Euclidean components for a class of self-injective algebras
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 $\mit\Gamma $ with a commutative semisimple group algebra to have components with Euclidean tree class, depending on the components of the Auslander–Reiten quiver of $\mit\Gamma $.