Abstract: Representation theory of groups, rings, and algebras describes and compares categories of modules or representations, or related categories such as derived or stable categories. Equivalences between such categories need to be studied in order to reduce new examples to known ones and to exhibit new connections between seemingly different situations.
Given an artin algebra (e.g. finite dimensional algebra over a field) A, we usually associate A with three categories: the (finitely generated) module category modA, which is an abelian category, the bounded derived category D^b(modA), which is triangulated and the stable category stmodA, which is also triangulated in case A is self-injective. While the classical notion of Morita equivalence is well-known, much less is known about equivalences of derived or stable categories, which are source of major current problems and conjectures.
In this talk, I will review the study on derived and stable equivalences for artin algebras and report some recent advances in this field. My talk will contain the following:
1. The various examples of derived or stable equivalences naturally arising in the representation theory of finite groups and finite dimensional algebras;
2. Connections and comparisons on the constructions and (algebraic and homological) invariants between derived and stable equivalences;
3. Examples of stable equivalences (of Morita type) which are not lifted to derived equivalences.