Euclidean components for a class of self-injective algebras

Volume 115 / 2009

Sarah Scherotzke Colloquium Mathematicum 115 (2009), 219-245 MSC: 16G70, 16G10, 16S40, 16E20. DOI: 10.4064/cm115-2-7

Abstract

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 $.

Authors

  • Sarah ScherotzkeMathematical Institute
    University of Oxford
    24-29 St. Giles
    Oxford OX1 3LB, United Kingdom
    e-mail

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