Introduction

If A is a finite dimensional algebra then its blocks are in one-to-one correspondence with its primitive central idempotents. The aim of this paper is to study this interaction for a class of noetherian algebras arising naturally in representation theory. This class includes the universal enveloping algebra of a reductive Lie algebra in positive characteristic and its quantised counterpart, the quantised enveloping algebra of a Borel subalgebra and the quantised function algebra of a semisimple algebraic group at roots of unity.

More generally this paper is concerned with the role the centre of these algebras plays in their representation theory. The techniques used fall into two categories: local and global. The local approach is concerned principally with the behaviour of certain finite dimensional factors of these noetherian algebras whilst the global approach focuses on general properties of these algebras. The aim in both cases is to understand the structure of these finite dimensional factor algebras. In the first case we use a little deformation theory to piece things together whilst in the second case we can use some geometric tools before passing to the factors.

In Section 2 we introduce the class of algebras we wish to study and present some general properties these have in common. In the following three sections we apply this theory to the study of enveloping algebras and quantised enveloping algebras of Lie algebras and to quantised function algebras.