Standard objects, filtered categories and representations of boxes

点击翻译按钮获取中文摘要

There are many examples of standard objects: Verma modules of Lie algebras, Weyl modules of algebraic groups, exceptional coherent sheaves, ... The category of objects with standard filtration is used in the context of Ringel duality and of standardisation and for constructing Morita or derived equivalences with modules over quasi-hereditary algebras. In joint work with Julian Külshammer and Sergiy Ovsienko, it has been shown that this category is equivalent to the category of representations of a box. As an application, the problem of existence of exact Borel subalgebras of quasi-hereditary algebras (up to Morita equivalence) could be solved positively and thus an analogue of the theorem of Poincare, Birkhoff and Witt could be established.This project aims at developing and extending this approach and using it for investigating Ringel duality and tilting modules as well as for determining representation types and for computing representations and cohomology.