Graded semisimple algebras are symmetric

Semisimple Strongly Graded Rings

Let G be a finite group and R a strongly G-graded ring. The question of when R is semisimple (meaning in this paper semisimple artinian) has been studied by several authors. The most classical result is Maschke’s Theorem for group rings. For crossed products over fields there is a satisfactory answer given by Aljadeff and Robinson [3]. Another partial answer for skew group rings was given by Al...

Small Semisimple Subalgebras of Semisimple Lie Algebras

The goal of Section 2 is to provide a proof of Theorem 2.0.1. Section 3 introduces the necessary facts about Lie algebras and representation theory, with the goal being the proof of Proposition 3.5.7 (ultimately as an application of Theorem 2.0.1), and Proposition 3.3.1. In Section 4 we prove the main theorem, using Propositions 3.3.1 and 3.5.7. In Section 5, we apply the theorem to the special...

Almost semisimple algebras

Hz-algebra Spectra Are Differential Graded Algebras

We construct Quillen equivalences on the Quillen model categories of rings, modules and algebras over Z-graded chain complexes and HZ-module spectra. A Quillen equivalence of Quillen model categories is the most highly structured notion of equivalence between homotopy theories. We use these equivalences in turn to produce algebraic models for rational stable model categories.

Graded self-injective algebras “are” trivial extensions

Article history: Received 20 March 2009 Available online 9 June 2009 Communicated by Michel Van den Bergh Dedicated to Professor Helmut Lenzing on the occasion of his seventieth birthday