Filtrations, Stratifications and Applications

摘要

Finite dimensional algebras and their representations and cohomology are known to play a crucial role in many areas of mathematics. Frequently, in dealing with a potential application, one encounters families of finite dimensional algebras which are defined in some explicit way or by explicit properties, specific to this particular area. Does it make sense to extract abstract properties of such algebras, to use these properties for denning interesting abstract classes of finite dimensional algebras, to study such classes of algebras with the tools available within representation theory of finite dimensional algebras, and then to go back and apply the knowledge obtained in this way? In other words, does the general theory matter in explicit situations, and does it help to solve problems coming up in other areas? Many situations are now known for the answer to be positive.