Quasihereditary algebras with exact Borel subalgebras

Exact Borel subalgebras of quasihereditary algebras emulate the role of classic Borel subalgebras of complex semisimple Lie algebras. Not every quasihereditary algebra $A$ has an exact Borel subalgebra. However, a theorem by Koenig, Külshammer and Ovsienko establishes that there always exists a quasihereditary algebra Morita equivalent to $A$ that has a (regular) exact Borel subalgebra. Despite that, an explicit characterisation of such “special” Morita representatives is not directly obtainable from Koenig, Külshammer and Ovsienko’s work. In this talk, I shall present a criterion to decide whether a quasihereditary algebra contains a regular exact Borel subalgebra, and a method to compute all Morita representatives of $A$ that have a regular exact Borel subalgebra. We shall also see that the Cartan matrix of a regular exact Borel subalgebra of a quasihereditary algebra $A$ only depends on the composition factors of the standard and costandard $A$-modules, and on the dimension of the Hom-spaces between standard $A$-modules. I will conclude the talk with a characterisation of the basic quasihereditary algebras that admit a regular exact Borel subalgebra.