Ringel self-duality via relative dominant dimension

Abstract: Quasi-hereditary algebras provide an abstract framework for the homological structure of Schur algebras and the BGG category $O$ of a semi-simple Lie algebra and they always appear in pairs via Ringel duality.

In this talk, we discuss a generalisation of dominant dimension using relative homological algebra. This homological invariant is compatible with the tools from integral representation theory and it increases our understanding of classical dominant dimension.

In particular, this homological invariant provides tools to deduce that quasi-hereditary covers formed by quasi-hereditary algebras with a simple preserving duality with large enough dominant dimension also appear in pairs. As an application, we give a new proof of Ringel self-duality of the blocks of the BGG category $O$ of a complex semi-simple Lie algebra.

Mathematics

Audience: researchers in the topic

Greek Algebra & Number Theory Seminar