Tuesday, March 3 and March 5, 14:00, 7.527.

Anna Rodriguez Rasmussen, Three talks on some aspects of exact Borel subalgebras:

Abstract:
Exact Borel subalgebras of quasi-hereditary algebras are a notion defined by König, in analogy to Borel subalgebras of Lie algebras. These subalgebras encompass a lot of information about the quasi-hereditary algebra and can be a useful tool to understand its structure. Often, one considers exact Borel subalgebras with additional properties; here, I would like to focus on two such properties: Strong and regular. These are useful, but very different notions, and I would like to, in a broad sense, speak about their advantages and disadvantages such as I have happened to encounter them thus far.

Talk 1: Quasi-hereditary monomial algebras and their exact Borel subalgebras

This talk will deal with the question of actually finding exact Borel subalgebras, in a very easy, namely the monomial, case. In this setting, a simple characterisation of quasi-hereditary algebras due to Green and Schroll is known. Moreover, one can show that every quasi-hereditary algebra has a unique exact Borel subalgebra with a basis given by paths. Since a monomial algebra is basic by assumption, this exact Borel subalgebra is strong. However, in general it is not regular.

Talk 2: Exact Borel subalgebras under some on constructions on quasi-hereditary algebras

Quasi-hereditary algebras are stable under several constructions, such as taking tensor products (a result due to Chan), constructing skew group algebras with respect to a compatible group action, and, under some assumptions, forming triangular matrix rings (a result due to Zhu) and even tensor algebras of generalised species. I would like to speak about some of these constructions and how strong/regular exact Borel subalgebras behave with respect to them.

Talk 3: Uniqueness of exact Borel subalgebras

A hallmark result in the study of quasi-hereditary algebras, due to König, Külshammer and Ovsienko, asserts that every quasi-hereditary algebra is Morita equivalent to a quasi-hereditary algebra admitting a basic regular exact Borel subalgebra, while for strong exact Borel subalgebras, no such result holds. In this talk, I would like to speak about uniqueness of regular exact Borel subalgebras (in particular, about results due to Conde, Külshammer-Miemietz, and myself), and the absence of such results for strong exact Borel subalgebras.