Ringel duality revisited

• Applicant: Professor Dr. Steffen Koenig
• Subject Area: Mathematics
• Term: from 2020 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 430932201

• Final Report Year: 2023

Final Report Abstract

Quasi-hereditary algebras and highest weight categories occur in representation theory, algebraic Lie theory, invariant theory, mathematical physics and other areas. Ringel duality associates another quasi-hereditary algebra to a given one and turns a category of modules with standard filtrations into one with costandard filtrations. In previous work, categories of modules with (co)standard filtrations have been shown to be categories of representations of a bocs, that is of a generalisation of an algebra. In this context, Ringel duality becomes Burt-Butler duality. The bocs associated to a given quasihereditary algebra Λ can be seen as a ring extension that consists of a quasi-hereditary algebra A Morita equivalent to Λ and a subalgebra B that is a basic regular exact Borel subalgebra of A - in analogy to the situation of the Poincaré-Birkhoff-Witt theorem in Lie theory. To find and to desribe A and B and the inclusion is difficult. To describe Ringel duality and Ringel self-duality is difficult, too. A main outcome of work on these problems is an article by Teresa Conde, showing that A is unique up to isomorphism and describing A and B in terms of an algorithm that only uses elementary data on quasi-hereditary structures. Given an algebra Morita equivalent to Λ, the algorithm decides if this algebra is the unknown A, up to isomorphism, if it does contain a (not necessarily basic) regular exact Borel subalgebra, and if all algebras in this Morita equivalence class do contain a regular exact Borel subalgebra. Building on this article, characterisations have been given, when an exact Borel subalgebra of a basic quasi-hereditary algebra even is regular. Another main outcome is that canonical operations on quasi-hereditary algebras, such as taking good quotients and good subalgebras, preserve A∞ -structures. This is work by Teresa Conde and Julian Külshammer. Bocses can be seen as corings. In this context, it has been shown by Teresa Conde that bocses (or corings) coming with exact Borel subalgebras always are normal, which means that their counits are surjective. Corings in general have been shown to be Burt-Butler self-dual if and only if they are Frobenius corings. The project is being continued by adding further algebraic tools such as semialgebras to complement the use of corings and bocses, and to improve and to extend the results obtained so far to describe exact categories such as categories of modules with (co)standard filtrations.

Publications

• From quasi‐hereditary algebras with exact Borel subalgebras to directed bocses. Bulletin of the London Mathematical Society, 52(2), 367-378.
  Brzeziński, Tomasz; Koenig, Steffen & Külshammer, Julian
  
• All exact Borel subalgebras and all directed bocses are normal. Journal of Algebra, 579, 106-113.
  Conde, Teresa
  
• All quasihereditary algebras with a regular exact Borel subalgebra. Advances in Mathematics, 384, 107751.
  Conde, Teresa
  
• Dominant and global dimension of blocks of quantised Schur algebras. Mathematische Zeitschrift, 300(1), 463-473.
  Fang, Ming; Hu, Wei & Koenig, Steffen
  
• On derived equivalences and homological dimensions. Journal für die reine und angewandte Mathematik (Crelles Journal), 2021(770), 59-85.
  Fang, Ming; Hu, Wei & Koenig, Steffen
  
• A functorial approach to rank functions on triangulated categories. Journal für die reine und angewandte Mathematik (Crelles Journal), 0(0).
  Conde, Teresa; Gorsky, Mikhail; Marks, Frederik & Zvonareva, Alexandra