Euclidean components for a class of self-injective algebras

TL;DR

This paper characterizes the Euclidean components in the Auslander-Reiten quivers of certain self-injective algebras, revealing conditions for their occurrence and periodicity, with applications to Lie algebra blocks and smash products.

Contribution

It determines the length of composition series for modules in G-transitive algebras with Euclidean components and explores their periodicity, extending to specific algebra classes.

Findings

Modules with certain composition series lengths are periodic.

G-transitive blocks of universal enveloping algebras have Euclidean components only when p=2.

Conditions for Euclidean components in smash products depending on the basic algebra's AR quiver.

Abstract

We determine the length of composition series of projective modules of G-transitive algebras with an Auslander-Reiten component of Euclidean tree class. 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 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 with a commutative semi-simple group algebra to have components with Euclidean tree class, depending on the components of the Auslander-Reiten quiver of the basic algebra. We also develop some properties of the bilinear form dim Hom_A(-,-) for the representation ring G(A) of any finite-dimensional algebra A.