Euclidean components for a class of self-injective algebras

Sarah Scherotzke¹

¹ Mathematical Institute University of Oxford 24-29 St. Giles Oxford OX1 3LB, United Kingdom

Colloquium Mathematicum, Tome 115 (2009) no. 2, pp. 219-245

Résumé

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 $.

Détail

DOI : 10.4064/cm115-2-7

Keywords: determine length composition series projective modules g transitive algebras auslander reiten component euclidean tree class thereby correct generalize result farnsteiner math nachr furthermore modules certain length composition series periodic apply these results g transitive blocks universal enveloping algebras restricted p lie algebras prove g transitive principal blocks only allow components euclidean tree class finally deduce conditions smash product local basic algebra mit gamma commutative semisimple group algebra have components euclidean tree class depending components auslander reiten quiver mit gamma