On the structure of nonsemisimple Hopf algebras

Abstract

The left module structure of finite-dimensional quantum algebras is analysed using the theory of primitive idempotents. In particular, a complete structural result in terms of principal (projective) indecomposable modules (p.i.m.) is given in the case of (q a root of 1) by finding a complete set of primitive idempotents. The structure of p.i.m. is analysed in detail. The Jacobson radical of the algebra is investigated and its significance in the study of nonsemisimple symmetries in physical systems is discussed.