Ringel duality revisited

• Applicant: Professor Dr. Steffen Koenig
• Subject Area: Mathematics
• Term: from 2020 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 430932201

This is an extension of the project "Ringel duality revisited" and it will build on the progress made during the two years of that project obtaining funding (ending in March 2022). The basis of this project continues to be the description of quasi-hereditary algebras and modules with standard filtrations in terms of bocses and the A-infinity categories behind these. This has been supplemented in my recent joint work with Brzezinski and Külshammer with a translation into the language of corings. The description in terms of bocses itself has been strengthened much by Conde's solution of Ovsienko's problem in a stronger form than stated as objective in the first proposal. In particular, she precisely described which quasi-hereditary algebras have an exact Borel subalgebra, given by a bocs, and she obtained very precise structural and numerical information about bocses and exact Borel subalgebras. While solving this problem, she also put together a precise functorial framework for Ringel duality, which here gets identified with Burt-Butler duality.In the third year, there are three main aims left to be considered, all continuing and extending the investigation (objective A) of ring structure and homological structure of Ringel self-dual algebras, especially with respect to simple-preserving dualities.Part 1 (objective B): Extend and work out in detail Conde's framework for Ringel self-duality by adding a third duality known as co- and contramodules correspondence in such a way that the framework also covers stratified algebras and infinite highest weight categories.Part 2 (objective B, continued): In the original proposal, it has been conjectured that Ringel self-duality of an algebra implies the existence of a simple preserving duality on ist module category. This will solve many problems stated in the first proposal. Using the framework in part 1 together with the machinery of Frobenius functors and also results on equivalences between comodules and contramodules over corings (bocses), this conjecture and the problems motivating it will be approached.Part 3 (objectives C and D): The planned work on A-infinity structure has already started, and is to get extended to the general categorical framework. The final results will clarify the behaviour of A-infinity structures of Yoneda extension algebras under passage to "good" quotient algebras or "good" subalgebras of quasi-hereditary algebras (or more general categories) and on compatibility of exact Borel subalgebras with such operations.

• DFG Programme: Research Grants
• Co-Investigator: Professorin Dr. Anne Henke