115-2-7,
author = {Sarah Scherotzke},
title = {Euclidean components for a class of self-injective algebras},
journal = {Colloquium Mathematicae},
volume = {116},
year = {2009},
pages = {219-245},
zbl = {1203.16018},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm115-2-7}
}
Sarah Scherotzke. Euclidean components for a class of self-injective algebras. Colloquium Mathematicae, Tome 116 (2009) pp. 219-245. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm115-2-7/