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Seminar by Kevin Coulembier (Sydney University)

Title: Borelic pairs for diagram algebras

2016.05.04 | Christine Dilling

Date Thu 12 May
Time 14:15 15:15
Location 1531-219 (Aud-D4)

Quasi-hereditary algebras were introduced to study category O and modular representation theory of semisimple algebraic groups, and have interesting connections with cellular algebras (such as Iwahori-Hecke algebras). Some, but not all, quasi-hereditary algebras posses an exact Borel subalgebra, which is an analogue of the Borel subalgebra of a Lie algebra. Recently König, Külshammer and Ovsienko proved that every Morita class of quasi-hereditary algebras contains one with exact Borel subalgebra. However, there are not much concrete examples available. I will give an elementary introduction to the above concepts and then comment on some recent contributions, which are joint work with Ruibin Zhang.

We introduce the notion of a Borelic pair of an arbitrary algebra. If such a pair satisfies certain conditions, the algebra is quasi-hereditary and has an exact Borel subalgebra. We use this to generate examples of quasi-hereditary algebras, with exact Borel subalgebra, which are Morita equivalent to certain diagram algebras such as (walled) Brauer, Jones and Temperley-Lieb algebras and also to prove that Auslander-Dlab-Ringel algebras have exact Borel subalgebras.

Other conditions on the pair imply that the algebra is based-stratified. This is a new concept we introduce as a generalisation and simplification of the recent theory of cellularly stratified algebras. We use this concept to determine when the cell modules of cellular diagram algebras behave as the standard modules of a quasi-hereditary algebra. Such surprising behaviour of cell modules was first discovered for Iwahori-Hecke algebras by Hemmer and Nakano, and extended to other cases by Hartmann, Henke, König and Paget.